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A converse of asymptotic formulae in simultaneous approximation
Applied Mathematics and Computation 217 (2010) 2676–2683
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A converse of asymptotic formulae in simultaneous approximation q P. Garrancho, D. Cárdenas-Morales * Departamento de Matemáticas, Universidad de Jaén, 23071 Jaén, Spain
a r t i c l e
i n f o
a b s t r a c t In the general setting of simultaneous approximation by sequences of linear shape preserving operators, this paper contains a sort of converse result of Voronovskaya-type asymptotic formulae. As a by-product a saturation result is derived. Applications to some very well-known approximation processes are also presented. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Asymptotic formula Shape preserving approximation Simultaneous approximation Saturation
1. Introduction In 1932 Voronovskaya [19] proved the following famous asymptotic formula for the classical Bernstein polynomials Bn: if f is a real bounded function on [0, 1] and f00 (x) exists, then
lim 2nðBn f ðxÞ f ðxÞÞ ¼ xð1 xÞf 00 ðxÞ:
n!1
Many well-known approximation processes by sequences of linear operators fulfill as well a formula of this type, as the Kantorovich operators, the Schurer–Szász–Mirakyan operators, the Meyer–König and Zeller operators and the Bleimann–Butzer and Hahn operators (for a definition of these sequences see for instance the excellent monograph [3]). Specifically, if Ln is such a sequence, then there exist a sequence of real numbers kn ? +1, and two polynomial functions p, q such that, under certain assumptions, 00
0
lim kn ðLn f f ÞðxÞ ¼ ðpf þ qf ÞðxÞ:
n!1
ð1Þ
On the other hand, in the last few years some results have appeared in the setting of the so-called simultaneous approximation proving that these formulae remain valid if certain linear differential operator, say D, is applied to both sides of (1), i.e. 00
0
lim kn DðLn f f ÞðxÞ ¼ Dðpf þ qf ÞðxÞ
n!1
ð2Þ
(see [1,13,8,2,14,15]). Actually, D is narrowly connected with the shape preserving properties that these operators posses, which at the time represent an essential tool to prove the validity of (2). These properties can be written in short as follows: Df P 0 implies DLn f P 0. Roughly speaking, as we have proceeded till this point, our aim with this work is to prove a converse result of (2). It can be considered as a generalization of a result of the authors [11], and follows the line that began with the pioneer paper of Amel’kovic˘ [4] and found further development in the papers of Mühlbach [17] and Berens [6]. As a by-product we derive
q
This work is partially supported by Junta de Andalucı´a (FQM-0178) and by Ministerio de Ciencia y Tecnologı´a (MTM 2006-14590). * Corresponding author. Tel.: +34 953 21 21 44; fax: +34 953 21 18 71. E-mail address: [email protected] (D. Cárdenas-Morales).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.08.005
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
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a general saturation theorem that extends and reinterprets the ones of [16,9] (see also [10]), and whose application allows to achieve new results for some of the aforementioned particular sequences of operators. Here it is an outline of the paper: in the next section we fix the notation and precise the framework; this includes a definition of the operator D and formal statements of the asymptotic formulae and the shape preserving properties we shall deal with. Some auxiliary results are also proved. Sections 3 and 4 are devoted to the main result and applications, while the last one contains the aforesaid results on saturation. 2. General setting For a real interval I R and a non-negative integer i 2 N0 ¼ N [ f0g, as usual we denote by Ci(I) the space of all real valued i-times continuously differentiable functions defined on I and by Di the classical i-th differential operator (C0(I) = C(I) is the space of all continuous functions on I and D0 ¼ I is the identity operator). C 1 ðIÞ ¼ \i2N C i ðIÞ, C iB ðIÞ denotes the subspace of Ci(I) formed by the bounded functions. A function f 2 Ci(I) is said to be i-convex if Dif P 0 on I and a linear operator is said to be i-convex if it maps i-convex functions onto i-convex functions. Now we shall consider a sequence of linear operators fulfilling a shape preserving property (H1) and an asymptotic formula (H2) (see below). To this end, we start presenting the domain and range of the operators and the differential operator D referred in the previous section. Let us fix an interval I R, a number k 2 N0 and a function w 2 C1(I) such that w(t) – 0 for all t 2 I. Now we take any function u 2 C1(I) such that D1u = w and define the space
C k;w ðIÞ ¼ ff 2 C k ðIÞ : 9c1 ; c2 2 Rþ =8t 2 I; jf ðtÞj < c1 ec2 uðtÞ g: Besides, we consider the m-th iterate of the operator
D0;w ¼ I;
D1;w ¼
1 1 D ; w
Dmþ1;w ¼ D1;w Dm;w ;
1 1 D , w
i.e.
m 2 N:
Finally, let us consider a sequence of linear operators
Ln : C k;w ðIÞ ! C k;w ðIÞ fulfilling the following hypotheses: (H1) if f 2 Ck,w(I) is such that Dk,wf P 0 on I, then for each n 2 N, Dk,wLnf P 0 on I, (H2) there exist a sequence of real numbers kn? +1 and two functions p, q 2 Ck(I), p being strictly positive on Int (I), such that for f 2 Ck,w(I), k + 2-times differentiable at a point x 2 Int(I),
lim kn Dk;w ðLn f f ÞðxÞ ¼ Dk;w ðpD2 f þ qD1 f ÞðxÞ:
n!þ1
ð3Þ
Besides, if k 2 N, we assume in addition that (H3) there exist three strictly positive functions wi 2 C2i(Int(I)), i = 0, 1, 2, such that for f 2 Ck,w(I), whenever it is meaningful,
Dk;w ðpD2 f þ qD1 f Þ ¼
1 1 1 1 1 k;w D D D f w2 w1 w0
(see the right-hand side of (3)). Some important remarks have to be pointed out. They are basic but essential for what follows. First of all, observe that the definition of Ck,w(I) does not depend on how u is chosen from w. Notice also that if I is compact or u is bounded on I, then C k;w ðIÞ ¼ C kB ðIÞ. Besides, if w(t) = 1 for all t 2 I, then (H1) means that for each n Ln is k-convex. Secondly, an easy check shows that for k 2 N, t 2 I and i 2 {0, 1, . . . , k 1} Dk,wui(t) = 0 (ui is the i-th power of u). From this and (H1), Dk,wLnui(t) = 0 for all t 2 I, and from (H2), Dk,w(pD2ui + qDui) 0. Hence the linear differential equation in the unknown function v
1 1 1 1 1 k;w 0 D D D v w2 w1 w0
ð4Þ
has a fundamental system of solutions of the form {1, u , . . . , uk1, v1, v2}. The change of variable z = Dk,wv makes Eq. (4) become the following one of second order:
Lz ¼ LðzÞ :¼
1 1 1 1 z 0: D D w2 w1 w0
In the sequel, if not specified in other sense, solutions of Eqs. (4) and (5) are understood on Int(I).
ð5Þ
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Now, if we consider a bounded subinterval J I and fix a point c 2 J, it is well-known that the functions w0(t) and Rt w0 ðtÞ c w1 ðsÞds form in J an extended complete Tchebychev system (see for instance [12]) and a fundamental set of solutions of Lz 0 as well. Thus, following Bonsall [7] we are in a position to talk about sub-ðLÞ functions in J. In the sequel this notion plays an important role. At this respect we follow the notation of [7] and for a function f defined in J and for t1, t2 2 J, we denote by F(f, t1, t2) the unique solution of (5) taking the values f(t1) and f(t2) respectively at t1 and t2. We end this section proving some auxiliary results which also have interest by themselves. First of all we recover [11, Lemma 1] which extends [5, Lemma], the key result of the so-called parabola technique. For the sake of completeness we include here a particular proof applied to the differential Eq. (5). Lemma 1. Let J be a bounded open subinterval of I. Let g, h 2 C(J) and t0, t1, t2 2 J such that t0 2 (t1, t2), g(t1) = g(t2) = 0 and g(t0) > 0. Then there exist a real number a < 0, a solution of the differential equation (5) on J, say z, and a point x 2 (t1, t2) such that for all t 2 [t1, t2], ah(t) + z(t) P g(t) and at the point x, ah(x) + z(x) = g(x). Proof. Let y be a solution of (5) strictly positive on (t1, t2) and let be small enough so that g(t0) (F(h, t1, t2)(t0) h(t0)) > 0. Then the function
g ðFðh; t 1 ; t2 Þ hÞ y is continuous on [t1, t2], vanishes at both end points of that interval and is strictly positive at t0, hence it reaches its maximum M at a point x 2 (t1, t2). Thus, on [t1, t2]
ðFðh; t1 ; t2 Þ hÞ þ My P g and at the point x
ðFðh; t1 ; t2 ÞðxÞ hðxÞÞ þ MyðxÞ ¼ gðxÞ: Finally the proof is over taking a = and z = eF(h, t1, t2) + My. h Next lemma is a type of localization result for the approximation processes we are dealing with, while Lemma 3 is a nice generalization of the well-known statement which assures the convexity of a function f 2 C[0, 1] provided lim supn?1 n(Bnf f)(x) P 0, x 2 (0,1). Lemma 2. Let h 2 Ck,w(I) and x 2 Int(I). Assume that there exists a neighborhood Nx of x where Dk,wh P 0. Then
Dk;w Ln hðxÞ P 0 þ o k1 : n Proof. Let x1, x2 2 Nx with x1 < x < x2 and let u1, u2 2 h1, u1, . . . , uki such that for j = 1, 2 and 0 6 i 6 k, Di,wuj(xj) = Di,wh(xj) e 2 C k;w ðIÞ be defined as: (notice that {1, u1, . . . , uk} is a Tchebychev system). Let h
8 > < u1 ðtÞ t < x1 ; e x1 6 t 6 x 2 ; hðtÞ ¼ hðtÞ > : u2 ðtÞ x2 < t: e hÞ ¼ 0. Indeed, it suffices to recall that for i = 0, 1, . . . , k 1, Dk,wui = 0 h P 0 and on (x1, x2), Dk;w ð h Then on the whole I, Dk;w e k;w k,w k e Dk;w Ln hðxÞ ¼ oðk1 and observe that D u = k!. Finally, (H2) yields D Ln hðxÞ n Þ, and from (H1)
~ ¼ Dk;w Ln hðxÞ þ oðk1 0 6 Dk;w Ln hðxÞ n Þ:
Lemma 3. Let f 2 Ck,w(I) and let J be a bounded open subinterval of I. If for each t 2 J
lim sup kn Dk;w ðLn f f ÞðtÞ P 0; n!1
then D
k,w
f is sub-ðLÞ in J.
Proof. Assuming the contrary, one can find t0, t1, t2 2 J with t0 2 (t1, t2) such that Dk,wf(t0) > F(Dk,wf, t1, t2)(t0). Let v 2 Ck,w(I) Rt Rs such that for t 2 [t1, t2] Dk;w v ðtÞ ¼ w0 ðtÞ t1 w1 ðsÞ t1 w2 ðmÞdmds. Then Lemma 1, applied to g = Dk,wf F(Dk,wf, t1, t2) and h = Dk,wv, yields the existence of a solution of (5) on J, say z, a real number a < 0 and a point x 2 (t1, t2) such that on [t1, t2]
aDk;w v ðtÞ þ zðtÞ Dk;w f ðtÞ þ FðDk;w f ; t1 ; t2 ÞðtÞ P 0 and at the point x
ð6Þ
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
aDk;w v ðxÞ þ zðxÞ Dk;w f ðxÞ þ FðDk;w f ; t1 ; t2 ÞðxÞ ¼ 0:
2679
ð7Þ
Let vz, vF 2 Ck,w(I) solutions of (4) on (t1, t2) such that on this interval Dk,wvz = z and Dk,wvF = F(Dk,wf, t1, t2). Thus, applying Lemma 2 to h = av + vz f + vF and Nx = (t1, t2) we obtain from (6) and (7) that
akn Dk;w ðLn v v ÞðxÞ þ kn Dk;w ðLn v z v z ÞðxÞ kn Dk;w ðLn f f ÞðxÞ þ kn Dk;w ðLn v F v F ÞðxÞ P 0 þ oð1Þ: Finally, it suffices to use (H2) with the functions v, vz and vF to obtain the following inequality which contradicts the hypothesis of this lemma:
kn Dk;w ðLn f f ÞðxÞ 6 a þ oð1Þ:
3. The main result Under the general setting of the previous section we can state the main result of the paper, Theorem 1, which represents a converse result of assumption (H2). To prove it we need the following lemma. In the sequel, we shall make use of the funcRt tions W i ðtÞ ¼ a wi ðsÞds, i = 1, 2. Lemma 4. Let (a, b) be a bounded subinterval of I and let x 2 (a, b). Besides, let h 2 C[a, b] and let H 2 Ck,w(I) such that for all Rt t 2 (a, b), Dk;w HðtÞ ¼ w0 ðtÞ a hðsÞw1 ðsÞds. Then
lim sup kn Dk;w ðLn H HÞðxÞ 6 lim sup n!1
t!x
hðtÞ hðxÞ W 2 ðtÞ W 2 ðxÞ
and
lim inf
t!x
hðtÞ hðxÞ 6 lim inf kn Dk;w ðLn H HÞðxÞ: n!1 W 2 ðtÞ W 2 ðxÞ
Proof. We shall only prove the first inequality, since one may proceed analogously for the other. Let
l ¼ lim sup t!x
hðtÞ hðxÞ hðtÞ hðxÞ ¼ lim sup : W 2 ðtÞ W 2 ðxÞ t!x ðt xÞw2 ðxÞ
If l = +1, then there is nothing to prove. Otherwise, for every h(t) h(x) 6 ( + l)(t x)w2(x) and directly
Z
t
ðhðsÞ hðxÞÞw1 ðsÞds ¼
x
Z
t
hðsÞw1 ðsÞds hðxÞ
x
6 ð þ lÞw2 ðxÞ
Z
Z
> 0,
t
w1 ðsÞds ¼
x
there exists d = d(, x) such that if jt xj < d, then
Dk;w HðtÞ Dk;w HðxÞ hðxÞ w0 ðtÞ w0 ðxÞ
Z
t
w1 ðsÞds
x
t
ðs xÞw1 ðsÞds:
x
Now, multiplying by w0(t) and taking v0, v1, v2 2 Ck,w(I) such that for each t 2 (x d, x + d), Dk,wv0(t) = w0(t), Rt Dk,wv1(t) = w0(t)W1(t) and Dk;w v 2 ðtÞ ¼ w2 ðxÞw0 ðtÞ x ðs xÞw1 ðsÞds, one has that on (x d, x + d), k;w
D
Dk;w HðxÞv 0 H hðxÞðv 1 W 1 ðxÞv 0 Þ w0 ðxÞ
!
6 ð þ lÞDk;w v 2 :
Applying Lemma 2 and introducing the zero term
Dk;w HðxÞ þ
Dk;w HðxÞ k;w D v 0 ðxÞ Dk;w v 1 ðxÞ þ W 1 ðxÞDk;w v 0 ðxÞ; w0 ðxÞ
it is derived that
Dk;w HðxÞ k;w D ðLn v 0 v 0 ÞðxÞ hðxÞðDk;w ðLn v 1 v 1 ÞðxÞ W 1 ðxÞðDk;w ðLn v 0 v 0 ÞðxÞÞÞ w0 ðxÞ : 6 ð þ lÞDk;w Ln v 2 ðxÞ þ o k1 n
Dk;w ðLn H HÞðxÞ
Finally, to end the proof it suffices to apply (H2) to v0, v1 and v2, observing that v0 and v1 are solutions of (4) on (x d, x + d). It is obtained that for all > 0
lim sup kn ðDk;w ðLn H HÞðxÞÞ 6 þ l: n!1
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Theorem 1. Let f 2 Ck,w(I), let (a, b) be a bounded subinterval of I and let w be a finitely-valued Lebesgue-integrable function on (a, b) such that for each x 2 (a, b)
lim inf kn Dk;w ðLn f f ÞðxÞ 6 wðxÞ 6 lim sup kn Dk;w ðLn f f ÞðxÞ: n!1
n!1
Then, almost everywhere on (a, b)
w ¼ Dk;w ðpD2 f þ qD1 f Þ: Proof. Let WðtÞ ¼ w0 ðtÞ
Rt a
w1 ðsÞ
Rs a
wðmÞw2 ðv Þdmds, and let G 2 Ck,w(I) such that for all t 2 (a, b)
Dk;w GðtÞ ¼ Dk;w f ðtÞ WðtÞ: For q 2 N, let mq and Mq be respectively the minor and major functions of w with respect to w2 such that
Z t 1 mq ðtÞ < ; wðsÞw ðsÞds 2 q a Z t 1 < ; Mq ðtÞ wðsÞw ðsÞds 2 q a
t 2 ða; bÞ; t 2 ða; bÞ;
whose existence is guaranteed from the theory of Lebesgue integration (see e.g. [18]). In particular it follows that
lim sup t!x
mq ðtÞ mq ðxÞ M q ðtÞ Mq ðxÞ 6 wðxÞ 6 lim inf : t!x W 2 ðtÞ W 2 ðxÞ W 2 ðtÞ W 2 ðxÞ
From the assumptions and Lemma 4, if we consider mq 2 C k;w ðIÞ such that for all t 2 (a, b) Dk;w mq ðtÞ ¼ w0 ðtÞ we have that
lim sup kn ðDk;w Ln mq ðxÞ Dk;w mq ðxÞÞ 6 lim sup n!1
t!x
Rt a
mq ðsÞw1 ðsÞds,
mq ðtÞ mq ðxÞ 6 wðxÞ 6 lim sup kn Dk;w ðLn f f ÞðxÞ; W 2 ðtÞ W 2 ðxÞ n!1
hence
lim sup kn Dk;w ðLn ðf mq Þ ðf mq ÞÞðxÞ P 0: n!1
Now Lemma 3 yields that for each q 2 NDk;w ðf mq Þ is sub-ðLÞ in (a, b). Letting q tend to infinity we derive that Dk,wG is subðLÞ. If we proceed this way with Mq we conclude that Dk,wG is sub-ðLÞ in (a, b) as well. Hence, in this interval LðDk;w GÞ ¼ 0 and consequently, almost everywhere on (a, b)
LðDk;w f Þ ¼ LðWÞ; from where the proof follows recalling the definition of W at the top of the proof, the one of L in (5), and finally using (H3). h
4. Applications In this section we apply Theorem 1 to the sequences of linear operators mentioned in the introduction, i.e. the Bernstein operators Bn, the Kantorovich operators Kn, the Schurer–Szász–Mirakyan operators Sv,n, v being a non-negative integer parameter, the Meyer-König and Zeller operators Mn and the Bleimann–Butzer and Hahn operators Hn. Now these operators are considered to be defined and taking values on Ck,w(I) where for each case the corresponding interval I and function w are written in Table 1. In the sequel we show that these sequences fit the general setting we have dealt with in the paper, that is to say they fulfill (H1), (H2) and (H3). For the sake of clarity and brevity we have completed Table 1 with the required information. As far as (H1) is concerned, w(t) = 1 for Bn, Kn and Sv,n, so in these three cases this assumption means that the operators are k-convex for all k 2 N0 , what is a very well-known fact. As for Mn and Hn we refer the reader to [15, Corollaries 14, 23]. At this respect it is important to point out that if u 2 C1(I) is such that D1u = w, then the differential operator Dku defined as
Dku f ðtÞ :¼ Dk ðf u1 ÞðuðtÞÞ
ð8Þ
and used in [15] coincides with our operator Dk,w, and consequently our assumption (H1) is equivalent to the u -convexity used there (see [15, Definitions 3, 4 ]). This latter notion was introduced in [13] and further used in [14,15]. These last references are appropriate to check as well the validity of (H2). Table 1 shows for each case the values of kn, p and q for which this hypothesis is fulfilled.
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
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Table 1 Information under which the listed classical operators fulfill hypotheses (H1), (H2) and (H3). Bn
Kn
Sv,n
Mn
Hn
I w(t) kn p(t) q(t) w0(t)
[0, 1] 1 2n t (1 t) 0
[0, 1] 1 2(n + 1) t (1 t) 1 2t
[0, 1) 1 2n t 2vt
1 t k1
1 tk
1 e2tv
[0, 1) 1/(1 t)2 2n t(1 t)2 0 (1 t)k+1
[0, 1) 1/(1 + t)2 2n t(1 + t)2 0 (1 + t)k+1
w1(t)
t k2 ð1tÞk
t 2k ðtð1tÞÞkþ1
e2tv tk
1 ð1tÞ2 t k
1 ð1þtÞ2 t k
w2(t)
(1 t)k1
(1 t)k
tk1
t k1 ð1tÞkþ1
t k1 ð1þtÞkþ1
Finally, easy but tedious computations allow to check that (H3) holds true for each k 2 N0 taking the corresponding functions w0, w1, w2 described in Table 1. Thus we are in a position to apply the results of the paper to our five illustrative sequences of operators. This is done in the following corollaries. We state them without proof as they are direct applications of Theorem 1. The only work to be done is to compute for each case the quantity Dk,w(pD2f + qD1f)(t). When w(t) = 1 (Dk,w = Dk) it suffices to use the Leibniz generalized product rule. When Ln = Mn and Ln = Hn we have used as well the following formula, derived directly from the Faá di Bruno one, which relates Dku ¼ Dk;w (see (8)) with the classical Dk (for details, see [15, Lemma 9 ]): for g 2 Ck(I), t 2 I, u 2 C1(I) such that D1u = w,
Dk;w gðtÞ ¼ Dku gðtÞ ¼
k X Ds gðtÞ k s D ht ðtÞ; s! s¼0
where ht(z) = u1(z) t. Throughout the following corollaries k is a non-negative integer, (a, b) is a bounded subinterval of the corresponding interval I and w is a finitely-valued Lebesgue-integrable function on (a, b). Corollary 1. (Bn and Kn) Let I = [0, 1] and f 2 C k ðIÞ ¼ C kB ðIÞ. If for each x 2 (a, b)
lim inf 2nDk ðBn f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk ðBn f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b)
wðtÞ ¼ tð1 tÞDkþ2 f ðtÞ þ kð1 2tÞDkþ1 f ðtÞ kðk 1ÞDk f ðtÞ: If for each x 2 (a, b)
lim inf 2ðn þ 1ÞDk ðK n f f ÞðxÞ 6 wðxÞ 6 lim sup 2ðn þ 1ÞDk ðK n f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b)
wðtÞ ¼ tð1 tÞDkþ2 f ðtÞ þ ðk þ 1Þð1 2tÞDkþ1 f ðtÞ kðk þ 1ÞDk f ðtÞ:
Corollary 2. (Sv,n) Let
v 2 N0 , let I = [0, 1), let w(t) = 1 and f 2 Ck,w(I). If for each x 2 (a, b)
lim inf 2nDk ðSv ;n f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk ðSv ;n f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b) k
wðtÞ ¼ tDkþ2 f ðtÞ þ ðk þ 2v tÞDkþ1 f ðtÞ þ 2kv D f ðtÞ: Corollary 3. (Mn) Let I = [0, 1), let w(t) = 1/(1 t)2 and f 2 Ck,w(I). If for each x 2 (a, b)
lim inf 2nDk;w ðMn f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk;w ðMn f f ÞðxÞ; n!1
then, for almost every t 2 (a, b)
n!1
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wðtÞ ¼
k X k ðk 1Þ! s¼0
s ðs 1Þ!
ðt 1Þkþs
n o tð1 tÞ2 Dsþ2 f ðtÞ þ sð3t 2 4t þ 1ÞDsþ1 f ðtÞ þ sðs 1Þð3t 2ÞDs f ðtÞ þ sðs 1Þðs 2ÞDs1 f ðtÞ : Corollary 4. (Hn) Let I = [0, 1), let w(t) = 1/(1 + t)2 and f 2 C k;w ðIÞ ¼ C kB ðIÞ. If for each x 2 (a, b)
lim inf 2nDk;w ðHn f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk;w ðHn f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b)
wðtÞ ¼
k X k ðk 1Þ! s¼0
s ðs 1Þ!
ð1 þ tÞkþs
n o tð1 þ tÞ2 Dsþ2 f ðtÞ þ sð3t 2 þ 4t þ 1ÞDsþ1 f ðtÞ þ sðs 1Þð3t þ 2ÞDs f ðtÞ þ sðs 1Þðs 2ÞDs1 f ðtÞ : 5. Saturation This section deals with the saturation of the approximation process of Dk,wLnf towards Dk,wf. Its content is directly obtained from the results of the previous sections. Proposition 1 is a direct consequence of Lemma 3 and provides us with information about the so-called trivial class. Proposition 2 goes further and solves completely the saturation problem. Proposition 1. Let f 2 Ck,w(I) and let (a, b) be a bounded open subinterval of I. Then for each t 2 (a, b)
kn Dk;w Ln f ðtÞ Dk;w f ðtÞ ¼ oð1Þ; if and only if f is a solution of (4) in J. Proposition 2. Let f 2 Ck,w(I), let (a, b) be a bounded subinterval of I and let M P 0. Then for each x 2 (a, b)
kn Dk;w Ln f ðxÞ Dk;w f ðxÞ 6 M þ oð1Þ; if and only if, for almost every t 2 (a, b)
jDk;w ðpD2 f þ qD1 f ÞðtÞj 6 M: Proof. The ‘only if’ part follows directly from Theorem 1 taking w(x) = lim infn?+1kn(Dk,wLnf(x) Dk,wf(x)). Rt For the converse, choose h 2 C[a, b] such that for each t 2 (a, b), Dk;w f ðtÞ ¼ w0 ðtÞ a hðsÞw1 ðsÞds and apply Lemma 4 with H = f. It is obtained that
hðtÞ hðxÞ W 2 ðtÞ W 2 ðxÞ
ð9Þ
hðtÞ hðxÞ 6 lim inf kn Dk;w ðLn f f ÞðxÞ: n!1 W 2 ðtÞ W 2 ðxÞ
ð10Þ
lim sup kn Dk;w ðLn f f ÞðxÞ 6 lim sup n!1
t!x
and
lim inf
t!x
On the other hand, the hypothesis and (H3) yield that almost everywhere on (a, b)
1 1 1 1 1 k;w D 6 M; D D f w w1 w0 2
which, by taking into account [16, Section 3], is equivalent to
Z t 1 1 1 k;w 1 1 1 k;w 6M D D f ðtÞ D D f ðxÞ w2 ðsÞds w w0 w1 w0 1 x and also to
M 6 lim inf
t!x
hðtÞ hðxÞ hðtÞ hðxÞ 6 lim sup 6 M; W 2 ðtÞ W 2 ðxÞ t!x W 2 ðtÞ W 2 ðxÞ
just observing that on (a, b), h ¼ w11 D1
1 w0
ð11Þ
Dk;w f and recalling the definition of W2. The proof is over directly from (9)–(11). h
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
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It goes without saying that these two propositions can be applied to the particular approximation processes we have considered above. Some well-known statements are then recovered (see [16,6,9,10]), but further new results appear. We close the paper with two instances. Corollary 5. Let I = [0, 1), let w(t) = 1/(1 t)2 and f 2 Ck,w(I). Then for each x 2 (a, b)
2nðDk;w M n f ðxÞ Dk;w f ðxÞÞ ¼ oð1Þ; if and only if the restriction of f to the interval (a, b) belongs to the linear space spanned by the set of functions
1; uðtÞ; u2 ðtÞ; . . . ; uk1 ðtÞ; v 1 ðtÞ; v 2 ðtÞ ; where u(t) = t/(1 t),
t v1(t) = t and v 2 ðtÞ ¼ t logð1t Þ.
Corollary 6. Let I = [0, 1), let w(t) = 1/(1 + t)2 and f 2 C k;w ðIÞ ¼ C kB ðIÞ. Let (a, b) be a bounded subinterval of I and let M P 0. Then for each x 2 (a, b)
2njDk;w Hn f ðxÞ Dk;w f ðxÞj 6 M þ oð1Þ; if and only if, for almost every t 2 (a, b)
X k n k ðk 1Þ! ð1 þ tÞkþs tð1 þ tÞ2 Dsþ2 f ðtÞ þ sð3t2 þ 4t þ 1ÞDsþ1 f ðtÞ s¼0 s ðs 1Þ! o s s1 þsðs 1Þð3t þ 2ÞD f ðtÞ þ sðs 1Þðs 2ÞD f ðtÞ 6 M: References [1] U. Abel, Asymptotic approximation with Kantorovich polynomial, Approx. Theory Appl. 3 (14) (1998) 106–116. [2] U. Abel, M. Ivan, Asymptotic expansion of the Jakimovski–Leviatan operators and their derivatives, in: Functions, Series, Operators (Budapest, 1999), János Bolyai Math. Soc., Budapest, 2002, pp. 103–119. [3] F. Altomare, M. Campiti, Koronkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter, Belin-New York, 1994. [4] V.G. Amel’kovic˘, A theorem converse to a theorem of Voronovskaja type, Teor. Funkc˘iı˘, Funkc˘ional. Anal. i Priloz˘en 2 (1966) 67–74. [5] B. Bajsanski, R. Bojanic´, A note on approximation by Bernstein polynomials, Bull. Am. Math. Soc. 70 (1964) 675–677. [6] H. Berens, Pointwise saturation of positive operators, J. Approx. Theory 6 (1972) 135–146. [7] F.F. Bonsall, The characterization of generalized convex functions, Q. J. Math. Oxford Ser. 2 (1) (1950) 100–111. [8] D. Cárdenas-Morales, P. Garrancho, F.J. Muñoz-Delgado, A result on asymptotic formulae for linear k-convex operators, Int. J. Differ. Equ. Appl. 2 (3) (2001) 335–347. [9] D. Cárdenas-Morales, P. Garrancho, Local saturation of conservative operators, Acta Math. Hungar. 100 (1–2) (2003) 83–95. [10] D. Cárdenas-Morales, P. Garrancho, F.J. Muñoz-Delgado, Addendum and corrigendum to ‘Local saturation of conservative operators’, Acta Math. Hungar. 105 (3) (2004) 257–259. [11] D. Cárdenas-Morales and P. Garrancho, On saturation in conservative approximation, in: M. Gasca (Ed.), Multivariate Approximation and Interpolation with Applications, Monografı´as de la Academia de Ciencias de Zaragoza 20, Real Academia de Ciencias de Zaragoza, Zaragoza, 2002, pp. 59–67. [12] S.J. Karlin, W.J. Studden, Tchebycheff Systems, Interscience, New York, 1966. [13] A.J. López-Moreno, Expresiones y estimaciones de operadores lineales conservativos, Ph.D. thesis, University of Jaén, Spain, 2001. [14] A.J. López-Moreno, J. Martı´nez-Moreno, F.J. Muñoz-Delgado, Asymptotic expression of derivatives of Bernstein type operators, Suppl. Cir. Mat. Palermo Ser. II 68 (2002) 615–624. [15] A.J. López-Moreno, F.J. Muñoz-Delgado, Asymptotic expansion of multivariate conservative linear operators, J. Comput. Appl. Math. 150 (2003) 219– 251. [16] G.G. Lorentz, L.L. Schumaker, Saturation of positive operators, J. Approx. Theory 5 (1972) 413–424. [17] G. Mühlbach, Operatoren vom Bernsteinschen Typ, J. Approx. Theory 3 (1970) 274–292. [18] I. Pesin, Classical and Modern Integration Theories, Academic Press, New York, 1951. [19] E. Voronovskaya, Détermination de la forme asymptotique d’approximation des fonctions par les polynômes de S. Bernstein, Dokl. Akad. Nauk. USSR A (1932) 79–85.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A converse of asymptotic formulae in simultaneous approximation q P. Garrancho, D. Cárdenas-Morales * Departamento de Matemáticas, Universidad de Jaén, 23071 Jaén, Spain
a r t i c l e
i n f o
a b s t r a c t In the general setting of simultaneous approximation by sequences of linear shape preserving operators, this paper contains a sort of converse result of Voronovskaya-type asymptotic formulae. As a by-product a saturation result is derived. Applications to some very well-known approximation processes are also presented. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Asymptotic formula Shape preserving approximation Simultaneous approximation Saturation
1. Introduction In 1932 Voronovskaya [19] proved the following famous asymptotic formula for the classical Bernstein polynomials Bn: if f is a real bounded function on [0, 1] and f00 (x) exists, then
lim 2nðBn f ðxÞ f ðxÞÞ ¼ xð1 xÞf 00 ðxÞ:
n!1
Many well-known approximation processes by sequences of linear operators fulfill as well a formula of this type, as the Kantorovich operators, the Schurer–Szász–Mirakyan operators, the Meyer–König and Zeller operators and the Bleimann–Butzer and Hahn operators (for a definition of these sequences see for instance the excellent monograph [3]). Specifically, if Ln is such a sequence, then there exist a sequence of real numbers kn ? +1, and two polynomial functions p, q such that, under certain assumptions, 00
0
lim kn ðLn f f ÞðxÞ ¼ ðpf þ qf ÞðxÞ:
n!1
ð1Þ
On the other hand, in the last few years some results have appeared in the setting of the so-called simultaneous approximation proving that these formulae remain valid if certain linear differential operator, say D, is applied to both sides of (1), i.e. 00
0
lim kn DðLn f f ÞðxÞ ¼ Dðpf þ qf ÞðxÞ
n!1
ð2Þ
(see [1,13,8,2,14,15]). Actually, D is narrowly connected with the shape preserving properties that these operators posses, which at the time represent an essential tool to prove the validity of (2). These properties can be written in short as follows: Df P 0 implies DLn f P 0. Roughly speaking, as we have proceeded till this point, our aim with this work is to prove a converse result of (2). It can be considered as a generalization of a result of the authors [11], and follows the line that began with the pioneer paper of Amel’kovic˘ [4] and found further development in the papers of Mühlbach [17] and Berens [6]. As a by-product we derive
q
This work is partially supported by Junta de Andalucı´a (FQM-0178) and by Ministerio de Ciencia y Tecnologı´a (MTM 2006-14590). * Corresponding author. Tel.: +34 953 21 21 44; fax: +34 953 21 18 71. E-mail address: [email protected] (D. Cárdenas-Morales).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.08.005
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
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a general saturation theorem that extends and reinterprets the ones of [16,9] (see also [10]), and whose application allows to achieve new results for some of the aforementioned particular sequences of operators. Here it is an outline of the paper: in the next section we fix the notation and precise the framework; this includes a definition of the operator D and formal statements of the asymptotic formulae and the shape preserving properties we shall deal with. Some auxiliary results are also proved. Sections 3 and 4 are devoted to the main result and applications, while the last one contains the aforesaid results on saturation. 2. General setting For a real interval I R and a non-negative integer i 2 N0 ¼ N [ f0g, as usual we denote by Ci(I) the space of all real valued i-times continuously differentiable functions defined on I and by Di the classical i-th differential operator (C0(I) = C(I) is the space of all continuous functions on I and D0 ¼ I is the identity operator). C 1 ðIÞ ¼ \i2N C i ðIÞ, C iB ðIÞ denotes the subspace of Ci(I) formed by the bounded functions. A function f 2 Ci(I) is said to be i-convex if Dif P 0 on I and a linear operator is said to be i-convex if it maps i-convex functions onto i-convex functions. Now we shall consider a sequence of linear operators fulfilling a shape preserving property (H1) and an asymptotic formula (H2) (see below). To this end, we start presenting the domain and range of the operators and the differential operator D referred in the previous section. Let us fix an interval I R, a number k 2 N0 and a function w 2 C1(I) such that w(t) – 0 for all t 2 I. Now we take any function u 2 C1(I) such that D1u = w and define the space
C k;w ðIÞ ¼ ff 2 C k ðIÞ : 9c1 ; c2 2 Rþ =8t 2 I; jf ðtÞj < c1 ec2 uðtÞ g: Besides, we consider the m-th iterate of the operator
D0;w ¼ I;
D1;w ¼
1 1 D ; w
Dmþ1;w ¼ D1;w Dm;w ;
1 1 D , w
i.e.
m 2 N:
Finally, let us consider a sequence of linear operators
Ln : C k;w ðIÞ ! C k;w ðIÞ fulfilling the following hypotheses: (H1) if f 2 Ck,w(I) is such that Dk,wf P 0 on I, then for each n 2 N, Dk,wLnf P 0 on I, (H2) there exist a sequence of real numbers kn? +1 and two functions p, q 2 Ck(I), p being strictly positive on Int (I), such that for f 2 Ck,w(I), k + 2-times differentiable at a point x 2 Int(I),
lim kn Dk;w ðLn f f ÞðxÞ ¼ Dk;w ðpD2 f þ qD1 f ÞðxÞ:
n!þ1
ð3Þ
Besides, if k 2 N, we assume in addition that (H3) there exist three strictly positive functions wi 2 C2i(Int(I)), i = 0, 1, 2, such that for f 2 Ck,w(I), whenever it is meaningful,
Dk;w ðpD2 f þ qD1 f Þ ¼
1 1 1 1 1 k;w D D D f w2 w1 w0
(see the right-hand side of (3)). Some important remarks have to be pointed out. They are basic but essential for what follows. First of all, observe that the definition of Ck,w(I) does not depend on how u is chosen from w. Notice also that if I is compact or u is bounded on I, then C k;w ðIÞ ¼ C kB ðIÞ. Besides, if w(t) = 1 for all t 2 I, then (H1) means that for each n Ln is k-convex. Secondly, an easy check shows that for k 2 N, t 2 I and i 2 {0, 1, . . . , k 1} Dk,wui(t) = 0 (ui is the i-th power of u). From this and (H1), Dk,wLnui(t) = 0 for all t 2 I, and from (H2), Dk,w(pD2ui + qDui) 0. Hence the linear differential equation in the unknown function v
1 1 1 1 1 k;w 0 D D D v w2 w1 w0
ð4Þ
has a fundamental system of solutions of the form {1, u , . . . , uk1, v1, v2}. The change of variable z = Dk,wv makes Eq. (4) become the following one of second order:
Lz ¼ LðzÞ :¼
1 1 1 1 z 0: D D w2 w1 w0
In the sequel, if not specified in other sense, solutions of Eqs. (4) and (5) are understood on Int(I).
ð5Þ
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Now, if we consider a bounded subinterval J I and fix a point c 2 J, it is well-known that the functions w0(t) and Rt w0 ðtÞ c w1 ðsÞds form in J an extended complete Tchebychev system (see for instance [12]) and a fundamental set of solutions of Lz 0 as well. Thus, following Bonsall [7] we are in a position to talk about sub-ðLÞ functions in J. In the sequel this notion plays an important role. At this respect we follow the notation of [7] and for a function f defined in J and for t1, t2 2 J, we denote by F(f, t1, t2) the unique solution of (5) taking the values f(t1) and f(t2) respectively at t1 and t2. We end this section proving some auxiliary results which also have interest by themselves. First of all we recover [11, Lemma 1] which extends [5, Lemma], the key result of the so-called parabola technique. For the sake of completeness we include here a particular proof applied to the differential Eq. (5). Lemma 1. Let J be a bounded open subinterval of I. Let g, h 2 C(J) and t0, t1, t2 2 J such that t0 2 (t1, t2), g(t1) = g(t2) = 0 and g(t0) > 0. Then there exist a real number a < 0, a solution of the differential equation (5) on J, say z, and a point x 2 (t1, t2) such that for all t 2 [t1, t2], ah(t) + z(t) P g(t) and at the point x, ah(x) + z(x) = g(x). Proof. Let y be a solution of (5) strictly positive on (t1, t2) and let be small enough so that g(t0) (F(h, t1, t2)(t0) h(t0)) > 0. Then the function
g ðFðh; t 1 ; t2 Þ hÞ y is continuous on [t1, t2], vanishes at both end points of that interval and is strictly positive at t0, hence it reaches its maximum M at a point x 2 (t1, t2). Thus, on [t1, t2]
ðFðh; t1 ; t2 Þ hÞ þ My P g and at the point x
ðFðh; t1 ; t2 ÞðxÞ hðxÞÞ þ MyðxÞ ¼ gðxÞ: Finally the proof is over taking a = and z = eF(h, t1, t2) + My. h Next lemma is a type of localization result for the approximation processes we are dealing with, while Lemma 3 is a nice generalization of the well-known statement which assures the convexity of a function f 2 C[0, 1] provided lim supn?1 n(Bnf f)(x) P 0, x 2 (0,1). Lemma 2. Let h 2 Ck,w(I) and x 2 Int(I). Assume that there exists a neighborhood Nx of x where Dk,wh P 0. Then
Dk;w Ln hðxÞ P 0 þ o k1 : n Proof. Let x1, x2 2 Nx with x1 < x < x2 and let u1, u2 2 h1, u1, . . . , uki such that for j = 1, 2 and 0 6 i 6 k, Di,wuj(xj) = Di,wh(xj) e 2 C k;w ðIÞ be defined as: (notice that {1, u1, . . . , uk} is a Tchebychev system). Let h
8 > < u1 ðtÞ t < x1 ; e x1 6 t 6 x 2 ; hðtÞ ¼ hðtÞ > : u2 ðtÞ x2 < t: e hÞ ¼ 0. Indeed, it suffices to recall that for i = 0, 1, . . . , k 1, Dk,wui = 0 h P 0 and on (x1, x2), Dk;w ð h Then on the whole I, Dk;w e k;w k,w k e Dk;w Ln hðxÞ ¼ oðk1 and observe that D u = k!. Finally, (H2) yields D Ln hðxÞ n Þ, and from (H1)
~ ¼ Dk;w Ln hðxÞ þ oðk1 0 6 Dk;w Ln hðxÞ n Þ:
Lemma 3. Let f 2 Ck,w(I) and let J be a bounded open subinterval of I. If for each t 2 J
lim sup kn Dk;w ðLn f f ÞðtÞ P 0; n!1
then D
k,w
f is sub-ðLÞ in J.
Proof. Assuming the contrary, one can find t0, t1, t2 2 J with t0 2 (t1, t2) such that Dk,wf(t0) > F(Dk,wf, t1, t2)(t0). Let v 2 Ck,w(I) Rt Rs such that for t 2 [t1, t2] Dk;w v ðtÞ ¼ w0 ðtÞ t1 w1 ðsÞ t1 w2 ðmÞdmds. Then Lemma 1, applied to g = Dk,wf F(Dk,wf, t1, t2) and h = Dk,wv, yields the existence of a solution of (5) on J, say z, a real number a < 0 and a point x 2 (t1, t2) such that on [t1, t2]
aDk;w v ðtÞ þ zðtÞ Dk;w f ðtÞ þ FðDk;w f ; t1 ; t2 ÞðtÞ P 0 and at the point x
ð6Þ
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
aDk;w v ðxÞ þ zðxÞ Dk;w f ðxÞ þ FðDk;w f ; t1 ; t2 ÞðxÞ ¼ 0:
2679
ð7Þ
Let vz, vF 2 Ck,w(I) solutions of (4) on (t1, t2) such that on this interval Dk,wvz = z and Dk,wvF = F(Dk,wf, t1, t2). Thus, applying Lemma 2 to h = av + vz f + vF and Nx = (t1, t2) we obtain from (6) and (7) that
akn Dk;w ðLn v v ÞðxÞ þ kn Dk;w ðLn v z v z ÞðxÞ kn Dk;w ðLn f f ÞðxÞ þ kn Dk;w ðLn v F v F ÞðxÞ P 0 þ oð1Þ: Finally, it suffices to use (H2) with the functions v, vz and vF to obtain the following inequality which contradicts the hypothesis of this lemma:
kn Dk;w ðLn f f ÞðxÞ 6 a þ oð1Þ:
3. The main result Under the general setting of the previous section we can state the main result of the paper, Theorem 1, which represents a converse result of assumption (H2). To prove it we need the following lemma. In the sequel, we shall make use of the funcRt tions W i ðtÞ ¼ a wi ðsÞds, i = 1, 2. Lemma 4. Let (a, b) be a bounded subinterval of I and let x 2 (a, b). Besides, let h 2 C[a, b] and let H 2 Ck,w(I) such that for all Rt t 2 (a, b), Dk;w HðtÞ ¼ w0 ðtÞ a hðsÞw1 ðsÞds. Then
lim sup kn Dk;w ðLn H HÞðxÞ 6 lim sup n!1
t!x
hðtÞ hðxÞ W 2 ðtÞ W 2 ðxÞ
and
lim inf
t!x
hðtÞ hðxÞ 6 lim inf kn Dk;w ðLn H HÞðxÞ: n!1 W 2 ðtÞ W 2 ðxÞ
Proof. We shall only prove the first inequality, since one may proceed analogously for the other. Let
l ¼ lim sup t!x
hðtÞ hðxÞ hðtÞ hðxÞ ¼ lim sup : W 2 ðtÞ W 2 ðxÞ t!x ðt xÞw2 ðxÞ
If l = +1, then there is nothing to prove. Otherwise, for every h(t) h(x) 6 ( + l)(t x)w2(x) and directly
Z
t
ðhðsÞ hðxÞÞw1 ðsÞds ¼
x
Z
t
hðsÞw1 ðsÞds hðxÞ
x
6 ð þ lÞw2 ðxÞ
Z
Z
> 0,
t
w1 ðsÞds ¼
x
there exists d = d(, x) such that if jt xj < d, then
Dk;w HðtÞ Dk;w HðxÞ hðxÞ w0 ðtÞ w0 ðxÞ
Z
t
w1 ðsÞds
x
t
ðs xÞw1 ðsÞds:
x
Now, multiplying by w0(t) and taking v0, v1, v2 2 Ck,w(I) such that for each t 2 (x d, x + d), Dk,wv0(t) = w0(t), Rt Dk,wv1(t) = w0(t)W1(t) and Dk;w v 2 ðtÞ ¼ w2 ðxÞw0 ðtÞ x ðs xÞw1 ðsÞds, one has that on (x d, x + d), k;w
D
Dk;w HðxÞv 0 H hðxÞðv 1 W 1 ðxÞv 0 Þ w0 ðxÞ
!
6 ð þ lÞDk;w v 2 :
Applying Lemma 2 and introducing the zero term
Dk;w HðxÞ þ
Dk;w HðxÞ k;w D v 0 ðxÞ Dk;w v 1 ðxÞ þ W 1 ðxÞDk;w v 0 ðxÞ; w0 ðxÞ
it is derived that
Dk;w HðxÞ k;w D ðLn v 0 v 0 ÞðxÞ hðxÞðDk;w ðLn v 1 v 1 ÞðxÞ W 1 ðxÞðDk;w ðLn v 0 v 0 ÞðxÞÞÞ w0 ðxÞ : 6 ð þ lÞDk;w Ln v 2 ðxÞ þ o k1 n
Dk;w ðLn H HÞðxÞ
Finally, to end the proof it suffices to apply (H2) to v0, v1 and v2, observing that v0 and v1 are solutions of (4) on (x d, x + d). It is obtained that for all > 0
lim sup kn ðDk;w ðLn H HÞðxÞÞ 6 þ l: n!1
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Theorem 1. Let f 2 Ck,w(I), let (a, b) be a bounded subinterval of I and let w be a finitely-valued Lebesgue-integrable function on (a, b) such that for each x 2 (a, b)
lim inf kn Dk;w ðLn f f ÞðxÞ 6 wðxÞ 6 lim sup kn Dk;w ðLn f f ÞðxÞ: n!1
n!1
Then, almost everywhere on (a, b)
w ¼ Dk;w ðpD2 f þ qD1 f Þ: Proof. Let WðtÞ ¼ w0 ðtÞ
Rt a
w1 ðsÞ
Rs a
wðmÞw2 ðv Þdmds, and let G 2 Ck,w(I) such that for all t 2 (a, b)
Dk;w GðtÞ ¼ Dk;w f ðtÞ WðtÞ: For q 2 N, let mq and Mq be respectively the minor and major functions of w with respect to w2 such that
Z t 1 mq ðtÞ < ; wðsÞw ðsÞds 2 q a Z t 1 < ; Mq ðtÞ wðsÞw ðsÞds 2 q a
t 2 ða; bÞ; t 2 ða; bÞ;
whose existence is guaranteed from the theory of Lebesgue integration (see e.g. [18]). In particular it follows that
lim sup t!x
mq ðtÞ mq ðxÞ M q ðtÞ Mq ðxÞ 6 wðxÞ 6 lim inf : t!x W 2 ðtÞ W 2 ðxÞ W 2 ðtÞ W 2 ðxÞ
From the assumptions and Lemma 4, if we consider mq 2 C k;w ðIÞ such that for all t 2 (a, b) Dk;w mq ðtÞ ¼ w0 ðtÞ we have that
lim sup kn ðDk;w Ln mq ðxÞ Dk;w mq ðxÞÞ 6 lim sup n!1
t!x
Rt a
mq ðsÞw1 ðsÞds,
mq ðtÞ mq ðxÞ 6 wðxÞ 6 lim sup kn Dk;w ðLn f f ÞðxÞ; W 2 ðtÞ W 2 ðxÞ n!1
hence
lim sup kn Dk;w ðLn ðf mq Þ ðf mq ÞÞðxÞ P 0: n!1
Now Lemma 3 yields that for each q 2 NDk;w ðf mq Þ is sub-ðLÞ in (a, b). Letting q tend to infinity we derive that Dk,wG is subðLÞ. If we proceed this way with Mq we conclude that Dk,wG is sub-ðLÞ in (a, b) as well. Hence, in this interval LðDk;w GÞ ¼ 0 and consequently, almost everywhere on (a, b)
LðDk;w f Þ ¼ LðWÞ; from where the proof follows recalling the definition of W at the top of the proof, the one of L in (5), and finally using (H3). h
4. Applications In this section we apply Theorem 1 to the sequences of linear operators mentioned in the introduction, i.e. the Bernstein operators Bn, the Kantorovich operators Kn, the Schurer–Szász–Mirakyan operators Sv,n, v being a non-negative integer parameter, the Meyer-König and Zeller operators Mn and the Bleimann–Butzer and Hahn operators Hn. Now these operators are considered to be defined and taking values on Ck,w(I) where for each case the corresponding interval I and function w are written in Table 1. In the sequel we show that these sequences fit the general setting we have dealt with in the paper, that is to say they fulfill (H1), (H2) and (H3). For the sake of clarity and brevity we have completed Table 1 with the required information. As far as (H1) is concerned, w(t) = 1 for Bn, Kn and Sv,n, so in these three cases this assumption means that the operators are k-convex for all k 2 N0 , what is a very well-known fact. As for Mn and Hn we refer the reader to [15, Corollaries 14, 23]. At this respect it is important to point out that if u 2 C1(I) is such that D1u = w, then the differential operator Dku defined as
Dku f ðtÞ :¼ Dk ðf u1 ÞðuðtÞÞ
ð8Þ
and used in [15] coincides with our operator Dk,w, and consequently our assumption (H1) is equivalent to the u -convexity used there (see [15, Definitions 3, 4 ]). This latter notion was introduced in [13] and further used in [14,15]. These last references are appropriate to check as well the validity of (H2). Table 1 shows for each case the values of kn, p and q for which this hypothesis is fulfilled.
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Table 1 Information under which the listed classical operators fulfill hypotheses (H1), (H2) and (H3). Bn
Kn
Sv,n
Mn
Hn
I w(t) kn p(t) q(t) w0(t)
[0, 1] 1 2n t (1 t) 0
[0, 1] 1 2(n + 1) t (1 t) 1 2t
[0, 1) 1 2n t 2vt
1 t k1
1 tk
1 e2tv
[0, 1) 1/(1 t)2 2n t(1 t)2 0 (1 t)k+1
[0, 1) 1/(1 + t)2 2n t(1 + t)2 0 (1 + t)k+1
w1(t)
t k2 ð1tÞk
t 2k ðtð1tÞÞkþ1
e2tv tk
1 ð1tÞ2 t k
1 ð1þtÞ2 t k
w2(t)
(1 t)k1
(1 t)k
tk1
t k1 ð1tÞkþ1
t k1 ð1þtÞkþ1
Finally, easy but tedious computations allow to check that (H3) holds true for each k 2 N0 taking the corresponding functions w0, w1, w2 described in Table 1. Thus we are in a position to apply the results of the paper to our five illustrative sequences of operators. This is done in the following corollaries. We state them without proof as they are direct applications of Theorem 1. The only work to be done is to compute for each case the quantity Dk,w(pD2f + qD1f)(t). When w(t) = 1 (Dk,w = Dk) it suffices to use the Leibniz generalized product rule. When Ln = Mn and Ln = Hn we have used as well the following formula, derived directly from the Faá di Bruno one, which relates Dku ¼ Dk;w (see (8)) with the classical Dk (for details, see [15, Lemma 9 ]): for g 2 Ck(I), t 2 I, u 2 C1(I) such that D1u = w,
Dk;w gðtÞ ¼ Dku gðtÞ ¼
k X Ds gðtÞ k s D ht ðtÞ; s! s¼0
where ht(z) = u1(z) t. Throughout the following corollaries k is a non-negative integer, (a, b) is a bounded subinterval of the corresponding interval I and w is a finitely-valued Lebesgue-integrable function on (a, b). Corollary 1. (Bn and Kn) Let I = [0, 1] and f 2 C k ðIÞ ¼ C kB ðIÞ. If for each x 2 (a, b)
lim inf 2nDk ðBn f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk ðBn f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b)
wðtÞ ¼ tð1 tÞDkþ2 f ðtÞ þ kð1 2tÞDkþ1 f ðtÞ kðk 1ÞDk f ðtÞ: If for each x 2 (a, b)
lim inf 2ðn þ 1ÞDk ðK n f f ÞðxÞ 6 wðxÞ 6 lim sup 2ðn þ 1ÞDk ðK n f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b)
wðtÞ ¼ tð1 tÞDkþ2 f ðtÞ þ ðk þ 1Þð1 2tÞDkþ1 f ðtÞ kðk þ 1ÞDk f ðtÞ:
Corollary 2. (Sv,n) Let
v 2 N0 , let I = [0, 1), let w(t) = 1 and f 2 Ck,w(I). If for each x 2 (a, b)
lim inf 2nDk ðSv ;n f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk ðSv ;n f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b) k
wðtÞ ¼ tDkþ2 f ðtÞ þ ðk þ 2v tÞDkþ1 f ðtÞ þ 2kv D f ðtÞ: Corollary 3. (Mn) Let I = [0, 1), let w(t) = 1/(1 t)2 and f 2 Ck,w(I). If for each x 2 (a, b)
lim inf 2nDk;w ðMn f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk;w ðMn f f ÞðxÞ; n!1
then, for almost every t 2 (a, b)
n!1
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wðtÞ ¼
k X k ðk 1Þ! s¼0
s ðs 1Þ!
ðt 1Þkþs
n o tð1 tÞ2 Dsþ2 f ðtÞ þ sð3t 2 4t þ 1ÞDsþ1 f ðtÞ þ sðs 1Þð3t 2ÞDs f ðtÞ þ sðs 1Þðs 2ÞDs1 f ðtÞ : Corollary 4. (Hn) Let I = [0, 1), let w(t) = 1/(1 + t)2 and f 2 C k;w ðIÞ ¼ C kB ðIÞ. If for each x 2 (a, b)
lim inf 2nDk;w ðHn f f ÞðxÞ 6 wðxÞ 6 lim sup 2nDk;w ðHn f f ÞðxÞ; n!1
n!1
then, for almost every t 2 (a, b)
wðtÞ ¼
k X k ðk 1Þ! s¼0
s ðs 1Þ!
ð1 þ tÞkþs
n o tð1 þ tÞ2 Dsþ2 f ðtÞ þ sð3t 2 þ 4t þ 1ÞDsþ1 f ðtÞ þ sðs 1Þð3t þ 2ÞDs f ðtÞ þ sðs 1Þðs 2ÞDs1 f ðtÞ : 5. Saturation This section deals with the saturation of the approximation process of Dk,wLnf towards Dk,wf. Its content is directly obtained from the results of the previous sections. Proposition 1 is a direct consequence of Lemma 3 and provides us with information about the so-called trivial class. Proposition 2 goes further and solves completely the saturation problem. Proposition 1. Let f 2 Ck,w(I) and let (a, b) be a bounded open subinterval of I. Then for each t 2 (a, b)
kn Dk;w Ln f ðtÞ Dk;w f ðtÞ ¼ oð1Þ; if and only if f is a solution of (4) in J. Proposition 2. Let f 2 Ck,w(I), let (a, b) be a bounded subinterval of I and let M P 0. Then for each x 2 (a, b)
kn Dk;w Ln f ðxÞ Dk;w f ðxÞ 6 M þ oð1Þ; if and only if, for almost every t 2 (a, b)
jDk;w ðpD2 f þ qD1 f ÞðtÞj 6 M: Proof. The ‘only if’ part follows directly from Theorem 1 taking w(x) = lim infn?+1kn(Dk,wLnf(x) Dk,wf(x)). Rt For the converse, choose h 2 C[a, b] such that for each t 2 (a, b), Dk;w f ðtÞ ¼ w0 ðtÞ a hðsÞw1 ðsÞds and apply Lemma 4 with H = f. It is obtained that
hðtÞ hðxÞ W 2 ðtÞ W 2 ðxÞ
ð9Þ
hðtÞ hðxÞ 6 lim inf kn Dk;w ðLn f f ÞðxÞ: n!1 W 2 ðtÞ W 2 ðxÞ
ð10Þ
lim sup kn Dk;w ðLn f f ÞðxÞ 6 lim sup n!1
t!x
and
lim inf
t!x
On the other hand, the hypothesis and (H3) yield that almost everywhere on (a, b)
1 1 1 1 1 k;w D 6 M; D D f w w1 w0 2
which, by taking into account [16, Section 3], is equivalent to
Z t 1 1 1 k;w 1 1 1 k;w 6M D D f ðtÞ D D f ðxÞ w2 ðsÞds w w0 w1 w0 1 x and also to
M 6 lim inf
t!x
hðtÞ hðxÞ hðtÞ hðxÞ 6 lim sup 6 M; W 2 ðtÞ W 2 ðxÞ t!x W 2 ðtÞ W 2 ðxÞ
just observing that on (a, b), h ¼ w11 D1
1 w0
ð11Þ
Dk;w f and recalling the definition of W2. The proof is over directly from (9)–(11). h
P. Garrancho, D. Cárdenas-Morales / Applied Mathematics and Computation 217 (2010) 2676–2683
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It goes without saying that these two propositions can be applied to the particular approximation processes we have considered above. Some well-known statements are then recovered (see [16,6,9,10]), but further new results appear. We close the paper with two instances. Corollary 5. Let I = [0, 1), let w(t) = 1/(1 t)2 and f 2 Ck,w(I). Then for each x 2 (a, b)
2nðDk;w M n f ðxÞ Dk;w f ðxÞÞ ¼ oð1Þ; if and only if the restriction of f to the interval (a, b) belongs to the linear space spanned by the set of functions
1; uðtÞ; u2 ðtÞ; . . . ; uk1 ðtÞ; v 1 ðtÞ; v 2 ðtÞ ; where u(t) = t/(1 t),
t v1(t) = t and v 2 ðtÞ ¼ t logð1t Þ.
Corollary 6. Let I = [0, 1), let w(t) = 1/(1 + t)2 and f 2 C k;w ðIÞ ¼ C kB ðIÞ. Let (a, b) be a bounded subinterval of I and let M P 0. Then for each x 2 (a, b)
2njDk;w Hn f ðxÞ Dk;w f ðxÞj 6 M þ oð1Þ; if and only if, for almost every t 2 (a, b)
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