On Chebyshev-type integral quasi-interpolation operators

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Mathematics and Computers in Simulation 79 (2009) 3478–3491

On Chebyshev-type integral quasi-interpolation operators夽 M.A. Fortes a , M.J. Ibá˜nez b,∗ , M.L. Rodríguez b a

Departamento de Matemática Aplicada, Universidad de Granada, E.T.S. de Ingenieros de Caminos, Canales y Puertos, Campus de Fuentenueva s/n, 18071-Granada, Spain b Departamento de Matemática Aplicada, Universidad de Granada, Facultad de Ciencias, Campus de Fuentenueva s/n, 18071-Granada, Spain Received 31 January 2008; received in revised form 19 February 2009; accepted 9 April 2009 Available online 23 April 2009

Abstract Spline quasi-interpolants on the real line are approximating splines to given functions with optimal approximation orders. They are called integral quasi-interpolants if the coefficients in the spline series are linear combinations of weighted mean values of the function to be approximated. This paper is devoted to the construction of new integral quasi-interpolants with compactly supported piecewise polynomial weights. The basic idea consists of minimizing an expression appearing in an estimate for the quasi-interpolation error. It depends on how well the quasi-interpolation operator approximates the first non-reproduced monomial. Explicit solutions as well as some numerical tests in the B-spline case are given. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. PACS: 41A05; 41A15; 65D05; 65D07 Keywords: B-splines; Integral quasi-interpolants; Quasi-interpolation error; Chebyshev-type integral quasi-interpolants

1. Introduction  Let φ be a nonnegative univariate piecewise polynomial function with a compact support and normalized by i ∈ Z φ(· − i) = 1. Let S:=span(φ(· − i))i ∈ Z be the cardinal spline space spanned by the shifts of φ. As quoted in [6, chapter III], a quasi-interpolation operator Q for S is a linear map into S which is local, bounded (in a relevant norm) and reproduces some (nontrivial) polynomial space. The standard structure for a quasi-interpolant Q(f ):=Qf for f is given by the expression  (1) Qf := λf (· + i)φ(· − i), i∈Z

λ being a linear functional. Usually, λf :=λ(f ) is a linear combination of values of f and some of its derivatives at some points in some neighborhood of the support of φ, or a linear combination of values of f at some points in this neighborhood. The resulting operators are called differential or discrete quasi-interpolants, respectively. 夽

Research supported in part by Ministerio de Ciencia e Innovación (Research project MTM2008-00671) and Junta de Andalucía (Research group FQM/191). ∗ Corresponding author. E-mail addresses: [email protected] (M.A. Fortes), [email protected] (M.J. Ibá˜nez), [email protected] (M.L. Rodríguez). 0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.04.014

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The linear functional involved in the definition of Q is constructed to reproduce the polynomials in S and so the scaled operator Qh :=σh Qσ1/ h , with σh f :=f (·/ h) achieves the optimal approximation order of S. If Pd is the space of polynomials reproduced by Q, and Sh :=σh (S), then that property means that dist(f, Sh ) = O(hd+1 ) for all sufficiently smooth f. In this paper, we are interested in integral quasi-interpolants (iQIs). The associated linear functional λ acting on f is a linear combination of weighted mean values of f. More precisely,  λf := γj f, ψ(· + j), (2) j∈J

where J is a finite subset of Z, γ:=(γj )j ∈ J is a vector to be determined, ψ is another nonnegative univariate piecewise  polynomial function with a compact support and normalized by i ∈ Z ψ(· − i) = 1, and ·, · stands for the usual inner product. This kind of quasi-interpolants has been used for nonparametric density estimation on the real line (see [5]) and in connection with the construction of histopolating splines (cf. [8,18] and references therein). We also refer to [10,14–16]. In [12], a method is proposed to define discrete quasi-interpolants based on B-splines yielding small constants in an appropriate estimate of the quasi-interpolation error. That estimate involves the error for the first non-reproduced monomial, depending on the vector γ that defines the discrete quasi-interpolant, and it is quite natural to miminize it under the constraints yielding the exactness of the discrete quasi-interpolant. In this paper, we extend the idea to the integral case, but in such a way that the solutions are characterized in terms of some splines which do not depend on λ. Therefore, this technique can be also used to solve the problem addressed in [12,11]. On the other hand, the function φ is not restricted to be a classical B-spline. The article is organized as follows. In Section 2, we characterize the exactness of an iQI. Section 3 is devoted to derive an appropriate expression for the quasi-interpolation error. In Section 4, we formulate the problem and we prove that it always has a solution related to the best uniform approximation by constants of a spline that can be expressed in terms of the Schoenberg operator. In Section 5, we present explicit results for cubic and quintic B-splines φ using a general B-spline ψ as weight function, and some general comments concerning the problem when an even degree B-spline is used as φ. Finally, in Section 6, we give some numerical and graphical results. 2. Exactness conditions In this section we obtain conditions on γ equivalent to the exactness on Pd of the integral quasi-interpolant Q given by (1) and (2). Define m (x):=x /!,  ≥ 0. Let (g )≥0 be the sequence defined by recurrence as follows: g0 :=1,

g :=m −



φ(j)

j∈Z

−1 

m−k (−j)gk ,

 ≥ 1.

k=0

Lemma 1. The iQI Q defined by (1) and (2)is exact on Pd if and only if the linear form λ given by (2) coincides on Pd with  d : f → g (0)f () (0). (3) ≥0

Remark 2. Note that (g )≥0 is the Appell sequence for the linear functional μ defined by μf := () μgk



j φ(−j)f (j),

i.e.

= δk,l , where δ stands for the Kronecker’s symbol (cf. [4, p. 117], [5, p. 19]).

Next result follows easily from Lemma 1. Proposition 3. The iQI Q given by (1) and (2)is exact on Pd if and only if λm = g (0),

0 ≤  ≤ d.

(4)

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For any solution γ to this system, the associated iQI Q is exact on Pd and the same is true for the scaled iQI Qh , h > 0. Moreover, for all interval I, we have

f − Qh f L∞ (I) = O(hn ) for a function f of class Cn (Iε ) in a open set Iε containing I, and so Qh achieves the optimal approximation order of S. Note that Qh f can be expressed as follows: ⎛ ⎞ ·   ⎝ γj f (h·), ψ(· − i + j)⎠ φ −i . (5) Qh f = h j∈J

i∈Z

3. Quasi-interpolation error The main goal of the construction we propose in this paper is to define integral quasi-interpolants exact on Pd in such a way that they yield small constants in the error estimates for regular enough functions. It extends to the integral case the construction proposed in [12] for discrete QIs (see also [11]). Therefore, we start from an appropriate representation for the quasi-interpolation error Eh f :=f − Qh f when f ∈ Cd+2 (R). Note that usually the condition f ∈ Cd+1 (R) is required to obtain standard error estimates for Eh f when the operator is exact on Pd . Proposition 4. Let f ∈ Cd+2 (R) and Ih,k :=[kh, (k + 1)h] for some k ∈ Z. If Q is an iQI given by (1) and (2)which is exact on Pd , then there exist both a constant C > 0 independent of f and h, and a neighborhood V = Vh,k independent of f such that

Eh f ∞,Ih,k ≤ hd+1 md+1 − Qmd+1 ∞,[0,1] f (d+1) ∞,V +

C hd+2 f (d+2) ∞,V . (d + 2)!

(6)

Proof. For each i ∈ Z, we have

 cj f (· + i), ψ(· + j) = f (t)H(t − i) dt, λf (· + i) = j∈J

where H(t):=



R

cj ψ(t + j).

j∈J

Hence, Qf =

R

with K(t, x):=

f (t)K(t, ·) dt,



H(t − i)φ(· − i).

i∈Z

Therefore, for the scaled quasi-interpolation operator Qh we get



 · 1 t · dt = , dt. f (ht)K t, f (t) K Qh f = h h h h R R In summary, Qh can be expressed in integral form from a shift-invariant kernel having sufficient decay and reproducing Pd , namely (1/ h)K((t/ h), (·/ h)). Thus, by Theorem 1.1 in [7], we conclude that there exist both a neighborhood V = Vh,k independent of f and a constant C > 0 independent of f and h such that

Eh f ∞,Ih,k ≤ hd+1 ed,φ,ψ ∞,[0,1] f (d+1) ∞,V +

C hd+2 f (d+2) ∞,V , (d + 2)!

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where ed,φ,ψ (x):=Q(md+1 (· − x))(x) =

d+1 

Q(m )(x)md+1− (−x).

=0

Then, by the exactness of Q on Pd , we get ed,φ,ψ (x) =

d 

m (x)md+1− (−x) + Q(md+1 )(x) =

=0

d+1 

m (x)md+1− (−x) − md+1 (x) + Q(md+1 )(x)

=0

= −(md+1 (x) − Q(md+1 )(x)) 

and the claim follows.

The constant md+1 − Qmd+1 ∞,[0,1] in the leading part of (6) can be expressed in an appropriate form using the sequence (g )≥0 . Lemma 5. Let Q be an iQI given by (1) and (2)which is exact on Pd . Then md+1 − Qmd+1 = Gd+1 − (λmd+1 − gd+1 (0)), where Gd+1 :=md+1 −



gd+1 (i)φ(· − i).

(7)

i∈Z

Proof. Taking into account that md+1 (· + i) =

d+1 

m (i)md+1− ,

=0

we can write λmd+1 (· + i) =

d+1 

m (i)λmd+1− .

=0

By (4), we get λmd+1 (· + i) =

d 

gd+1− (0)m (i) + λmd+1 =

=0

d+1 

gd+1− (0)m (i) + λmd+1 − gd+1 (0).

=0

Therefore (cf. [4, p. 120]), λmd+1 (· + i) = gd+1 (i) + λmd+1 − gd+1 (0), and finally we have  Qmd+1 = gd+1 (i)φ(· − i) + λmd+1 − gd+1 (0), i∈Z

and the proof is complete.



We recall that the Schoenberg operator S:=Sφ associated with the B-spline φ is defined by  Sf := f (i)φ(· − i), i∈Z

and so Gd+1 = md+1 − Sgd+1 . Note that Gd+1 does not depend on the weight function ψ.

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4. The new integral quasi-interpolants Now we are in position to formulate the minimization problem. Problem 6. Find γ ∈ R#J to minimize Qmd+1 − md+1 ∞,[0,1] subject to conditions (4). By Lemma 5, γ is a solution of Problem 6 if and only if λmd+1 − gn (0) is the best constant uniform approximation to Gd+1 on [0, 1]. Therefore, we have the following result (cf. [19]). Proposition 7. γ is a solution of Problem 6 if and only if 1 λmd+1 = gd+1 (0) + (max[0,1] Gd+1 + minGd+1 ). [0,1] 2

(8)

If γ fulfils (4) and (8)  then the corresponding iQI is said to be a Chebyshev iQI relative to J. Let μk :=μk (ψ):= R xk ψ(x) dx be the moment of order k of ψ. After some computations, for each 0 ≤  ≤ d + 1 we obtain  1  Γk μ−k (m), λm = ! k k=0  with Γ := j ∈ J γj (−j) , and so Eqs. (4) and (8) give a linear system AΓ = b, with A the lower triangular matrix ⎞ ⎛ μ0 ⎟ ⎜ μ0 ⎟ ⎜ μ1 ⎟ ⎜ ⎟ ⎜ μ2 2μ μ 1 0 ⎟ ⎜ ⎟, ⎜ . .. .. .. ⎟ ⎜ . . . . ⎟ ⎜ . ⎟ ⎜

⎟ ⎜ d+1 ⎠ ⎝ μd+1 (d + 1)μd μd−1 · · · μ0 2 T 1 b = (g0 (0), g1 (0), . . . , d!gd (0), (d + 1)!(gd+1 (0) + (max[0,1] Gd+1 + minGd+1 ))) , [0,1] 2

and Γ :=(Γ0 , . . . , Γd+1 )T .

 Remark 8. Note that all diagonal entries in A are equal to 1 due to the normalization condition R ψ = 1 imposed to the compactly supported function ψ. Let c:=(c0 , . . . , cd+1 )T = A−1 b. The conditions characterizing the solutions of Problem 6 can be written as Γ = c ,

 = 0, . . . , d + 1,

that is to say, we have a linear system on γ. If #J = d + 2 and the points in the subset J are pairwise distinct, then this Vandermonde system has a unique solution γ˜ giving the coefficients of the linear form that defines the Chebyshev iQI ˜ relative to J. Q The iQIs associated with the solutions to Problem 6 are optimal for functions of class Cd+1 (R) w.r.t. the estimate we have considered because the smallest constant in the leading term of (6) is achieved. In practice, we need an estimate for f in Cd (R) instead of Cd+1 (R). Next result provides a useful result under the hypothesis above and the exactness of the operator on Pd . Proposition 9. Let f ∈ Cd+1 (R) and Ih,k and Q as in Proposition 4. Then there exist both a neighborhood W = Wh,k,φ,ψ i ndependent of f and a constant C > 0 independent of f and h such that

Eh f ∞,Ih,k ≤

hd+1 maxΨ f (d+1) W , (d + 1)! [0,1]

M.A. Fortes et al. / Mathematics and Computers in Simulation 79 (2009) 3478–3491

where Ψ (ξ):=



3483

| · −ξ + i|d+1 , ψ|Φ(ξ − i)|, ξ ∈ [0, 1],

i ∈ F0

with F0 :={i ∈ Z : suppΦ(· − i) ∩ (0, 1) = / ∅}. Proof. By (5), we can write ⎛ ⎞ · · 1   ·   ⎝ Qh f = f, ψ γj f (h·), ψ(· − i + j)⎠ φ −i = −i Φ −i , h h h h i∈Z

where Φ:=



j∈J

i∈Z

γj φ(· − j)

(9)

j∈J

is the fundamental functional associated with Q. Using the Taylor expansion f (z) =

d 

f () (x)m (z − x) +

=0

1 f (d+1) (θx,z )md+1 (z − x), (d + 1)!

with θx,z a real number between z and x, we get Qh f (x) = P(x) + R(x), where

d

·  x 1   ()  P(x):= f (x) m (· − x), ψ −i Φ −i h h h i∈Z

and R(x):=

=0

·   x 1   (d+1) f (θx,· )md+1 (· − x), ψ −i Φ −i . h h h i∈Z

It is an easy matter to derive the equality P(x) =

d 

f () (x)Qh [m (· − x)](x),

=0

and therefore the exactness of Q on Pd implies that P(x) =

d 

f () (x)m (0) = f (x).

=0

To estimate R(x), write x = h(k + ξ) for some ξ ∈ [0, 1]. Define  ·  F := i ∈ Z : suppΦ − i ∩ (kh, (k + 1)h) = / ∅ . h It is a finite subset due to the compactness of the support of ψ. Moreover, F = F0 − :={i −  : i ∈ F0 }. Taking into account that

 · z   (d+1)  (θx,· )(· − x)d+1 , ψ |z − x|d+1 ψ − i  ≤ f (d+1) h(i+suppψ) − i dz,  f h h R

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after variable changes in the involved integral and summation, we get |R(x)| ≤

hd+1  (d+1)

f

h(i+suppψ) | · −ξ + i|d+1 , ψ|Φ(ξ − i)|. (d + 1)! i ∈ F0

Finally, |R(x)| ≤

 hd+1

f (d+1) W,h,φ,ψ | · −ξ + i|d+1 , ψ|Φ(ξ − i)|, (d + 1)! i ∈ F0

with W,h,φ,ψ :=



(h(i + suppψ)),

i ∈ F0

and the proof is complete.



5. Integral quasi-interpolants based on B-splines In this section, we consider integral quasi-interpolants based on the B-spline Mn centered at the origin, i.e. φ = Mn , n ≥ 1. To compute the polynomials g ,  ≥ 0 in the Appell sequence we need the values of Mn at the integers. They can be computed from the recurrence relation satisfied by some polynomials defined from that values (see e.g. [9, pp. 410–411], [17, p. 22 and p. 25]). The moments μk,m :=μk (Mm ) of the B-spline Mm , m ≥ 1 can be determined by recurrence using the central factorial numbers of second kind T (p, q) ([3, p. 423]). More explicitly ([3, p. 446, Eq. (4.2.8)]), we have T (k + m, m)

. μk,m = k+m m ˆ m of Mm is given by However, it is well known ([17, p. 12]) that the Fourier transform M  m ˆ m (u) = 2 sin u , M u 2 ˆ m in Taylor series and taking into account that and the moments of Mm result by expanding M ˆ m (u) = M

∞  =0

(−1) μ2,m

u2 . (2)!

In particular, the following equalities hold for the even order moments of the B-spline Mm : μ0,m = 1, m μ2,m = , 12 1 μ4,m = m(5m − 2), 240 1 μ6,m = m(35m2 − 42m + 16). 4032 The symmetry of Mn results in the following recurrence relation for computing the polynomials g . ⎞ ⎛ n−1  2  1 ⎟ ⎜  g0 = 1, g = m − 2 Mn (j)j −k ⎠ gk ,  ≥ 1. ⎝ ( − k)! 0≤k≤−1−keven

j=1

(10)

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Finally, the function Ψ in Proposition 9 can be expressed in a more explicit form for an even d = 2δ and ψ = Mm by expanding | · −ξ + i|d , Mm  in terms of the moments of Mm . More precisely, δ

  2δ 2 (i − ξ) μ2δ−2,m |Φ(ξ − i)|, ξ ∈ [0, 1]. (11) Ψ (ξ) = 2 i∈F =0 0

5.1. Odd degree integral quasi-interpolants In this section we detail the cubic and quintic cases, with the B-spline Mm as weight function. 5.1.1. The cubic case Let φ be the cubic B-spline M4 , i.e. n = 4, and ψ = Mm , m ≥ 1. By (10), after some computations we get 1 1 1 1 g2 = m2 − , g3 = m3 − m1 , and g4 = m4 − m2 + . 6 6 6 72 The spline function G4 = m4 − S4 g4 on [0, 1] in (7) is the polynomial (1/6)m2 (· − 1)m2 , whence   1 1 max[0,1] G4 = G4 = and minG4 = G4 (0) = 0. [0,1] 2 384 g0 = 1,

g1 = m1 ,

It follows that g4 (0) +

1 2



 max[0,1] G4 + minG4 [0,1]

=

35 . 2304

Therefore, γ is a solution to Problem 6 if and only if ⎞ ⎛ 1 ⎛ ⎞⎛ ⎞ ⎟ ⎜ Γ0 1 ⎜ 0⎟ ⎟ ⎜ ⎟⎜ ⎟ ⎜ 1 ⎜ 0 ⎟ ⎜ Γ1 ⎟ ⎜ 1 ⎟ ⎜ ⎟⎜ ⎟ ⎜− ⎟ ⎜ μ2,m ⎟ ⎜ Γ2 ⎟ = ⎜ 3 ⎟ . 0 1 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ 3μ2,m 0 1 ⎝ 0 ⎠ ⎝ Γ3 ⎠ ⎜ 0 ⎟ ⎟ ⎜ ⎝ 35 ⎠ μ4,m 0 6μ2,m 0 1 Γ4

(12)

96 Equivalently, 1 1 (m + 4), and Γ4 = (10m2 + 84m + 175). 12 480 Recall that the latter is a linear system on γ. When J = {−2, −1, 0, 1, 2}, it has the unique solution γ˜ m :=(γ˜ −2,m , γ˜ −1,m , γ˜ 0,m , γ˜ 1,m , γ˜ 2,m ) given by Γ0 = 1,

Γ1 = Γ3 = 0,

Γ2 = −

1 (2895 + 284m + 10m2 ), 1920 1 γ˜ 1,m = γ˜ −1,m = − (815 + 224m + 10m2 ), 2880 1 γ˜ 2,m = γ˜ −2,m = (335 + 124m + 10m2 ). 11520 By (9), the Chebyshev iQI relative to this subset J is then given by  ˜ 4,m = ˜ 4,m (· − i), Q f, Mm (· − i)L γ˜ 0,m =

i∈Z

˜ 4,m is the even function where the fundamental function L ˜ 4,m = γ˜ 0,m M4 + γ˜ 1,m (M4 (· + 1) + M4 (· − 1)) + γ˜ 2,m (M4 (· + 2) + M4 (· − 2)). L

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Table 1 ∗ and Ψ ˜ 4,m for 1 ≤ m ≤ 4. Maximum values of Ψ4,m m ∗ max[0,1] Ψ4,m

max[0,1] Ψ˜ 4,m

1

2

3

4

1.52 3.04

2.18 4.68

2.98 6.76

3.93 9.35

Table 2 Coefficients of the polynomials g in the basis (m )0≤≤6. 

Coefficients of g

0 1

1 0

1

2

− 41

0

3

0

− 41

0

4

1 30

0

− 41

0

5

0

1 30

0

− 41

0

1

6

1 − 320

0

1 30

0

− 41

0

1 1 1 1

˜ 4,m , we need to compare them with the iQIs Q∗ exact on P3 and having also In order to test the performance of Q 4,m even fundamental functions. The coefficient sequence γ ∗m of Q∗4,m is characterized by the first four equalities in (12) and comes from the choice J = {−2, −1, 0, 1} (or {−1, 0, 1, 2}). Their nonzero components are ∗ γ0,m =

In summary, Q∗4,m =

m+6 , 12 

∗ ∗ = γ−1,m =− and γ1,m

m+4 . 24

f, Mm (· − i)L∗4,m (· − i),

i∈Z

with ∗ ∗ L∗4,m = γ0,m M4 + γ1,m (M4 (· + 1) + M4 (· − 1)). ∗ and Ψ ˜ 4,m , respectively, ˜ 4,m associated with Q∗ and Q Table 1 shows the maximum values of the Ψ -functions Ψ4,m 4,m for a few values of m. A major drawback of the method used to derive the error estimate in Proposition 9, based on Taylor expansions, is that it strongly depends on the size of the support of the fundamental function of the iQI. ˜ 4,m are pessimistic when compared with the ones for Q∗ . Note Therefore, the estimates obtained for the iQIs Q 4,m ˜ that, for every m the estimate corresponding to Q4,m is about twice the one of Q∗4,m . A more precise, but complicated method consists of analyzing the quasi-interpolation error by means of the Peano’s kernel theorem (see e.g. [13]).

Remark 10. Taking into account that ∗ ∗ |L∗4,m | ≤ γ0,m M4 − γ1,m (M4 (· + 1) + M4 (· − 1)) =: Λ∗4,m , ∗ from the B-coefficients of the polynomial Λ∗ , that it is an easy matter to determine an upper bound for max[0,1] Ψ4,m 4,m ∗ is to say, from the coordinates of Λ4,m in the Bernstein basis. An analogous result holds for max[0,1] Ψ˜ 4,m .

5.1.2. Quintic quasi-interpolants In the quintic case (n = 6) we shortly present the results. The construction proposed requires the polynomials g in the Appell sequence. Their coordinates in the basis (m )0≤≤6 are given in Table 2. It can be easily proved that G6 = m6 − Sg6 =

1 m2 (· − 1)m2 (−m0 − 2m1 + 4m2 ), 360

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whence max[0,1] G6 = 0

and

minG6 = − [0,1]

1 , 15, 360

and so the solutions to Problem 6 are characterized by the linear system ⎛

1 ⎜ ⎜ 0 ⎜ ⎜ μ2 (m) ⎜ ⎜ ⎜ 0 ⎜ ⎜ μ4 (m) ⎜ ⎜ ⎝ 0 μ6 (m)

⎞⎛

1 0 3μ2 (m)

0

1

0

6μ2 (m)

0

5μ4 (m)

0 15μ4 (m)

0

1 1

10μ2 (m) 0 0 15μ2 (m)

1 0

⎞

⎛

⎜ Γ0 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ Γ1 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ Γ2 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ Γ3 ⎟ = ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ Γ4 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎠ ⎝ Γ5 ⎠ ⎜ ⎜ ⎝ 1 Γ6

−

⎞ 1 0⎟ ⎟ ⎟ 1⎟ − ⎟ 2⎟ ⎟ 0⎟. ⎟ 4⎟ ⎟ 5⎟ ⎟ 0⎟ ⎟ 291 ⎠

(13)

128

Therefore, γ is a solution to Problem 6 if and only if Γ0 = 1, Γ6 = −

Γ2 = −

1 (m + 6), 12

Γ4 =

1 (5m2 + 62m + 192), 240

1 (70m3 + 1344m2 + 8600m + 18,333), 8064

Γ1 = Γ3 = Γ5 = 0.

(14)

˜ 6,m having an even fundamental function with minimal support comes from J = The quintic Chebyshev-type iQI Q {−3, −2, −1, 0, 1, 2, 3}. The coefficients of its linear functional (depending on m) are provided by the unique solution γ˜ m of the linear system (14). Explicitly, they are 1 (2, 982, 609 + 353, 464m + 18, 480m2 + 350m3 ), 1, 451, 520 1 (1, 236, 753 + 299, 368m + 17, 640m2 + 350m3 ), = γ˜ −1,m = − 1, 935, 360 1 (119, 133 + 35, 480m + 3024m2 + 70m3 ), = γ˜ −2,m = 967, 680 1 = γ˜ −3,m = − (66, 717 + 21, 704m + 2184m2 + 70m3 ). 5, 806, 080

γ˜ 0,m = γ˜ 1,m γ˜ 2,m γ˜ 3,m

The first six equations in (13) characterize the exactness on P5 of the operator Q given by (1) and (2). The choices J = {−3, −2, −1, 0, 1, 2} or {−2, −1, 0, 1, 2, 3} provide such iQIs Q∗6,m with even fundamental functions supported on [−5, 5]. Their sequences γ ∗m have the following nonzero coefficients: 1 (1752 + 162m + 5m2 ), 960 1 ∗ (672 + 142m + 5m2 ), = γ−1,m =− 1440 1 ∗ = γ−2,m = (312 + 82m + 5m2 ). 5760

∗ = γ0,m ∗ γ1,m ∗ γ2,m

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Table 3 ∗ and Ψ ˜ 6,m for 1 ≤ m ≤ 6. Maximum values of Ψ6,m m

1

2

3

4

5

6

∗ max[0,1] Ψ6,m

34.43

49.92

69.66

94.28

124.46

160.95

max[0,1] Ψ˜ 6,m

67.53

98.43

138.46

189.97

256.21

339.13

∗ and Ψ ˜ 6,m , respectively ˜ 6,m in Proposition 9 for Q∗ and Q As in the cubic case, we can estimate the constants Ψ6,m 6,m ˜ 6,m from the corresponding fundamental functions. They are shown in Table 3. Again, the constants corresponding to Q ∗ are about twice the ones of Q6,m .

5.2. Even degree integral quasi-interpolants We have derived two new odd degree iQIs. In both cases, the additional condition (8) is established from a spline function (8)Gn , n = 4, 6, verifying the property Gn ((1/2) − ξ) = Gn ((1/2) + ξ), ξ ∈ [0, (1/2)], and this is a general property satisfied by Gn when n is even. However, Gn ((1/2) − ξ) = −Gn ((1/2) + ξ), ξ ∈ [0, (1/2)], when n is odd. Therefore, max[0,1] Gn = −minGn [0,1]

and then 1 (max[0,1] Gn + minGn ) = 0. [0,1] 2 On the other hand, it holds gn (0) = 0 when n is odd. Thus, γ is a solution to Problem 6 if and only if its associated iQI is exact on Pn−1 . As a conclusion, in order to obtain new iQIs also in the even degree case, we can minimize the errors mn+r − Qmn+r , 0 ≤ r ≤ s for some s ≥ 1 (see [12] for the discrete case). 6. Numerical tests In this section we illustrate the performance of the cubic iQIs we have constructed. As a first test function, consider the chirp f1 (x) = cos(x − sin(x2 )) ˜ 4,2,h f1 . Note that and their scaled integral quasi-interpolants Q∗4,2,h f1 and Q    3 1 f1 (i) − f1 (i ± 1) M4 (· − i), Q∗4,2 f1 = 2 4 i∈Z    3503 1343 623 ˜ Q4,2 f1 = f1 (i) − f1 (i ± 1) + f1 (i ± 2) M4 (· − i). 1920 2880 11, 520 i∈Z

When |x| tends to infinity the graph of f exhibits oscillations with increasing amplitude, as shown in Fig. 1. A small h is needed to achieve good approximants. ˜ 4,2,h f1 | on the interval [−8, 8] with h = Fig. 2 shows the quasi-interpolation errors |f1 − Q∗4,2,h f1 | and |f1 − Q (1/64). Both errors present a similar behavior, but the corresponding one to the classical quasi-interpolant is about seven times the error obtained from the new iQI. The values of the integrals in the iQIs are computed numerically by using the Gaussian quadrature formula exact on P3 √ √

1 6 6 1 f − +f M2 (x)f (x)dx ≈ 2 6 6 −1

M.A. Fortes et al. / Mathematics and Computers in Simulation 79 (2009) 3478–3491

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Fig. 1. Plot of the chirp f1 .

Fig. 2. Plots of the cubic quasi-interpolation errors for the chirp f1 , corresponding to the integral quasi-interpolation operators Q∗4,2,h (left) and ˜ 4,2,h with h = (1/64). Q

and the required formulas derived from it. Smaller errors are obtained if a more precise Gaussian formula is used. As a second example, we consider the exponentially decaying function given by f2 (x) =

x cos2 (x2 − sin x) . 1 + exp x

˜ 4,2,h f2 on [0, 8]. Now the maximum and Figs. 3 and 4 show the quasi-interpolation errors f2 − Q∗4,2,h f2 and f2 − Q the minimum errors are attained near the midpoint of the interval. Also in this case the errors are smaller for the new iQI.

Fig. 3. Quasi-interpolation errors |Q∗4,2,h f1 − f1 | with steplength h = 1/2n , 0 ≤ n ≤ 5.

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˜ 4,2,h f2 − f2 | with steplength h = 1/2n , 0 ≤ n ≤ 5. Fig. 4. Quasi-interpolation errors |Q

7. Conclusion A quite natural method for constructing integral quasi-interpolants on uniforms partitions of the real line has been proposed. Instead of constructing such operators by minimizing the infinity norm (see [1], and [2] in the bivariate case), we minimize the quasi-interpolation error for the first non-reproduced monomial since that error appears in an estimate for the quasi-interpolation error for a regular enough function. This method does not produce new iQIs (with minimally supported fundamental functions) when an even degree B-spline is used. In fact, in this case an iQI exact on the space of polynomials of the maximal order included in the space spanned by the integer shifts and defined from a subset J symmetrical w.r.t. zero is also a Chebyshev-type iQI. However, when a B-spline Mn of even order is used, new iQIs are obtained. The numerical tests show a good performance of the new iQIs, and the (pessimistic) constants in the estimates for the associated errors suggest that the new iQIs can give better results than the classical ones. Conclusive results can be derived by using the Peano’s kernel theorem. The proposed method can be easily extended to the multivariate case. References [1] D. Barrera, M.J. Ibá˜nez, P. Sablonnière, D. Sbibih, Near minimally normed spline quasi-interpolants on uniform partitions, J. Comput. Appl. Math. 181 (2005) 211–233. [2] D. Barrera, M.J. Ibá˜nez, P. Sablonnière, D. Sbibih, Near-best quasi-interpolants associated with H-splines on a three-direction mesh, J. Comput. Appl. Math. 183 (2005) 133–152. [3] P.L. Butzer, M. Schmidt, E.L. Stark, L. Vogt, Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optimiz. 10 (5&6) (1989) 419–488. [4] C.K. Chui, Multivariate Splines, SIAM, Philadelphia, 1988. [5] Z. Ciesielski, Asymptotic nonparametric spline density estimation, Probab. Math. Stat. 12 (1991) 1–24. [6] C. de Boor, K. Höllig, S. Riemenschneider, Box Splines, Springer-Verlag, New York, 1993. [7] S. Dekel, D. Leviatan, On measuring the efficiency of kernel operators Lp (Rd ), Adv. Comput. Math. 20 (2004) 53–65. [8] E.J. Delhez, A spline interpolation technique that preserves mass budgets, Appl. Math. Lett. 16 (2003) 17–26. [9] R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. [10] T.N.T. Goodman, A. Sharma, A modified Bernstein–Schoenberg operator, in: Construction Theory of Functions’87, Bulgarian Academy of Sciences, Sofia, 1988, pp. 168–173. [11] M.J. Ibá˜nez Pérez, Quasi-interpolantes spline discretos de norma casi mínima. Teoría y aplicaciones, Doctoral Dissertation, University of Granada, 2003. [12] M.J. Ibá˜nez Pérez, On Chebyshev-type discrete quasi-interpolants, Math. Comput. Simul. 77 (2008) 218–227. [13] G.M. Phillips, Interpolation and Approximation by Polynomials, Springer-Verlag, New-York, 2003. [14] P. Sablonnière, Positive spline operators and orthogonal splines, J. Approx. Theory 52 (1988) 28–42.

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P. Sablonnière, D. Sbibih, Spline integral operators exact on polynomials, Approx. Theory Appl. 10 (3) (1994) 56–73. D. Sbibih, Approximations des fonctions d’une ou deux variables par des opérateurs splines intégraux, Thèse, Université de Rennes, 1987. I.J. Schoenberg, Cardinal Spline Interpolation, SIAM, Philadelphia, 1973. R. Siewer, Histopolating splines, J. Comput. Appl. Math. 220 (2008) 261–673. G.A. Watson, Approximation Theory and Numerical Methods, Wiley, Chichester, 1980.