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Local approximation of functions with several variables
Nonlinear Analysis 73 (2010) 1569–1584
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Local approximation of functions with several variables M.A. Navascués ∗ Departamento de Matemática Aplicada, Universidad de Zaragoza, C/ María de Luna, 3, 50018 Zaragoza, Spain
article
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Article history: Received 8 May 2009 Accepted 27 April 2010 Keywords: Taylor’s formula Non-differentiability of functions of several variables Müntz powers Real-valued functions
abstract This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave–convex functions are obtained as well. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The procedures proposed in this article generalize the linear (Fréchet) approximation of scalar functions of several variables provided by the classical formula of Taylor of first order. The fundamental hypothesis is that the rate with which a function changes if the point is varied in some direction is of exponential type, beyond the linear character displayed by the tangent. In this way, the method described extends the directional derivative of a map at a point of its domain including a new element of power type. The article proposes a kind of nonlinear differential map as well. The word differential is used here in the sense that it measures the variation (difference) of the mapping in neighboring points. The subject is a generalization of paper [1] to the multivariable case. The approximation studied enables the calculus of local and global extrema in order to include some functions which are not differentiable and/or the criterion of the hessian matrix is not applicable. The basic functions which provide the contact with the scalar field are Müntz powers. H. Müntz solved the problem of finding necessary and sufficient conditions on the exponents λ1 , λ2 , . . . (not necessarily integer) for the completeness of the system {xλ1 , xλ2 , . . .} in the space C [0, 1] [2]. This question had been previously raised by Bernstein [3,4]. Müntz’s Theorem was the contribution of the author in honour of his teacher H. Schwarz. An interesting essay about Müntz is written in Ref. [5]. Extensions of this Theorem and related results can be found for instance in [6–8]. 2. Directional variability exponents In this paragraph, the concept of directional derivative is generalized in order to include more general sets of maps. The local behavior of a multivariate function f is approached by means of non-polynomial basic functions. The procedure considers rates of change of power type for f with respect to linear changes in the domain coordinate.
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0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.063
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Definition 2.1. Let f be a scalar field, f : D ⊆ Rn → R, a ∈ D0 and v ∈ Rn . f admits a positive variability exponent (λv (a) > 0) along the vector v if the limit cv (a) = lim
f (a + hv) − f (a) hλv (a)
h→0+
exists, is real and non-null. In this case, cv (a) is the variability coefficient of f at the point a along the vector v . If for all λ ∈ R, lim
f (a + hv) − f (a) hλ
h→0+
=0
then, by definition, λv (a) = 1 and cv (a) = 0 (this is the case, for instance, if f is constant in a neighborhood of a). Example 1. Let us consider the mapping f (x, y) = x2 + 5y1/3 . This function does not own derivative at a = (0, 0) along v = (1, 1). However it possesses a variability exponent at this point along v since cv (0, 0) = lim
f (h, h) − f (0, 0) hλ
h→0+
= lim
h→0+
h2 + 5h1/3 hλ
.
Taking λ = λv (0, 0) = 1/3, cv (0, 0) = 5. Definition 2.2. The scalar field f admits a directional variability exponent at a ∈ D0 along the vector v ∈ Rn if the exponent λu (a) exists for u = v/kvk. Example 2. For the data of Example 1 the directional derivative is cu (0, 0) = lim
h→0+
1 h1/3
h
2
√
2
h
+5 √
2
1/3 !
= 5/21/6 .
The next results can be proved as in Propositions 2.5 and 2.3 of the Ref. [1]. Proposition 2.3. The exponent and coefficient of f at a along the vector v , if they exist, are unique. Proposition 2.4. If f admits a non-null derivative at the point a along the vector v (Dv f (a)) then λv (a) = 1 and cv (a) = Dv f (a). Note 1. If the exponent λv (a) exists, it provides a local approximation formula for f in the direction of v , f (a + hv) ' f (a) + cv (a)hλv (a) and the directional coefficient is a rate of the variation of f at a on the line a + hv with respect to the increase of hλv (a) at zero. Note 2. As in the differentiable case, the existence of positive exponents along any direction does not imply the continuity of f at the point a. A counterexample is given by the mapping f (x, y) =
xy2 x2
+ y4
if (x, y) 6= (0, 0), f (0, 0) = 0. For v = (α, β), α 6= 0 and β 6= 0, λv (0, 0) = 1 and cv (0, 0) = β 2 /α . For v = (0, β), λv (0, 0) = 1 and cv (0, 0) = 0. However, f is not continuous at (0, 0). Proposition 2.5. If f : D ⊆ Rn → R admits a positive variability exponent and cv (a) 6= 0, then λ = λv (a) is the supremum of the numbers α ∈ R such that La,α,v
f (a + hv) − f (a) =0 = lim h→0+ hα
(1)
and the infimum of the constants α ∈ R such that
f (a + hv) − f (a) = ∞. hα
La,α,v = lim h→0+
(2)
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Proof. For α ∈ R, and λ = λv (a) let us consider
f (a + hv) − f (a) = lim f (a + hv) − f (a) α α−λ λ h→0+ h h h
La,α,v = lim h→0+ La,α,v = lim
|cv (a)| hα−λ
h→0+
(3)
.
(4)
If α < λ then La,α,v = 0 and if α > λ then La,α,v = ∞.
Note 3. If cv (a) = 0 this result does not hold. For instance for a = 0 and the function f (t ) = e−1/t if t 6= 0, f (0) = 0, the exponent and coefficient are λ(0) = 1 and c (0) = 0. However, L0,α = 0, ∀α ∈ R and sup{α} = +∞ in R. In this sense we can say that mappings of this kind have an infinite variability exponent. 2
For f (t ) = log(t ),
log(t ) =0 t →0+ tα lim
if α < 0 and
log(t ) = +∞ t →0+ tα lim
if α ≥ 0. In this sense we could say that log(t ) has a null exponent at t = 0. 2.1. Directional formulae Assuming the existence of λv (a), let us denote for the sake of simplicity λ = λv (a) and ga,v (h) =
f (a + hv) − f (a) hλ
.
Then lim ga,v (h) = cv (a)
h→0+
and consider ga∗,v (h) = ga,v (h) − cv (a). The definition of exponent provides the variability formula of first order along the vector v : f (a + hv) = f (a) + cv (a)hλ + ga∗,v (h)hλ for h > 0 and ga∗,v a scalar function such that lim ga∗,v (h) = 0.
h→0+
This formula can be generalized taking exponents of higher order. In the Ref. [1] we defined superior coefficients for functions of a real variable. We can use this procedure here defining q q(h) = f (a + hv). In this way the coefficients of q at t = 0 (c1 (0)) agree with those of f at the point a along the vector f v (cv (a)). cvf (a) = lim
f (a + hv) − f (a) hλ
h→0+
= c1q (0).
Denoting λfv (a) = λ1 = λ, the second exponent and coefficient were defined as q(h) − q(0) − c1 (0)hλ1 q
q
c2 (0) = lim
hλ2
h→0+
and so on. These magnitudes provide the one-dimensional formula for h > 0, q(h) = q(0) + c1 (0)hλ1 + c2 (0)hλ2 + o(hλ2 ). q
f
q
q
f
q
Defining cv,1 (a) = c1 (0), cv,2 (a) = c2 (0), we have the second-order directional formula of f at the point a along the vector v , f (a + hv) = f (a) + cv,1 (a)hλ1 + cv,2 (a)hλ2 + o(hλ2 ), and if b = a + hv , with h = 1 and v = b − a, f
f
f
f
f (b) = f (a) + cv,1 (a) + cv,2 (a) + o(1).
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3. Variability map Definition 3.1. Let f be a scalar field, f : D ⊆ Rn → R, a ∈ D0 and v ∈ Rn . f admits a positive kth partial exponent of variability (λek (a) > 0) if λ = λek (a) exists according to Definition 2.1, where ek is the kth vector of the canonical basis of Rn . That is to say, the kth partial coefficient of f at a is given by the limit cek (a) = lim
f (a + hek ) − f (a) hλek (a)
h→0+
with the considerations of Definition 2.1 for cek (a). Other notations for cek (a) are ck (a) and cxk (a). Likewise the exponent will be represented by λk (a) and λxk (a) as well. We will include in this definition the exponents corresponding to −ek for k = 1, 2, . . . , n as well. The coefficients cek generalize the concept of kth partial derivative and contain other cases of non-differentiability, as illustrated in the following Examples. Example 3. Let f be defined by f (x, y) = x2/3 + y2/3 , f does not admit partial derivatives at (0, 0). However the partial coefficients are given by f (h, 0) − f (0, 0)
cx (0, 0) = lim
h2/3
h→0+
= 1.
Likewise cy (0, 0) = 1 and λx (0, 0) = λy (0, 0) = 2/3. Example 4. Let f be defined by f (x, y) = x2/3 + (xy)1/3 , f (h, 0) − f (0, 0)
cx (0, 0) = lim
h2/3
h→0+
h→0+
f (0, h) − f (0, 0)
cy (0, 0) = lim
hλ
h→0+
= lim
h2/3 h2/3
= 1,
= 0,
for all λ ∈ R. Using the definition of exponents cy (0, 0) = 0 and λy (0, 0) = 1, cx (0, 0) = 1 and λx (0, 0) = 2/3. Definition 3.2. f has a variability singularity in the kth partial exponent at a if there exists λk (x), ∀x ∈ B(a, r ) for r > 0 and λk (x) is not continuous at a. Example 5. Let f be defined now as f (x, y) = x2 + xy1/3 . According to Definition 2.1 cx (x, y) = 2x + y1/3 ,
λx (x, y) = 1
for any (x, y) such that 2x + y cx (x0 , −8x30 ) = lim
1/3
6= 0. For the rest of the points:
f (x0 + h, −8x30 ) − f (x0 , −8x30 ) hλ
h→0+
=1
for λ = λx = 2. As a consequence, f has a singularity in the x-variability on the line 2x + y1/3 = 0. Concerning the y-coefficient, cy (x, y) = xy−2/3 /3,
λy (x, y) = 1
for any (x, y) such that y 6= 0. For the rest of the points: cy (x0 , 0) = lim
h→0+
f (x0 , h) − f (x0 , 0) hλ
= x0
for λ = λy = 1/3 if x0 6= 0, λ = λy = 1 if x0 = 0. As a consequence, f has a singularity in the y-variability on the line y = 0. If the exponents λk (a) exist for k = 1, 2, . . . , n, we can consider the vector of partial variability exponents e(a) = (λ1 (a), λ2 (a), . . . , λn (a)) and the vector of partial coefficients c (a) = (c1 (a), c2 (a), . . . , cn (a)) which generalize the gradient of f at the point a.
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Proposition 3.3. If f admits non-null kth partial derivative at the point a, then cek (a) =
∂f (a), ∂ xk
c−ek (a) = −
∂f (a). ∂ xk
(5)
Proof. According to the definition of partial derivative
∂f f (a + hek ) − f (a) (a) = lim = cek (a) h→0+ ∂ xk h for λek (a) = 1. Likewise, c−ek (a) = lim
f (a − hek ) − f (a) hλ
h→0+
.
With the change h = −e h, c−ek (a) = lim
f (a + e hek ) − f (a)
e h→0−
for λ = 1.
(−e h)λ
=−
∂f (a) ∂ xk
As a consequence, ∂f
∂f
∂f
Corollary 3.4. If cek (a)c−ek (a) > 0 then either ∂ x (a) = 0 or ∂ x (a) does not exist. If cek (a)c−ek (a) = 0, then either ( ∂ x )+ k k k (a) = 0 or ( ∂∂xf )− (a) = 0. k
Proof. It is a straightforward result of the former proposition and Definition 2.1.
Definition 3.5. For a scalar field f : D ⊆ Rn → R and a ∈ D0 , let us assume that all the partial exponents and coefficients exist at the point a (λ±ek (a), c±ek (a) for k = 1, 2, . . . , n). Let us consider the variability map Pa : Rn → R defined as Pa (u1 , u2 , . . . , un ) =
n X
ck∗,u (a)|uk |
λ∗k,u (a)
k=1
where ck∗,u (a) =
cek (a) c−ek (a)
if uk ≥ 0 if uk < 0
and
λ∗k,u (a)
λ ( a) = ek λ−ek (a)
if uk ≥ 0 if uk < 0.
The map f admits a variability of Müntz type at the point a if lim
f (a + u) − f (a) − Pa (u)
k uk λ
u→0
=0
(6)
where λ = min{λek (a), λ−ek (a); k = 1, 2, . . . , n}. Note 4. This definition extends the concept of differentiability of scalar fields. In particular, bearing in mind Proposition 3.3, it is easy to check that if f admits non-null partial derivatives at the point a and f is differentiable at a, the differential map Df (a) agrees with Pa and f admits a variability of Müntz type (see (11)). Note 5. According to Definition 3.5,
• If λek (a) = λ−ek (a) = λk and cek (a) = c−ek (a) = ck , the variable xk contributes to Pa with the term ck |uk |λk . • If λek (a) = λ−ek (a) = λk , cek (a) = c−ek (a) = ck and hλk is an even function, the variable xk contributes to Pa with the λ
term ck uk k . • If λek (a) = λ−ek (a) = λk , cek (a) = −c−ek (a) = ck , and hλk is an odd function, the variable xk contributes to Pa with the λ
term ck uk k . (This is the case if the kth partial derivative exists and is non-null.) If all the variables satisfy one of these conditions then coefficients and exponents of Pa do not depend on u, and one can speak about an approximation surface of Müntz type of f at the point a, which generalizes the tangent plane.
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Note 6. If f is a Müntz polynomial of type f (x) =
n X
ck |xk |λk
k=1
where λk > 0 for all k = 1, 2, . . . , n, then f agrees with its variability map at the origin. Example 6. Let us consider the map f (x, y) = x2/3 + y2/3 . This function is not differentiable at the points (x, 0) and (0, y). However the partial coefficients exist on the whole domain. According to Note 6, P(0,0) = f . For the rest of points, the coefficients and exponents are cx (x, y) =
2x−1/3 /3, 1,
x 6= 0, λx (x, y) = 1 x = 0, λx (0, y) = 2/3
−2x−1/3 /3, x 6= 0, λ−x (x, y) = 1 1, x = 0, λ−x (0, y) = 2/3 −1/3 2y /3, y 6= 0, λy (x, y) = 1 cy (x, y) = 1, y = 0, λy (x, 0) = 2/3 −2y−1/3 /3, y 6= 0, λ−y (x, y) = 1 c−y (x, y) = 1, y = 0, λ−y (x, 0) = 2/3 c−x (x, y) =
where c−x = c−e1 , c−y = c−e2 and the same notation for the exponents. For points with non-null coordinates f is differentiable and Pa = Df (a) (see (11)). For (x, 0) where x 6= 0, 2 −1/3 2/3 x u1 + u2 . 3
P(x,0) (u1 , u2 ) =
Let us see if the definition of variability is satisfied: lim
f (x + u1 , u2 ) − f (x, 0) − P(x,0) (u1 , u2 )
kuk2/3 2/3 2/3 (x + u1 ) − x − 32 x−1/3 u1 = 0. lim u→0 kuk2/3 u→0
=0
Let us see that all the terms tend to zero as u → 0:
|x−1/3 u1 | kuk−2/3 ≤ |x−1/3 | kuk1−2/3 → 0. On the other hand, if u → 0,
(x + u1 )
2/3
−x
2/3
=x
2/3
x + u1
2/3
! −1 '
x
2 3
x2/3 log
x + u1 x
'
2 −1/3 x u1
3
then
|x−1/3 u1 | kuk−2/3 ≤ |x−1/3 | kuk1−2/3 → 0. The limit is null and f can be approximated by means of the variability map. The result for (0, y) where y 6= 0 is analogous. Example 7. The map f (x, y) = y2 + y|x|1/3 does not admit partial derivatives at the points (0, y) due to the double condition of the absolute value and x-exponent lower than 1. Consequently, an approximation of Taylor type is not possible at these points. However f has an approximation of Müntz type. Let us consider a = (0, y) where y 6= 0, cx (0, y) = lim
f (h, y) − f (0, y)
= y,
hλ f (−h, y) − f (0, y)
h→0+
c−x (0, y) = lim
h→0+
hλ
= y,
for λ = λx = λ−x = 1/3, c−x = c−e1 and λ−x = λ−e1 . In the same way, cy (0, y) = lim
f (0, y + h) − f (0, y)
h→0+
c−y (0, y) = lim
h→0+
for λ = λy = λ−y = 1.
= 2y,
hλ f (0, y − h) − f (0, y) hλ
= −2y,
M.A. Navascués / Nonlinear Analysis 73 (2010) 1569–1584
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Fig. 1. Surface corresponding to the map f (x, y) = y2 + y|x|1/3 .
Fig. 2. Müntz approximation of the map f (x, y) = y2 + y|x|1/3 (Fig. 1) at (0, 1).
The variability map is then (see Note 5): P(0,y) (u1 , u2 ) = y|u1 |1/3 + 2yu2 . Figs. 1 and 2 represent the surface corresponding to f and its Müntz approximation at the point (0, 1) respectively. Let us prove the variability condition for the points (0, y), y 6= 0: lim
f (u1 , y + u2 ) − f (0, y) − y|u1 |1/3 − 2yu2
kuk1/3
u→0
lim
u→0
u22 + u2 |u1 |1/3
kuk1/3
= 0,
= 0.
For (0, 0) it is easy to check that
cx (0, 0) = 0, c−x (0, 0) = 0,
λx (0, 0) = 1 λ−x (0, 0) = 1
cy (0, 0) = 1, c−y (0, 0) = 1,
λy (0, 0) = 2 λ−y (0, 0) = 2.
The variability map is P(0,0) (u1 , u2 ) = u22 and the condition (6) is satisfied as well. In this way, f admits an approximation of Müntz type ∀(x, y) ∈ R2 despite the lack of tangent plane at many points.
√
Example 8. The map f (x, y) = (x + x2 + y2 )3/2 / 2 admits partial derivatives at the origin and the differential application is null. The higher-order Taylor approximation cannot be defined due to the non-existence of some second derivatives. The
p
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variability map provides a much better approximation than Taylor’s formula. cx (0, 0) = lim
f (h, 0) − f (0, 0)
= 2,
hλx f (−h, 0) − f (0, 0)
h→0+
c−x (0, 0) = lim
hλ−x
h→0+
= 0,
for λx = 3/2, and λ−x = 1. In the same way, cy (0, 0) = lim
f (0, h) − f (0, 0) hλy
h→0+
c−y (0, 0) = lim
√ = 1/ 2,
f (0, −h) − f (0, 0) hλ−y
h→0+
√ = 1/ 2,
for λy = λ−y = 3/2. The variability map at (0, 0) is then P(0,0) (u1 , u2 ) = 2
u1 + sign(u1 )u1
3/2
2
|u2 |3/2 + √ 2
and the variability condition is obviously satisfied. 3.1. Variability formula The definition of Müntz variability provides a formula of approximation in a neighborhood of the point. Denoting x = a + u, according to the definition of variability (6), f (x) = f (a) + Pa (x − a) + kx − akλ Ea (x − a)
(7)
where λ = min{λxk (a), λ−xk (a); k = 1, 2, . . . , n} and lim Ea (x − a) = 0.
x→a
This formula extends Taylor’s approximation of first order: Let us consider the images of the elements of the canonical basis ej = (e1j , e2j , . . . , enj ) where ekj = δjk . Following Definition 3.5, for all j = 1, 2, . . . , n, Pa (ej ) =
n X
λ∗k,e (a)
ck∗,ej (a)|ekj |
j
= cej (a).
(8)
k=1
In the same way, Pa (−ej ) = c−ej (a).
(9)
For an arbitrary u ∈ Rn , let us consider Ju+ = {i ∈ N : 1 ≤ i ≤ n, ui ≥ 0} Ju− = {i ∈ N : 1 ≤ i ≤ n, ui < 0} then, by definition of Pa , (8) and (9), P a ( u) =
X
λei (a)
ui
Pa (ei ) +
+
X
(−ui )λ−ei (a) Pa (−ei ).
(10)
−
i∈Ju
i∈Ju
∂f
In the differentiable case, if ∂ x (a) 6= 0 for all i = 1, 2, . . . , n, λei (a) = λ−ei (a) = 1, i Pa (ei ) = cei (a) =
∂f (a), ∂ xi
Pa (−ei ) = c−ei (a) = −
∂f (a) ∂ xi
and from (10) Pa (u) = ∇ f (a) · u = Df (a)(u) generalizing the differential application. The next result provides a formula for cv (a) in terms of the partial coefficients.
(11)
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Proposition 3.6. If f has a variability of Müntz type at a ∈ D with positive exponents, let us consider λ = min{λek (a), λe−k (a); k = 1, 2, . . . , n}. For fixed v ∈ Rn let Jv be defined as Jv = {i ∈ N : 1 ≤ i ≤ n, λ∗i,v (a) = λ}. Then cv (a) =
X
λ∗i,v (a)
ci∗,v (a)|vi |
i∈Jv
assuming that Jv is non-empty and the right term of this equality is non-null. Proof. Let us consider v ∈ Rn and define u = hv for h > 0. Applying the variability formula (7), f (a + hv) = f (a) +
n X
ck∗,v (a)h
λ∗k,v (a)
|vk |λk,v (a) + hλ kvkλ Ea (hv) ∗
k=1
where λ = min{λek (a), λe−k (a); k = 1, 2, . . . , n}. Then f (a + hv) − f (a) hλ
=
n X
ck∗,v (a)h
λ∗k,v (a)−λ
|vk |λk,v (a) + kvkλ Ea (hv). ∗
k=1
Applying the definitions of coefficient along v and Jv , cv (a) = lim
f (a + hv) − f (a)
=
hλ
h→0+
X
λ∗i,v (a)
ci∗,v (a)|vi |
i∈Jv
assuming that the sum is non-null.
Note 7. If f is differentiable at the point a with non-null first derivatives, Jv = {i ∈ N ; 1 ≤ i ≤ n} and the formula agrees with the classical method to compute derivatives along vectors by means of the gradient of f : Dv f (a) = ∇ f (a) · v = Df (a)(v). Example 9. For the mapping f (x, y) = x2 + 5y1/3 , a = (0, 0) and v = (1, 1), it is clear that cx (0, 0) = 1, cy (0, 0) = 5, λx (0, 0) = 2, λy (0, 0) = 1/3. Applying the definition of derivative along vectors: cv (0, 0) = lim
f (h, h) − f (0, 0) hλ
h→0+
=5
for λv (0, 0) = 1/3. With the procedure provided by the previous proposition, Jv = {2} and λ∗ (0,0) cv (0, 0) = c2∗,v (0, 0)|v2 | 2,v = cy (0, 0) = 5.
Proposition 3.7. If f has a variability of Müntz type at the point a ∈ D, with positive partial exponents λ±ek (a) > 0 for all k = 1, 2, . . . , n, then f is continuous at a. Proof. According to the variability formula (7)–(9) f (a + u) = f (a) +
n X
|uk |λk,u (a) Pa (sign(uk )ek ) + kukλ Ea (u) ∗
k=1
where λ > 0 and Ea (u) → 0 as u → 0. Pa (sign(uk )ek ) agrees with cek (a) or c−ek (a), depending on the sign of uk , then it is a bounded magnitude. When u → 0, the second and third terms of the former equality tend to zero and f (a + u) tends to f (a). 4. Extrema of functions with Müntz variability In this section we develop some necessary and sufficient conditions for the existence of local extrema of functions with a variability of Müntz type. These conditions involve of course the value of partial coefficients and exponents. The results are extensions of those of the smooth case. Proposition 4.1. If f admits partial variability exponents at the point a ∈ D and a is a stationary point then λ±ek (a) ≥ 1 for all k = 1, 2, . . . , n.
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∂f
Proof. If a is stationary ∂ x (a) = 0 and k
f (a + hek ) − f (a) = 0. h→0+ h lim
h, According to Proposition 2.5 λek (a) ≥ 1. In the same way, with the change h = −e f (a − hek ) − f (a)
lim
h
h→0+
= lim
f (a + e hek ) − f (a)
=−
−e h
e h→0−
∂f (a) = 0. ∂ xk
Then
f (a + h(−ek )) − f (a) =0 lim h→0+ h and λ−ek (a) ≥ 1.
Definition 4.2. a ∈ D is a steady point of f if f admits partial exponents at a and cek (a)ce−k (a) ≥ 0 for all k = 1, 2, . . . , n. According to Corollary 3.4, if a is steady, then it is either a stationary point of f or some partial derivative does not exist at a. A stationary point is not necessarily a steady point and vice versa. Example 10. The map f (x, y) = y2 − x3 has a stationary point at a = (0, 0) but this point is not steady since cx (0, 0) = −1 and c−x (0, 0) = 1 for λx (0, 0) = λ−x (0, 0) = 3. Example 11. The map f (x, y) = x2/3 + y2/3 has a steady point at a = (0, 0) but this point is not stationary since the partial derivatives do not exist at (0, 0). If some additional conditions are imposed, one of the implications is valid. ∂f
Proposition 4.3. If a ∈ D is steady and for any k either λek (a) and λ−ek (a) > 1 or ∂ x (a) exists then a is a stationary point. k Proof. For k such that λek (a), λ−ek (a) > 1, let us consider that 0 = lim cek (a)h−1+λek (a) = lim h→0+
f (a + hek ) − f (a) h
h→0+
=
∂f ∂ xk
+
(a).
For the other side
∂f ∂ xk
−
(a) = lim
f (a + hek ) − f (a) h
h→0−
= lim e h→0+
f (a − e hek ) − f (a)
−e h
and h−1+λ−ek (a) = − 0 = lim c−ek (a)e e h→0+
∂f
∂f ∂ xk
−
(a). ∂f
If ∂ x (a) exists and a is steady, bearing in mind Proposition 3.3, ∂ x (a) = 0. k k In all the cases the derivatives are null. Proposition 4.4. If a is a local extreme of f and f admits partial exponents at a then a is steady. In this case, if cek (a) ≥ 0 the point is a minimum and if cek (a) ≤ 0 is a maximum. Proof. For instance, if a is a local minimum there exists δ > 0 such that f (a + hek ) − f (a) > 0 and f (a − hek ) − f (a) > 0 for all h ∈ (0, δ). Applying the definition of kth exponent cek (a) ≥ 0 and c−ek (a) ≥ 0. The steadiness is then a necessary condition to be a local extreme. Let us see a sufficient condition. Let us assume a point a such that cek (a) = c−ek (a) = ck and λek (a) = λ−ek (a) = λk > 0, for all k = 1, 2, . . . , n then the variability map at a is P a ( u1 , u2 , . . . , un ) =
n X
ck |uk |λk
k=1
and one has the following cases:
• ck > 0 for k = 1, 2, . . . , n then Pa has a global minimum at zero. • ck < 0 for k = 1, 2, . . . , n then Pa has a global maximum at zero. • ck > 0 for some k and cj < 0 for the rest, zero is a saddle point.
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The question is now if this character may be extended to f with respect to the point a. The following result provides sufficient conditions for the knowledge of the character of a steady point. Theorem 4.5. Let f be a scalar field with a variability of Müntz type at a ∈ D with coefficients cek (a) = c−ek (a) = ck and positive exponents λek (a) = λ−ek (a) = λ > 0, for all k = 1, 2, . . . , n. Then:
• If ck > 0 for all k, f has a local minimum at the point a. • If ck < 0 for all k, f has a local maximum at the point a. • If ci > 0 for some i and cj < 0 for the rest, a is a saddle point. Proof. With the conditions given the variability map is Pa (u) =
n X
ck |uk |λ .
k=1
In the first case (ck > 0) Pa (u) > 0 for all u 6= 0. Let S be the unit sphere S = {x ∈ Rn : kxk = 1}. Since S is compact and Pa continuous there exists p∗ ∈ R such that Pa (x) ≥ p∗ > 0 for all x ∈ S. Let u be a non-null vector, u ∈ Rn . Since u/kuk ∈ S,
Pa
u
k uk
n X
1
=
k uk λ
ck |uk |λ =
k=1
Pa (u)
k uk λ
≥ p∗
and Pa (u) ≥ kukλ p∗ . According to (7) and the previous inequality, f (a + u) − f (a) ≥ kukλ p∗ + kukλ Ea (u). Since lim Ea (u) = 0
u→0
for ε = p∗ /2 > 0 there exists δ > 0 such that for 0 < kuk < δ ,
−
p∗ 2
< Ea (u) <
p∗ 2
and f (a + u) − f (a) > kukλ p∗ − kukλ p∗ /2 > 0. As a consequence f has a local minimum at the point a. The second case is proved in the same way. Let us consider now the third item and i, j ∈ {1, 2, . . . , n} such that ci > 0 and cj < 0 respectively. Applying the variability formula for x = a + hei , f (a + hei ) − f (a) = Pa (hei ) + |h|λ Ea (hei ) f (a + hei ) − f (a) = ci |h|λ + |h|λ Ea (hei ).
Since lim Ea (hei ) = 0 for h → 0, there exists δ > 0 such that if 0 < |h| < δ < 1 then
|Ea (hei )| <
ci 2
.
Thus ci
ci
2
2
− |h|λ < |h|λ Ea (hei ) <
|h|λ
f (a + hei ) − f (a) > ci |h|λ −
ci 2
|h|λ > 0
and the curve f (a + hei ) has a local minimum at h = 0. In the same way it can be proved that f (a + hej ) has a local maximum. As a consequence, a is a saddle point.
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Fig. 3. Graph of the function f (x) = 1 − x4/3 +
√
1 − x4 along with its Müntz approximation at x = 0 (upper curve).
Fig. 4. Graph of the mapping f (x, y) = x2/3 + y2/3 − 10|x|y + x3 − y4 displaying a local minimum at the origin.
Example 12. Let f be defined as f (x, y) = 1 − x4/3 − y4/3 + 1 − x4 − y4 . The classical criterion of the hessian matrix is not applicable at (0, 0) because it does not exist. However, the former result is conclusive:
p
√ cx (0, 0) = −1 + lim
h→0+
1 − h4 − 1 h4/3
= −1.
Thus, cx (0, 0) = c−x (0, 0) = cy (0, 0) = c−y (0, 0) = −1
λx (0, 0) = λ−x (0, 0) = λy (0, 0) = λ−y (0, 0) = 4/3. These data correspond to the second item. One can check easily that f has variability of Müntz type at (0, 0) and f has local maximum at (0, 0). The Müntz approximation is M(0,0) (u1 , u2 ) = f (0, 0) + P(0,0) (u1 , u2 ) = 2 − |u1 |4/3 − |u2 |4/3 . (See Fig. 3 for the one-dimensional case.) Conditions on the coefficients of higher order are not needed. Using the criteria exposed in Theorem 4.5 one can deduce the existence of local extremes in the following cases. Example 13. The map f (x, y) = x2/3 +y2/3 −10|x|y+x3 −y4 has a local minimum at (0, 0). In this case cx (0, 0) = cy (0, 0) = 1 and λ = 2/3, corresponding to the first item (see Fig. 4). Example 14. The map f (x, y) = x2/3 − y2/3 − 10|x|y + x3 − y4 has a saddle at (0, 0). Here cx (0, 0) = −cy (0, 0) = 1 and λ = 2/3, corresponding to the third item (see Fig. 5). Let us consider now f : C ⊆ Rn → R where C is open and convex. Let us remind that f is concave if f (λx + (1 − λ)y) ≥ λf (x) + (1 − λ)f (y) for all x, y ∈ C and 0 < λ < 1. f is convex if −f is concave. The next result illustrates the position of the Müntz approximation with respect to the graph of f in several cases.
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Fig. 5. Surface f (x, y) = x2/3 − y2/3 − 10|x|y + x3 − y4 with a saddle at the origin.
Theorem 4.6. If f admits a variability of Müntz type for all a ∈ C , f is concave and λ±ek (a) = λ(a) ≥ 1, for all k = 1, 2, . . . , n then for any a, x ∈ C , f (x) ≤ f (a) + Pa (x − a).
(12)
If f is convex and λ±ek (a) = λ(a) ≤ 1, for all k = 1, 2, . . . , n then for any a, x ∈ C , f (x) ≥ f (a) + Pa (x − a). If f is strictly concave (resp. convex) the former inequalities are strict for x 6= a. Proof. Let us consider f concave, λ±ek (a) = λ(a) ≥ 1, and a, x ∈ C , v = x − a and 0 ≤ h < 1, then f (a + hv) = f (h(a + v) + (1 − h)a) ≥ hf (a + v) + (1 − h)f (a) and f (a + hv) − f (a) ≥ h(f (a + v) − f (a)) ≥ hλ(a) (f (a + v) − f (a)). If the quantity Pa (hv) = h 0 = lim
λ(a)
Pa (v) is subtracted from the former inequality, dividing by h
f (a + hv) − f (a) − Pa (hv) hλ(a)
h→0+
(13) λ(a)
and taking limits,
≥ f (a + v) − f (a) − Pa (v)
and f (a + v) ≤ f (a) + Pa (v).
(14)
The second inequality is proved in the same way. If f is strictly concave, in particular is concave. For all a, x ∈ C such that x 6= a, let us consider v = x − a. Let h be such that 0 < h < 1. By (12) f (a + hv) ≤ f (a) + Pa (hv).
(15)
Using (13) with strict inequality, f (a + hv) − f (a) > hλ(a) (f (a + v) − f (a)) and as Pa (hv) = h h
λ(a)
λ(a)
(16)
Pa (v), by (15) and (16),
Pa (v) ≥ f (a + hv) − f (a) > hλ(a) (f (a + v) − f (a)).
Consequently Pa (v) > f (a + v) − f (a) and the inequality of the statement of the theorem is strict.
Corollary 4.7. With the hypotheses of the previous theorem,
• If f is concave and cek (a), c−ek (a) ≤ 0 for all k then a is a global maximum. • If f is convex and cek (a), c−ek (a) ≥ 0 then a is a global minimum.
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Table 1 Absolute and relative errors (in modulus) corresponding to Example 15 provided by the trapeze rule and the variability formula for three different lengths of the interval. h
Trap. abs.
Trap. rel.
Var. abs.
Var. rel.
10−2 10−3 10−4
0.00746 0.00057 0.00004
3.40474 1.34522 0.64399
0.00542 0.00022 0.000008
2.47423 0.51141 0.13637
Proof. If f is concave and cek (a), c−ek (a) ≤ 0 for all k then Pa (u) ≤ 0, for all u ∈ Rn and f (x) ≤ f (a) + Pa (x − a) ≤ f (a) for all x ∈ C .
Note 8. With strict concavity (resp. convexity) the extrema are strict. Note 9. According to the previous Theorem, if f is concave and λ(a) ≥ 1, the graph of f remains below the Müntz approximation. The map of Example 12 is concave and for a = (0, 0), cx = cy = c−x = c−y = −1, λx = λy = λ−x = λ−y = 4/3. The hypotheses of the Corollary are met and f has at (0, 0) a global maximum. Fig. 3 illustrates the one-dimensional case of this Example. The upper curve is the Müntz approximation and the lower one represents the graph of the original mapping for one coordinate. 5. Applications to numerical integration near singularities 5.1. Quadratures in small intervals adjacent to singular points The one-dimensional formulae of variability may efficiently solve the problem of computing the integral t0 + h
Z Ih =
f (t )dt
(17)
t0
where h ' 0 and f has a singularity in the variability at t0 , with exponent λ0 = λ(t0 ) > 0 and right coefficient c0 = c (t0 ) [1]. Using the Müntz polynomial approximating f in a right neighborhood of t0 f (t0 + δ) ' f (t0 ) + c0 δ λ0
(18)
and denoting y0 = f (t0 ), the integral is approached by t0 + h
Z Ih =
f (t )dt ' y0 h +
t0
c0 hλ0 +1
λ0 + 1
.
(19)
Example 15. Let f be the function defined by the expression f (t ) = (1 − t 2/9 )9/2 f has a singularity at t = 0 with right exponent λ0 = 2/9 and coefficient c0 = −9/2 (see [1], Example 6). If one wishes to approximate numerically the integral h
Z
f (t )dt
Ih =
(20)
0
one can apply the formula (19), which is expressed in this case as Ih ' h −
81 22
h11/9
(21)
for small h > 0. The trapeze rule is Ih '
h 2
1 + (1 − h2/9 )9/2 .
(22)
Table 1 joins the relative errors (in modulus) corresponding to the trapeze rule and the variability formula for steps h = 0.01, 0.001, 0.0001. The best results correspond to the latter. The error of the trapeze is near five times that of the variability for h = 10−4 .
M.A. Navascués / Nonlinear Analysis 73 (2010) 1569–1584
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5.2. Numerical solution of ordinary differential equations with singular initial values Let us consider now the problem of finding a function x(t ) such that x˙ (t ) = f (x(t ))
(23)
for all t ∈ (t0 , b] and x(t0 ) = x0 , and let us assume that f goes to infinity at x0 . This kind of problems cannot be solved by the usual one-step methods due to the lack of a real value for f at x0 . This question faces the construction of trajectories of dynamical √ systems with an infinite derivative at some point of the path (let us think that a mapping as conventional as x(t ) = t cannot be a solution of a differential equation of type (23) on the interval I = [0, 1]). One even could look for a curve with a fixed nonlinear slope at one extreme of the trajectory, in the sense of the Ref. [1], and satisfying the differential equation in the rest of the domain. If f admits Müntz variability at x0 and f (x) > 0 close to this point, it can be locally approached as [1] f (x) ' cf (x0 )(x − x0 )λ ,
(24)
where cf (x0 ) ∈ R and λ = λf (x0 ) < 0. The solution x will have an expansion (in a neighborhood (t0 , t0 + δ)) as x(t ) ' x(t0 ) + cx (t0 )(t − t0 )β ,
(25)
where β = βx (t0 ) > 0 and cx (t0 ) ∈ R are the variability exponent and coefficient of x at t0 respectively. In this case, x˙ (t ) ' cx (t0 )β(t − t0 )β−1 .
(26)
Bearing in mind (23)–(26) and the unicity of coefficients and exponents, cx (t0 )β(t − t0 )β−1 = cf (x0 )cx (t0 )λ (t − t0 )βλ
(27)
and thus
β = 1/(1 − λ)
(28)
cx (t0 ) = ((1 − λ)cf (x0 ))
1/(1−λ)
.
(29)
These equalities relate the coefficients and exponents of f and x at x0 and t0 . If the problem admits a solution, the former expressions provide an approximation of the field f in terms of the time variable (26), (28), (29): f (x(t )) ' (cf (x0 ))1/(1−λ) (1 − λ)λ/(1−λ) (t − t0 )λ/(1−λ)
(30)
and this approach gives a way of obtaining a numerical value for the solution x at the first step t1 = t0 + h, t0 + h
Z x1 − x0 =
f (x(t ))dt ' (cf (x0 ))1/(1−λ) (1 − λ)λ/(1−λ)
t0
Z
t0 + h
(t − t0 )λ/(1−λ) dt ,
(31)
t0
and the value x1 = x(t1 ) can be computed as: x1 = x0 + (1 − λ)cf (x0 )h
1/(1−λ)
.
(32)
The rest of the discrete trajectory can be obtained using a standard method of numerical resolution of o.d.e. This procedure can be easily generalized to non-autonomous evolution equations. Example 16. Let us consider the differential equation x˙ = 2x−2 + x,
(33)
for t ∈ (0, 1]; x(0) = 0. The exponents and coefficients are (33), (28), (29):
λ = −2,
cf (0) = 2,
β = 1/3,
cx (0) = 61/3 ,
and the first step reads as x1 = x0 + (1 − λ)cf (x0 )h
1/(1−λ)
= x0 + (6h)1/3 .
For the remaining t values, Heun’s method has been used: xk+1 = xk +
h 2
(f (xk ) + f (xk + hf (xk )))
where k ≥ 1. The exact solution is x(t ) = (2(e3t − 1))1/3 . Table 2 joins the exact values of x(tk ) along with the approximates for a step of h = 0.1.
(34)
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Table 2 Values of the exact and approximate paths, for a step h = 0.1. tk
Exact solution
Approximate solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.88778 1.18029 1.42918 1.66794 1.90959 2.16154 2.42906 2.71651 3.02787 3.36701
0.84343 1.15505 1.40834 1.64840 1.89006 2.14124 2.40744 2.69310 3.00222 3.33871
If one wishes the construction of a trajectory with a fractional initial value of type dx
(dt )β
(t0 ) = c0 ,
x(t0 ) = x0 ,
(35)
that is to say, a curve with a prescribed nonlinear slope, the coefficients and exponents of x and f at t0 and x0 , respectively, need to satisfy relations of compatibility of type (28) and (29). Only in this case a solution will be possible. Note. The derivative in (35) does not agree with the fractional derivative of Riemann–Liouville (see for instance [9]). However, if x is α -differentiable (in the sense of fractional derivative), 0 < α < 1 and x admits a right exponent α at t0 with coefficient cx (t0 ) then ([9], Corollary 3.2) x(α) (t0 ) = Γ (1 + α)cx (t0 ). In this case, the variability elements may be good for all the current applications of fractional derivatives. The former relation enables the computation of coefficients by means of known formulae of fractional differentiation as well. References [1] M.A. Navascués, Local variability of non-smooth functions, Nonlinear Anal. TMA 70 (7) (2009) 2506–2518. [2] H. Müntz, Über den Approximationssatz von Weierstrass, H. A. Schwarz-Festschrift, Berlin, 1914, 303–312. [3] S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné, Mem. Cl. Sci. Acad. Roy. Belg. 4 (1912) 1–103. [4] S.N. Bernstein, Sur les recherches récentes relatives à la meilleure approximation des fonctions continues par des polynômes, in: E.W. Hobson, A.E.H. Love (Eds), Proc. Fifth. Int. Congress of Math., I, 1913, pp. 256–266. [5] E.L. Ortiz, A. Pinkus, Herman Müntz: A Mathematician’s Odyssey, Math. Int. 27 (2005) 22–30. [6] P. Borwein, T. Erdélyi, The full Müntz theorem in C [0, 1] and L1 [0, 1], J. Lond. Math. Soc. 54 (1996) 102–110. [7] P. Borwein, T. Erdélyi, J. Zhang, Müntz systems and orthogonal Müntz–Legendre polynomials, Trans. Amer. Math. Soc. 342 (2) (1994) 523–542. [8] D.S. Lubinsky, E.B. Saff, Zero distribution of Müntz extremal polynomials in Lp[0, 1], Proc. Amer. Math. Soc. 135 (2007) 427–435. [9] G. Jumarie, Generalized Fokker–Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α , Cent. Eur. J. Phys. 6 (3) (2008) 737–753.
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Local approximation of functions with several variables M.A. Navascués ∗ Departamento de Matemática Aplicada, Universidad de Zaragoza, C/ María de Luna, 3, 50018 Zaragoza, Spain
article
info
Article history: Received 8 May 2009 Accepted 27 April 2010 Keywords: Taylor’s formula Non-differentiability of functions of several variables Müntz powers Real-valued functions
abstract This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave–convex functions are obtained as well. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The procedures proposed in this article generalize the linear (Fréchet) approximation of scalar functions of several variables provided by the classical formula of Taylor of first order. The fundamental hypothesis is that the rate with which a function changes if the point is varied in some direction is of exponential type, beyond the linear character displayed by the tangent. In this way, the method described extends the directional derivative of a map at a point of its domain including a new element of power type. The article proposes a kind of nonlinear differential map as well. The word differential is used here in the sense that it measures the variation (difference) of the mapping in neighboring points. The subject is a generalization of paper [1] to the multivariable case. The approximation studied enables the calculus of local and global extrema in order to include some functions which are not differentiable and/or the criterion of the hessian matrix is not applicable. The basic functions which provide the contact with the scalar field are Müntz powers. H. Müntz solved the problem of finding necessary and sufficient conditions on the exponents λ1 , λ2 , . . . (not necessarily integer) for the completeness of the system {xλ1 , xλ2 , . . .} in the space C [0, 1] [2]. This question had been previously raised by Bernstein [3,4]. Müntz’s Theorem was the contribution of the author in honour of his teacher H. Schwarz. An interesting essay about Müntz is written in Ref. [5]. Extensions of this Theorem and related results can be found for instance in [6–8]. 2. Directional variability exponents In this paragraph, the concept of directional derivative is generalized in order to include more general sets of maps. The local behavior of a multivariate function f is approached by means of non-polynomial basic functions. The procedure considers rates of change of power type for f with respect to linear changes in the domain coordinate.
∗
Tel.: +34 976761983; fax: +34 976761886. E-mail address: [email protected].
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.063
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Definition 2.1. Let f be a scalar field, f : D ⊆ Rn → R, a ∈ D0 and v ∈ Rn . f admits a positive variability exponent (λv (a) > 0) along the vector v if the limit cv (a) = lim
f (a + hv) − f (a) hλv (a)
h→0+
exists, is real and non-null. In this case, cv (a) is the variability coefficient of f at the point a along the vector v . If for all λ ∈ R, lim
f (a + hv) − f (a) hλ
h→0+
=0
then, by definition, λv (a) = 1 and cv (a) = 0 (this is the case, for instance, if f is constant in a neighborhood of a). Example 1. Let us consider the mapping f (x, y) = x2 + 5y1/3 . This function does not own derivative at a = (0, 0) along v = (1, 1). However it possesses a variability exponent at this point along v since cv (0, 0) = lim
f (h, h) − f (0, 0) hλ
h→0+
= lim
h→0+
h2 + 5h1/3 hλ
.
Taking λ = λv (0, 0) = 1/3, cv (0, 0) = 5. Definition 2.2. The scalar field f admits a directional variability exponent at a ∈ D0 along the vector v ∈ Rn if the exponent λu (a) exists for u = v/kvk. Example 2. For the data of Example 1 the directional derivative is cu (0, 0) = lim
h→0+
1 h1/3
h
2
√
2
h
+5 √
2
1/3 !
= 5/21/6 .
The next results can be proved as in Propositions 2.5 and 2.3 of the Ref. [1]. Proposition 2.3. The exponent and coefficient of f at a along the vector v , if they exist, are unique. Proposition 2.4. If f admits a non-null derivative at the point a along the vector v (Dv f (a)) then λv (a) = 1 and cv (a) = Dv f (a). Note 1. If the exponent λv (a) exists, it provides a local approximation formula for f in the direction of v , f (a + hv) ' f (a) + cv (a)hλv (a) and the directional coefficient is a rate of the variation of f at a on the line a + hv with respect to the increase of hλv (a) at zero. Note 2. As in the differentiable case, the existence of positive exponents along any direction does not imply the continuity of f at the point a. A counterexample is given by the mapping f (x, y) =
xy2 x2
+ y4
if (x, y) 6= (0, 0), f (0, 0) = 0. For v = (α, β), α 6= 0 and β 6= 0, λv (0, 0) = 1 and cv (0, 0) = β 2 /α . For v = (0, β), λv (0, 0) = 1 and cv (0, 0) = 0. However, f is not continuous at (0, 0). Proposition 2.5. If f : D ⊆ Rn → R admits a positive variability exponent and cv (a) 6= 0, then λ = λv (a) is the supremum of the numbers α ∈ R such that La,α,v
f (a + hv) − f (a) =0 = lim h→0+ hα
(1)
and the infimum of the constants α ∈ R such that
f (a + hv) − f (a) = ∞. hα
La,α,v = lim h→0+
(2)
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Proof. For α ∈ R, and λ = λv (a) let us consider
f (a + hv) − f (a) = lim f (a + hv) − f (a) α α−λ λ h→0+ h h h
La,α,v = lim h→0+ La,α,v = lim
|cv (a)| hα−λ
h→0+
(3)
.
(4)
If α < λ then La,α,v = 0 and if α > λ then La,α,v = ∞.
Note 3. If cv (a) = 0 this result does not hold. For instance for a = 0 and the function f (t ) = e−1/t if t 6= 0, f (0) = 0, the exponent and coefficient are λ(0) = 1 and c (0) = 0. However, L0,α = 0, ∀α ∈ R and sup{α} = +∞ in R. In this sense we can say that mappings of this kind have an infinite variability exponent. 2
For f (t ) = log(t ),
log(t ) =0 t →0+ tα lim
if α < 0 and
log(t ) = +∞ t →0+ tα lim
if α ≥ 0. In this sense we could say that log(t ) has a null exponent at t = 0. 2.1. Directional formulae Assuming the existence of λv (a), let us denote for the sake of simplicity λ = λv (a) and ga,v (h) =
f (a + hv) − f (a) hλ
.
Then lim ga,v (h) = cv (a)
h→0+
and consider ga∗,v (h) = ga,v (h) − cv (a). The definition of exponent provides the variability formula of first order along the vector v : f (a + hv) = f (a) + cv (a)hλ + ga∗,v (h)hλ for h > 0 and ga∗,v a scalar function such that lim ga∗,v (h) = 0.
h→0+
This formula can be generalized taking exponents of higher order. In the Ref. [1] we defined superior coefficients for functions of a real variable. We can use this procedure here defining q q(h) = f (a + hv). In this way the coefficients of q at t = 0 (c1 (0)) agree with those of f at the point a along the vector f v (cv (a)). cvf (a) = lim
f (a + hv) − f (a) hλ
h→0+
= c1q (0).
Denoting λfv (a) = λ1 = λ, the second exponent and coefficient were defined as q(h) − q(0) − c1 (0)hλ1 q
q
c2 (0) = lim
hλ2
h→0+
and so on. These magnitudes provide the one-dimensional formula for h > 0, q(h) = q(0) + c1 (0)hλ1 + c2 (0)hλ2 + o(hλ2 ). q
f
q
q
f
q
Defining cv,1 (a) = c1 (0), cv,2 (a) = c2 (0), we have the second-order directional formula of f at the point a along the vector v , f (a + hv) = f (a) + cv,1 (a)hλ1 + cv,2 (a)hλ2 + o(hλ2 ), and if b = a + hv , with h = 1 and v = b − a, f
f
f
f
f (b) = f (a) + cv,1 (a) + cv,2 (a) + o(1).
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3. Variability map Definition 3.1. Let f be a scalar field, f : D ⊆ Rn → R, a ∈ D0 and v ∈ Rn . f admits a positive kth partial exponent of variability (λek (a) > 0) if λ = λek (a) exists according to Definition 2.1, where ek is the kth vector of the canonical basis of Rn . That is to say, the kth partial coefficient of f at a is given by the limit cek (a) = lim
f (a + hek ) − f (a) hλek (a)
h→0+
with the considerations of Definition 2.1 for cek (a). Other notations for cek (a) are ck (a) and cxk (a). Likewise the exponent will be represented by λk (a) and λxk (a) as well. We will include in this definition the exponents corresponding to −ek for k = 1, 2, . . . , n as well. The coefficients cek generalize the concept of kth partial derivative and contain other cases of non-differentiability, as illustrated in the following Examples. Example 3. Let f be defined by f (x, y) = x2/3 + y2/3 , f does not admit partial derivatives at (0, 0). However the partial coefficients are given by f (h, 0) − f (0, 0)
cx (0, 0) = lim
h2/3
h→0+
= 1.
Likewise cy (0, 0) = 1 and λx (0, 0) = λy (0, 0) = 2/3. Example 4. Let f be defined by f (x, y) = x2/3 + (xy)1/3 , f (h, 0) − f (0, 0)
cx (0, 0) = lim
h2/3
h→0+
h→0+
f (0, h) − f (0, 0)
cy (0, 0) = lim
hλ
h→0+
= lim
h2/3 h2/3
= 1,
= 0,
for all λ ∈ R. Using the definition of exponents cy (0, 0) = 0 and λy (0, 0) = 1, cx (0, 0) = 1 and λx (0, 0) = 2/3. Definition 3.2. f has a variability singularity in the kth partial exponent at a if there exists λk (x), ∀x ∈ B(a, r ) for r > 0 and λk (x) is not continuous at a. Example 5. Let f be defined now as f (x, y) = x2 + xy1/3 . According to Definition 2.1 cx (x, y) = 2x + y1/3 ,
λx (x, y) = 1
for any (x, y) such that 2x + y cx (x0 , −8x30 ) = lim
1/3
6= 0. For the rest of the points:
f (x0 + h, −8x30 ) − f (x0 , −8x30 ) hλ
h→0+
=1
for λ = λx = 2. As a consequence, f has a singularity in the x-variability on the line 2x + y1/3 = 0. Concerning the y-coefficient, cy (x, y) = xy−2/3 /3,
λy (x, y) = 1
for any (x, y) such that y 6= 0. For the rest of the points: cy (x0 , 0) = lim
h→0+
f (x0 , h) − f (x0 , 0) hλ
= x0
for λ = λy = 1/3 if x0 6= 0, λ = λy = 1 if x0 = 0. As a consequence, f has a singularity in the y-variability on the line y = 0. If the exponents λk (a) exist for k = 1, 2, . . . , n, we can consider the vector of partial variability exponents e(a) = (λ1 (a), λ2 (a), . . . , λn (a)) and the vector of partial coefficients c (a) = (c1 (a), c2 (a), . . . , cn (a)) which generalize the gradient of f at the point a.
M.A. Navascués / Nonlinear Analysis 73 (2010) 1569–1584
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Proposition 3.3. If f admits non-null kth partial derivative at the point a, then cek (a) =
∂f (a), ∂ xk
c−ek (a) = −
∂f (a). ∂ xk
(5)
Proof. According to the definition of partial derivative
∂f f (a + hek ) − f (a) (a) = lim = cek (a) h→0+ ∂ xk h for λek (a) = 1. Likewise, c−ek (a) = lim
f (a − hek ) − f (a) hλ
h→0+
.
With the change h = −e h, c−ek (a) = lim
f (a + e hek ) − f (a)
e h→0−
for λ = 1.
(−e h)λ
=−
∂f (a) ∂ xk
As a consequence, ∂f
∂f
∂f
Corollary 3.4. If cek (a)c−ek (a) > 0 then either ∂ x (a) = 0 or ∂ x (a) does not exist. If cek (a)c−ek (a) = 0, then either ( ∂ x )+ k k k (a) = 0 or ( ∂∂xf )− (a) = 0. k
Proof. It is a straightforward result of the former proposition and Definition 2.1.
Definition 3.5. For a scalar field f : D ⊆ Rn → R and a ∈ D0 , let us assume that all the partial exponents and coefficients exist at the point a (λ±ek (a), c±ek (a) for k = 1, 2, . . . , n). Let us consider the variability map Pa : Rn → R defined as Pa (u1 , u2 , . . . , un ) =
n X
ck∗,u (a)|uk |
λ∗k,u (a)
k=1
where ck∗,u (a) =
cek (a) c−ek (a)
if uk ≥ 0 if uk < 0
and
λ∗k,u (a)
λ ( a) = ek λ−ek (a)
if uk ≥ 0 if uk < 0.
The map f admits a variability of Müntz type at the point a if lim
f (a + u) − f (a) − Pa (u)
k uk λ
u→0
=0
(6)
where λ = min{λek (a), λ−ek (a); k = 1, 2, . . . , n}. Note 4. This definition extends the concept of differentiability of scalar fields. In particular, bearing in mind Proposition 3.3, it is easy to check that if f admits non-null partial derivatives at the point a and f is differentiable at a, the differential map Df (a) agrees with Pa and f admits a variability of Müntz type (see (11)). Note 5. According to Definition 3.5,
• If λek (a) = λ−ek (a) = λk and cek (a) = c−ek (a) = ck , the variable xk contributes to Pa with the term ck |uk |λk . • If λek (a) = λ−ek (a) = λk , cek (a) = c−ek (a) = ck and hλk is an even function, the variable xk contributes to Pa with the λ
term ck uk k . • If λek (a) = λ−ek (a) = λk , cek (a) = −c−ek (a) = ck , and hλk is an odd function, the variable xk contributes to Pa with the λ
term ck uk k . (This is the case if the kth partial derivative exists and is non-null.) If all the variables satisfy one of these conditions then coefficients and exponents of Pa do not depend on u, and one can speak about an approximation surface of Müntz type of f at the point a, which generalizes the tangent plane.
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Note 6. If f is a Müntz polynomial of type f (x) =
n X
ck |xk |λk
k=1
where λk > 0 for all k = 1, 2, . . . , n, then f agrees with its variability map at the origin. Example 6. Let us consider the map f (x, y) = x2/3 + y2/3 . This function is not differentiable at the points (x, 0) and (0, y). However the partial coefficients exist on the whole domain. According to Note 6, P(0,0) = f . For the rest of points, the coefficients and exponents are cx (x, y) =
2x−1/3 /3, 1,
x 6= 0, λx (x, y) = 1 x = 0, λx (0, y) = 2/3
−2x−1/3 /3, x 6= 0, λ−x (x, y) = 1 1, x = 0, λ−x (0, y) = 2/3 −1/3 2y /3, y 6= 0, λy (x, y) = 1 cy (x, y) = 1, y = 0, λy (x, 0) = 2/3 −2y−1/3 /3, y 6= 0, λ−y (x, y) = 1 c−y (x, y) = 1, y = 0, λ−y (x, 0) = 2/3 c−x (x, y) =
where c−x = c−e1 , c−y = c−e2 and the same notation for the exponents. For points with non-null coordinates f is differentiable and Pa = Df (a) (see (11)). For (x, 0) where x 6= 0, 2 −1/3 2/3 x u1 + u2 . 3
P(x,0) (u1 , u2 ) =
Let us see if the definition of variability is satisfied: lim
f (x + u1 , u2 ) − f (x, 0) − P(x,0) (u1 , u2 )
kuk2/3 2/3 2/3 (x + u1 ) − x − 32 x−1/3 u1 = 0. lim u→0 kuk2/3 u→0
=0
Let us see that all the terms tend to zero as u → 0:
|x−1/3 u1 | kuk−2/3 ≤ |x−1/3 | kuk1−2/3 → 0. On the other hand, if u → 0,
(x + u1 )
2/3
−x
2/3
=x
2/3
x + u1
2/3
! −1 '
x
2 3
x2/3 log
x + u1 x
'
2 −1/3 x u1
3
then
|x−1/3 u1 | kuk−2/3 ≤ |x−1/3 | kuk1−2/3 → 0. The limit is null and f can be approximated by means of the variability map. The result for (0, y) where y 6= 0 is analogous. Example 7. The map f (x, y) = y2 + y|x|1/3 does not admit partial derivatives at the points (0, y) due to the double condition of the absolute value and x-exponent lower than 1. Consequently, an approximation of Taylor type is not possible at these points. However f has an approximation of Müntz type. Let us consider a = (0, y) where y 6= 0, cx (0, y) = lim
f (h, y) − f (0, y)
= y,
hλ f (−h, y) − f (0, y)
h→0+
c−x (0, y) = lim
h→0+
hλ
= y,
for λ = λx = λ−x = 1/3, c−x = c−e1 and λ−x = λ−e1 . In the same way, cy (0, y) = lim
f (0, y + h) − f (0, y)
h→0+
c−y (0, y) = lim
h→0+
for λ = λy = λ−y = 1.
= 2y,
hλ f (0, y − h) − f (0, y) hλ
= −2y,
M.A. Navascués / Nonlinear Analysis 73 (2010) 1569–1584
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Fig. 1. Surface corresponding to the map f (x, y) = y2 + y|x|1/3 .
Fig. 2. Müntz approximation of the map f (x, y) = y2 + y|x|1/3 (Fig. 1) at (0, 1).
The variability map is then (see Note 5): P(0,y) (u1 , u2 ) = y|u1 |1/3 + 2yu2 . Figs. 1 and 2 represent the surface corresponding to f and its Müntz approximation at the point (0, 1) respectively. Let us prove the variability condition for the points (0, y), y 6= 0: lim
f (u1 , y + u2 ) − f (0, y) − y|u1 |1/3 − 2yu2
kuk1/3
u→0
lim
u→0
u22 + u2 |u1 |1/3
kuk1/3
= 0,
= 0.
For (0, 0) it is easy to check that
cx (0, 0) = 0, c−x (0, 0) = 0,
λx (0, 0) = 1 λ−x (0, 0) = 1
cy (0, 0) = 1, c−y (0, 0) = 1,
λy (0, 0) = 2 λ−y (0, 0) = 2.
The variability map is P(0,0) (u1 , u2 ) = u22 and the condition (6) is satisfied as well. In this way, f admits an approximation of Müntz type ∀(x, y) ∈ R2 despite the lack of tangent plane at many points.
√
Example 8. The map f (x, y) = (x + x2 + y2 )3/2 / 2 admits partial derivatives at the origin and the differential application is null. The higher-order Taylor approximation cannot be defined due to the non-existence of some second derivatives. The
p
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variability map provides a much better approximation than Taylor’s formula. cx (0, 0) = lim
f (h, 0) − f (0, 0)
= 2,
hλx f (−h, 0) − f (0, 0)
h→0+
c−x (0, 0) = lim
hλ−x
h→0+
= 0,
for λx = 3/2, and λ−x = 1. In the same way, cy (0, 0) = lim
f (0, h) − f (0, 0) hλy
h→0+
c−y (0, 0) = lim
√ = 1/ 2,
f (0, −h) − f (0, 0) hλ−y
h→0+
√ = 1/ 2,
for λy = λ−y = 3/2. The variability map at (0, 0) is then P(0,0) (u1 , u2 ) = 2
u1 + sign(u1 )u1
3/2
2
|u2 |3/2 + √ 2
and the variability condition is obviously satisfied. 3.1. Variability formula The definition of Müntz variability provides a formula of approximation in a neighborhood of the point. Denoting x = a + u, according to the definition of variability (6), f (x) = f (a) + Pa (x − a) + kx − akλ Ea (x − a)
(7)
where λ = min{λxk (a), λ−xk (a); k = 1, 2, . . . , n} and lim Ea (x − a) = 0.
x→a
This formula extends Taylor’s approximation of first order: Let us consider the images of the elements of the canonical basis ej = (e1j , e2j , . . . , enj ) where ekj = δjk . Following Definition 3.5, for all j = 1, 2, . . . , n, Pa (ej ) =
n X
λ∗k,e (a)
ck∗,ej (a)|ekj |
j
= cej (a).
(8)
k=1
In the same way, Pa (−ej ) = c−ej (a).
(9)
For an arbitrary u ∈ Rn , let us consider Ju+ = {i ∈ N : 1 ≤ i ≤ n, ui ≥ 0} Ju− = {i ∈ N : 1 ≤ i ≤ n, ui < 0} then, by definition of Pa , (8) and (9), P a ( u) =
X
λei (a)
ui
Pa (ei ) +
+
X
(−ui )λ−ei (a) Pa (−ei ).
(10)
−
i∈Ju
i∈Ju
∂f
In the differentiable case, if ∂ x (a) 6= 0 for all i = 1, 2, . . . , n, λei (a) = λ−ei (a) = 1, i Pa (ei ) = cei (a) =
∂f (a), ∂ xi
Pa (−ei ) = c−ei (a) = −
∂f (a) ∂ xi
and from (10) Pa (u) = ∇ f (a) · u = Df (a)(u) generalizing the differential application. The next result provides a formula for cv (a) in terms of the partial coefficients.
(11)
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Proposition 3.6. If f has a variability of Müntz type at a ∈ D with positive exponents, let us consider λ = min{λek (a), λe−k (a); k = 1, 2, . . . , n}. For fixed v ∈ Rn let Jv be defined as Jv = {i ∈ N : 1 ≤ i ≤ n, λ∗i,v (a) = λ}. Then cv (a) =
X
λ∗i,v (a)
ci∗,v (a)|vi |
i∈Jv
assuming that Jv is non-empty and the right term of this equality is non-null. Proof. Let us consider v ∈ Rn and define u = hv for h > 0. Applying the variability formula (7), f (a + hv) = f (a) +
n X
ck∗,v (a)h
λ∗k,v (a)
|vk |λk,v (a) + hλ kvkλ Ea (hv) ∗
k=1
where λ = min{λek (a), λe−k (a); k = 1, 2, . . . , n}. Then f (a + hv) − f (a) hλ
=
n X
ck∗,v (a)h
λ∗k,v (a)−λ
|vk |λk,v (a) + kvkλ Ea (hv). ∗
k=1
Applying the definitions of coefficient along v and Jv , cv (a) = lim
f (a + hv) − f (a)
=
hλ
h→0+
X
λ∗i,v (a)
ci∗,v (a)|vi |
i∈Jv
assuming that the sum is non-null.
Note 7. If f is differentiable at the point a with non-null first derivatives, Jv = {i ∈ N ; 1 ≤ i ≤ n} and the formula agrees with the classical method to compute derivatives along vectors by means of the gradient of f : Dv f (a) = ∇ f (a) · v = Df (a)(v). Example 9. For the mapping f (x, y) = x2 + 5y1/3 , a = (0, 0) and v = (1, 1), it is clear that cx (0, 0) = 1, cy (0, 0) = 5, λx (0, 0) = 2, λy (0, 0) = 1/3. Applying the definition of derivative along vectors: cv (0, 0) = lim
f (h, h) − f (0, 0) hλ
h→0+
=5
for λv (0, 0) = 1/3. With the procedure provided by the previous proposition, Jv = {2} and λ∗ (0,0) cv (0, 0) = c2∗,v (0, 0)|v2 | 2,v = cy (0, 0) = 5.
Proposition 3.7. If f has a variability of Müntz type at the point a ∈ D, with positive partial exponents λ±ek (a) > 0 for all k = 1, 2, . . . , n, then f is continuous at a. Proof. According to the variability formula (7)–(9) f (a + u) = f (a) +
n X
|uk |λk,u (a) Pa (sign(uk )ek ) + kukλ Ea (u) ∗
k=1
where λ > 0 and Ea (u) → 0 as u → 0. Pa (sign(uk )ek ) agrees with cek (a) or c−ek (a), depending on the sign of uk , then it is a bounded magnitude. When u → 0, the second and third terms of the former equality tend to zero and f (a + u) tends to f (a). 4. Extrema of functions with Müntz variability In this section we develop some necessary and sufficient conditions for the existence of local extrema of functions with a variability of Müntz type. These conditions involve of course the value of partial coefficients and exponents. The results are extensions of those of the smooth case. Proposition 4.1. If f admits partial variability exponents at the point a ∈ D and a is a stationary point then λ±ek (a) ≥ 1 for all k = 1, 2, . . . , n.
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∂f
Proof. If a is stationary ∂ x (a) = 0 and k
f (a + hek ) − f (a) = 0. h→0+ h lim
h, According to Proposition 2.5 λek (a) ≥ 1. In the same way, with the change h = −e f (a − hek ) − f (a)
lim
h
h→0+
= lim
f (a + e hek ) − f (a)
=−
−e h
e h→0−
∂f (a) = 0. ∂ xk
Then
f (a + h(−ek )) − f (a) =0 lim h→0+ h and λ−ek (a) ≥ 1.
Definition 4.2. a ∈ D is a steady point of f if f admits partial exponents at a and cek (a)ce−k (a) ≥ 0 for all k = 1, 2, . . . , n. According to Corollary 3.4, if a is steady, then it is either a stationary point of f or some partial derivative does not exist at a. A stationary point is not necessarily a steady point and vice versa. Example 10. The map f (x, y) = y2 − x3 has a stationary point at a = (0, 0) but this point is not steady since cx (0, 0) = −1 and c−x (0, 0) = 1 for λx (0, 0) = λ−x (0, 0) = 3. Example 11. The map f (x, y) = x2/3 + y2/3 has a steady point at a = (0, 0) but this point is not stationary since the partial derivatives do not exist at (0, 0). If some additional conditions are imposed, one of the implications is valid. ∂f
Proposition 4.3. If a ∈ D is steady and for any k either λek (a) and λ−ek (a) > 1 or ∂ x (a) exists then a is a stationary point. k Proof. For k such that λek (a), λ−ek (a) > 1, let us consider that 0 = lim cek (a)h−1+λek (a) = lim h→0+
f (a + hek ) − f (a) h
h→0+
=
∂f ∂ xk
+
(a).
For the other side
∂f ∂ xk
−
(a) = lim
f (a + hek ) − f (a) h
h→0−
= lim e h→0+
f (a − e hek ) − f (a)
−e h
and h−1+λ−ek (a) = − 0 = lim c−ek (a)e e h→0+
∂f
∂f ∂ xk
−
(a). ∂f
If ∂ x (a) exists and a is steady, bearing in mind Proposition 3.3, ∂ x (a) = 0. k k In all the cases the derivatives are null. Proposition 4.4. If a is a local extreme of f and f admits partial exponents at a then a is steady. In this case, if cek (a) ≥ 0 the point is a minimum and if cek (a) ≤ 0 is a maximum. Proof. For instance, if a is a local minimum there exists δ > 0 such that f (a + hek ) − f (a) > 0 and f (a − hek ) − f (a) > 0 for all h ∈ (0, δ). Applying the definition of kth exponent cek (a) ≥ 0 and c−ek (a) ≥ 0. The steadiness is then a necessary condition to be a local extreme. Let us see a sufficient condition. Let us assume a point a such that cek (a) = c−ek (a) = ck and λek (a) = λ−ek (a) = λk > 0, for all k = 1, 2, . . . , n then the variability map at a is P a ( u1 , u2 , . . . , un ) =
n X
ck |uk |λk
k=1
and one has the following cases:
• ck > 0 for k = 1, 2, . . . , n then Pa has a global minimum at zero. • ck < 0 for k = 1, 2, . . . , n then Pa has a global maximum at zero. • ck > 0 for some k and cj < 0 for the rest, zero is a saddle point.
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The question is now if this character may be extended to f with respect to the point a. The following result provides sufficient conditions for the knowledge of the character of a steady point. Theorem 4.5. Let f be a scalar field with a variability of Müntz type at a ∈ D with coefficients cek (a) = c−ek (a) = ck and positive exponents λek (a) = λ−ek (a) = λ > 0, for all k = 1, 2, . . . , n. Then:
• If ck > 0 for all k, f has a local minimum at the point a. • If ck < 0 for all k, f has a local maximum at the point a. • If ci > 0 for some i and cj < 0 for the rest, a is a saddle point. Proof. With the conditions given the variability map is Pa (u) =
n X
ck |uk |λ .
k=1
In the first case (ck > 0) Pa (u) > 0 for all u 6= 0. Let S be the unit sphere S = {x ∈ Rn : kxk = 1}. Since S is compact and Pa continuous there exists p∗ ∈ R such that Pa (x) ≥ p∗ > 0 for all x ∈ S. Let u be a non-null vector, u ∈ Rn . Since u/kuk ∈ S,
Pa
u
k uk
n X
1
=
k uk λ
ck |uk |λ =
k=1
Pa (u)
k uk λ
≥ p∗
and Pa (u) ≥ kukλ p∗ . According to (7) and the previous inequality, f (a + u) − f (a) ≥ kukλ p∗ + kukλ Ea (u). Since lim Ea (u) = 0
u→0
for ε = p∗ /2 > 0 there exists δ > 0 such that for 0 < kuk < δ ,
−
p∗ 2
< Ea (u) <
p∗ 2
and f (a + u) − f (a) > kukλ p∗ − kukλ p∗ /2 > 0. As a consequence f has a local minimum at the point a. The second case is proved in the same way. Let us consider now the third item and i, j ∈ {1, 2, . . . , n} such that ci > 0 and cj < 0 respectively. Applying the variability formula for x = a + hei , f (a + hei ) − f (a) = Pa (hei ) + |h|λ Ea (hei ) f (a + hei ) − f (a) = ci |h|λ + |h|λ Ea (hei ).
Since lim Ea (hei ) = 0 for h → 0, there exists δ > 0 such that if 0 < |h| < δ < 1 then
|Ea (hei )| <
ci 2
.
Thus ci
ci
2
2
− |h|λ < |h|λ Ea (hei ) <
|h|λ
f (a + hei ) − f (a) > ci |h|λ −
ci 2
|h|λ > 0
and the curve f (a + hei ) has a local minimum at h = 0. In the same way it can be proved that f (a + hej ) has a local maximum. As a consequence, a is a saddle point.
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Fig. 3. Graph of the function f (x) = 1 − x4/3 +
√
1 − x4 along with its Müntz approximation at x = 0 (upper curve).
Fig. 4. Graph of the mapping f (x, y) = x2/3 + y2/3 − 10|x|y + x3 − y4 displaying a local minimum at the origin.
Example 12. Let f be defined as f (x, y) = 1 − x4/3 − y4/3 + 1 − x4 − y4 . The classical criterion of the hessian matrix is not applicable at (0, 0) because it does not exist. However, the former result is conclusive:
p
√ cx (0, 0) = −1 + lim
h→0+
1 − h4 − 1 h4/3
= −1.
Thus, cx (0, 0) = c−x (0, 0) = cy (0, 0) = c−y (0, 0) = −1
λx (0, 0) = λ−x (0, 0) = λy (0, 0) = λ−y (0, 0) = 4/3. These data correspond to the second item. One can check easily that f has variability of Müntz type at (0, 0) and f has local maximum at (0, 0). The Müntz approximation is M(0,0) (u1 , u2 ) = f (0, 0) + P(0,0) (u1 , u2 ) = 2 − |u1 |4/3 − |u2 |4/3 . (See Fig. 3 for the one-dimensional case.) Conditions on the coefficients of higher order are not needed. Using the criteria exposed in Theorem 4.5 one can deduce the existence of local extremes in the following cases. Example 13. The map f (x, y) = x2/3 +y2/3 −10|x|y+x3 −y4 has a local minimum at (0, 0). In this case cx (0, 0) = cy (0, 0) = 1 and λ = 2/3, corresponding to the first item (see Fig. 4). Example 14. The map f (x, y) = x2/3 − y2/3 − 10|x|y + x3 − y4 has a saddle at (0, 0). Here cx (0, 0) = −cy (0, 0) = 1 and λ = 2/3, corresponding to the third item (see Fig. 5). Let us consider now f : C ⊆ Rn → R where C is open and convex. Let us remind that f is concave if f (λx + (1 − λ)y) ≥ λf (x) + (1 − λ)f (y) for all x, y ∈ C and 0 < λ < 1. f is convex if −f is concave. The next result illustrates the position of the Müntz approximation with respect to the graph of f in several cases.
M.A. Navascués / Nonlinear Analysis 73 (2010) 1569–1584
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Fig. 5. Surface f (x, y) = x2/3 − y2/3 − 10|x|y + x3 − y4 with a saddle at the origin.
Theorem 4.6. If f admits a variability of Müntz type for all a ∈ C , f is concave and λ±ek (a) = λ(a) ≥ 1, for all k = 1, 2, . . . , n then for any a, x ∈ C , f (x) ≤ f (a) + Pa (x − a).
(12)
If f is convex and λ±ek (a) = λ(a) ≤ 1, for all k = 1, 2, . . . , n then for any a, x ∈ C , f (x) ≥ f (a) + Pa (x − a). If f is strictly concave (resp. convex) the former inequalities are strict for x 6= a. Proof. Let us consider f concave, λ±ek (a) = λ(a) ≥ 1, and a, x ∈ C , v = x − a and 0 ≤ h < 1, then f (a + hv) = f (h(a + v) + (1 − h)a) ≥ hf (a + v) + (1 − h)f (a) and f (a + hv) − f (a) ≥ h(f (a + v) − f (a)) ≥ hλ(a) (f (a + v) − f (a)). If the quantity Pa (hv) = h 0 = lim
λ(a)
Pa (v) is subtracted from the former inequality, dividing by h
f (a + hv) − f (a) − Pa (hv) hλ(a)
h→0+
(13) λ(a)
and taking limits,
≥ f (a + v) − f (a) − Pa (v)
and f (a + v) ≤ f (a) + Pa (v).
(14)
The second inequality is proved in the same way. If f is strictly concave, in particular is concave. For all a, x ∈ C such that x 6= a, let us consider v = x − a. Let h be such that 0 < h < 1. By (12) f (a + hv) ≤ f (a) + Pa (hv).
(15)
Using (13) with strict inequality, f (a + hv) − f (a) > hλ(a) (f (a + v) − f (a)) and as Pa (hv) = h h
λ(a)
λ(a)
(16)
Pa (v), by (15) and (16),
Pa (v) ≥ f (a + hv) − f (a) > hλ(a) (f (a + v) − f (a)).
Consequently Pa (v) > f (a + v) − f (a) and the inequality of the statement of the theorem is strict.
Corollary 4.7. With the hypotheses of the previous theorem,
• If f is concave and cek (a), c−ek (a) ≤ 0 for all k then a is a global maximum. • If f is convex and cek (a), c−ek (a) ≥ 0 then a is a global minimum.
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Table 1 Absolute and relative errors (in modulus) corresponding to Example 15 provided by the trapeze rule and the variability formula for three different lengths of the interval. h
Trap. abs.
Trap. rel.
Var. abs.
Var. rel.
10−2 10−3 10−4
0.00746 0.00057 0.00004
3.40474 1.34522 0.64399
0.00542 0.00022 0.000008
2.47423 0.51141 0.13637
Proof. If f is concave and cek (a), c−ek (a) ≤ 0 for all k then Pa (u) ≤ 0, for all u ∈ Rn and f (x) ≤ f (a) + Pa (x − a) ≤ f (a) for all x ∈ C .
Note 8. With strict concavity (resp. convexity) the extrema are strict. Note 9. According to the previous Theorem, if f is concave and λ(a) ≥ 1, the graph of f remains below the Müntz approximation. The map of Example 12 is concave and for a = (0, 0), cx = cy = c−x = c−y = −1, λx = λy = λ−x = λ−y = 4/3. The hypotheses of the Corollary are met and f has at (0, 0) a global maximum. Fig. 3 illustrates the one-dimensional case of this Example. The upper curve is the Müntz approximation and the lower one represents the graph of the original mapping for one coordinate. 5. Applications to numerical integration near singularities 5.1. Quadratures in small intervals adjacent to singular points The one-dimensional formulae of variability may efficiently solve the problem of computing the integral t0 + h
Z Ih =
f (t )dt
(17)
t0
where h ' 0 and f has a singularity in the variability at t0 , with exponent λ0 = λ(t0 ) > 0 and right coefficient c0 = c (t0 ) [1]. Using the Müntz polynomial approximating f in a right neighborhood of t0 f (t0 + δ) ' f (t0 ) + c0 δ λ0
(18)
and denoting y0 = f (t0 ), the integral is approached by t0 + h
Z Ih =
f (t )dt ' y0 h +
t0
c0 hλ0 +1
λ0 + 1
.
(19)
Example 15. Let f be the function defined by the expression f (t ) = (1 − t 2/9 )9/2 f has a singularity at t = 0 with right exponent λ0 = 2/9 and coefficient c0 = −9/2 (see [1], Example 6). If one wishes to approximate numerically the integral h
Z
f (t )dt
Ih =
(20)
0
one can apply the formula (19), which is expressed in this case as Ih ' h −
81 22
h11/9
(21)
for small h > 0. The trapeze rule is Ih '
h 2
1 + (1 − h2/9 )9/2 .
(22)
Table 1 joins the relative errors (in modulus) corresponding to the trapeze rule and the variability formula for steps h = 0.01, 0.001, 0.0001. The best results correspond to the latter. The error of the trapeze is near five times that of the variability for h = 10−4 .
M.A. Navascués / Nonlinear Analysis 73 (2010) 1569–1584
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5.2. Numerical solution of ordinary differential equations with singular initial values Let us consider now the problem of finding a function x(t ) such that x˙ (t ) = f (x(t ))
(23)
for all t ∈ (t0 , b] and x(t0 ) = x0 , and let us assume that f goes to infinity at x0 . This kind of problems cannot be solved by the usual one-step methods due to the lack of a real value for f at x0 . This question faces the construction of trajectories of dynamical √ systems with an infinite derivative at some point of the path (let us think that a mapping as conventional as x(t ) = t cannot be a solution of a differential equation of type (23) on the interval I = [0, 1]). One even could look for a curve with a fixed nonlinear slope at one extreme of the trajectory, in the sense of the Ref. [1], and satisfying the differential equation in the rest of the domain. If f admits Müntz variability at x0 and f (x) > 0 close to this point, it can be locally approached as [1] f (x) ' cf (x0 )(x − x0 )λ ,
(24)
where cf (x0 ) ∈ R and λ = λf (x0 ) < 0. The solution x will have an expansion (in a neighborhood (t0 , t0 + δ)) as x(t ) ' x(t0 ) + cx (t0 )(t − t0 )β ,
(25)
where β = βx (t0 ) > 0 and cx (t0 ) ∈ R are the variability exponent and coefficient of x at t0 respectively. In this case, x˙ (t ) ' cx (t0 )β(t − t0 )β−1 .
(26)
Bearing in mind (23)–(26) and the unicity of coefficients and exponents, cx (t0 )β(t − t0 )β−1 = cf (x0 )cx (t0 )λ (t − t0 )βλ
(27)
and thus
β = 1/(1 − λ)
(28)
cx (t0 ) = ((1 − λ)cf (x0 ))
1/(1−λ)
.
(29)
These equalities relate the coefficients and exponents of f and x at x0 and t0 . If the problem admits a solution, the former expressions provide an approximation of the field f in terms of the time variable (26), (28), (29): f (x(t )) ' (cf (x0 ))1/(1−λ) (1 − λ)λ/(1−λ) (t − t0 )λ/(1−λ)
(30)
and this approach gives a way of obtaining a numerical value for the solution x at the first step t1 = t0 + h, t0 + h
Z x1 − x0 =
f (x(t ))dt ' (cf (x0 ))1/(1−λ) (1 − λ)λ/(1−λ)
t0
Z
t0 + h
(t − t0 )λ/(1−λ) dt ,
(31)
t0
and the value x1 = x(t1 ) can be computed as: x1 = x0 + (1 − λ)cf (x0 )h
1/(1−λ)
.
(32)
The rest of the discrete trajectory can be obtained using a standard method of numerical resolution of o.d.e. This procedure can be easily generalized to non-autonomous evolution equations. Example 16. Let us consider the differential equation x˙ = 2x−2 + x,
(33)
for t ∈ (0, 1]; x(0) = 0. The exponents and coefficients are (33), (28), (29):
λ = −2,
cf (0) = 2,
β = 1/3,
cx (0) = 61/3 ,
and the first step reads as x1 = x0 + (1 − λ)cf (x0 )h
1/(1−λ)
= x0 + (6h)1/3 .
For the remaining t values, Heun’s method has been used: xk+1 = xk +
h 2
(f (xk ) + f (xk + hf (xk )))
where k ≥ 1. The exact solution is x(t ) = (2(e3t − 1))1/3 . Table 2 joins the exact values of x(tk ) along with the approximates for a step of h = 0.1.
(34)
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Table 2 Values of the exact and approximate paths, for a step h = 0.1. tk
Exact solution
Approximate solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.88778 1.18029 1.42918 1.66794 1.90959 2.16154 2.42906 2.71651 3.02787 3.36701
0.84343 1.15505 1.40834 1.64840 1.89006 2.14124 2.40744 2.69310 3.00222 3.33871
If one wishes the construction of a trajectory with a fractional initial value of type dx
(dt )β
(t0 ) = c0 ,
x(t0 ) = x0 ,
(35)
that is to say, a curve with a prescribed nonlinear slope, the coefficients and exponents of x and f at t0 and x0 , respectively, need to satisfy relations of compatibility of type (28) and (29). Only in this case a solution will be possible. Note. The derivative in (35) does not agree with the fractional derivative of Riemann–Liouville (see for instance [9]). However, if x is α -differentiable (in the sense of fractional derivative), 0 < α < 1 and x admits a right exponent α at t0 with coefficient cx (t0 ) then ([9], Corollary 3.2) x(α) (t0 ) = Γ (1 + α)cx (t0 ). In this case, the variability elements may be good for all the current applications of fractional derivatives. The former relation enables the computation of coefficients by means of known formulae of fractional differentiation as well. References [1] M.A. Navascués, Local variability of non-smooth functions, Nonlinear Anal. TMA 70 (7) (2009) 2506–2518. [2] H. Müntz, Über den Approximationssatz von Weierstrass, H. A. Schwarz-Festschrift, Berlin, 1914, 303–312. [3] S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné, Mem. Cl. Sci. Acad. Roy. Belg. 4 (1912) 1–103. [4] S.N. Bernstein, Sur les recherches récentes relatives à la meilleure approximation des fonctions continues par des polynômes, in: E.W. Hobson, A.E.H. Love (Eds), Proc. Fifth. Int. Congress of Math., I, 1913, pp. 256–266. [5] E.L. Ortiz, A. Pinkus, Herman Müntz: A Mathematician’s Odyssey, Math. Int. 27 (2005) 22–30. [6] P. Borwein, T. Erdélyi, The full Müntz theorem in C [0, 1] and L1 [0, 1], J. Lond. Math. Soc. 54 (1996) 102–110. [7] P. Borwein, T. Erdélyi, J. Zhang, Müntz systems and orthogonal Müntz–Legendre polynomials, Trans. Amer. Math. Soc. 342 (2) (1994) 523–542. [8] D.S. Lubinsky, E.B. Saff, Zero distribution of Müntz extremal polynomials in Lp[0, 1], Proc. Amer. Math. Soc. 135 (2007) 427–435. [9] G. Jumarie, Generalized Fokker–Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α , Cent. Eur. J. Phys. 6 (3) (2008) 737–753.