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Saturation results for the truncated max-product sampling operators based on sinc and Fejer-type kernels.

Abstract

In this paper we obtain the saturation order, a strong localization result and local saturation results in the approximation of continuous positive functions by the truncated max-product sampling operators based on sinc (Wittaker-type) and Fejer-type kernels. The localization-type results obtained present important potential applications in signal theory.

Key words and phrases: Max-product truncated sampling operators, signal theory, sinc (Whitaker-type) kernel, Fejer-type kernel, saturation order, local inverse saturation result, localization result, local direct saturation result, Lipschitz function on subintervals.

2000 AMS Mathematics Subject Classification--Primary 94A20, 94A12, 41A36, 41A40; Secondary 41A27, 41A20.

1 Introduction

The sinc-approximation operators were first introduced and studied in [15], [5] and [20] under the name of cardinal function and of truncated cardinal function. Later on, these linear approximation operators were intensively studied in e.g. [1]-[3], [6]-[9], [13], [14], [16], [17], [18], [19] (see also the references cited there).

Based on the Open Problem 5.5.4, pp. 324-326 in [12], in a series of recent papers submitted for publication we have introduced and studied the so-called max-product operators attached to the Bernstein polynomials and to other linear Bernstein-type operators, like those of Favard-Szasz-Mirakjan operators (truncated and nontruncated case), Baskakov operators (truncated and nontruncated case), Meyer-Konig and Zeller operators and Bleimann-Butzer-Hahn operators.

This idea applied, for example, to the linear Bernstein operators [B.sub.n](f)(x) = [[SIGMA].sup.n.sub.k=0] [p.sub.n,k](x)f(k/n), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and f is positive, works as follows. Writing in the equivalent form [B.sub.n](f)(x) = [[SIGMA].sup.n.sub.k=0] [p.sub.n,k](x)f(k/)/[[SIGMA].sup.n.sub.k=0][p.sub.n,k](x) and then replacing the sum operator [SIGMA] by the maximum operator V, one obtains the nonlinear Bernstein operator of max-product kind

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the notation [V.sup.n.sub.k=0][p.sub.n,k](x) means max{[p.sub.n,k](x); k [member of] {0, ...,n}} and similarly for the numerator.

For this nonlinear max-product operator nice approximation and shape preserving properties were found in e.g. [4].

In the recent paper [10], Theorem 3.2 (iv), applying this idea to the classical Whittaker's cardinal series, we have obtained a Jackson-type estimate in terms of [[omega].sub.1][(f; 1/W).sub.R], in approximation of a continuous, positive and bounded function f on R, by the nonlinear max-product Whittaker sampling operator given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [phi] is the sinc kernel given by the formula [phi](t) = sinc(t) = sin([pi]t)/[pi]. Also, replacing above [phi](t) = 1/2 x [sinc.sup.2](t/2) (the Fejer-type kernel), in [11], Theorem 2.4, we get the same Jackson-type order [[omega].sub.1][(f; 1/W).sub.R] in the approximation of a continuous, positive and bounded function f on R. In order to be of interest, only functions f that are uniformly continuous on R must be considered, because they have the property that [[omega].sub.1][(f; t).sub.R] [right arrow] 0 as t [right arrow] 0.

It is worth noting that comparing with the classical linear sampling operators to which they were attached, when for f bounded and Lipschitz of order [alpha] [member of] (0, 1) the order of approximation is log(n)/[n.sup.[alpha]] (see Butzer [6] or Butzer-Stens [9]), these two kinds of nonlinear max-product sampling operators give the approximation order 1/[n.sup.[alpha]], which is an essential improvement.

Going further and applying the max-product idea to the truncated Whittaker series defined by (see e.g. Borel [5], Whittaker [20])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and to the truncated sampling series based on the Fejer's kernel and defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the very recent paper [11], we introduced the truncated max-product Whittaker operator given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the truncated max-product operator based on the Fej4r-type kernel, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for which the Jackson-type order [[omega].sub.1][(f; 1/n).sub.[0,[pi]]] in the approximation of f was obtained. It is worth noting that the situation is completely different in the case of approximation by their linear counterparts [W.sub.n] and [T.sub.n], when for the uniform convergence in compacts subintervals in (0, [pi]) and pointwise convergence in [0, [pi]], additional conditions are required like [lim.sub.n[right arrow][infinity]] [[omega].sub.1][(f; 1/n).sub.[0,[pi]]] x log(n) = 0 (see Trynin [19]), contrariwise divergence properties may appear (see Sklyarov [161).

Also, the interpolation property is preserved by the operators [W.sup.(M).sub.n] and [T.sup.(M).sub.n], i.e. we have [W.sup.(M).sub.n](f)(j/[pi]/n) = [T.sup.(M).sub.n](f)(j[pi]/n) = f(j[pi]/n), j [member of] {0, 1, ..., n} (see [11]).

Due to the better approximation properties, it is natural to look for a complete characterization of the approximation properties of the nonlinear maxproduct sampling operators [T.sup.(M).sub.n] and [W.sup.(M).sub.n]. In this sense, the goal of the present paper is to obtain for them the saturation order, a strong localization result and local inverse and local direct saturation results.

It is worth noting, for example, that the strong localization results in the Theorems 5.1 and 6.1 and the local shape preserving properties in the Corollaries 5.3 and 6.4, clearly show the important advantage that may have the truncated max-product operators [T.sup.(M).sub.n] (f) and [T.sup.(M).sub.n] (f) in the local representation with great accuracy of a continuous non-smooth signal f, if we compare them with the linear sampling operators [T.sub.n](f) and [W.sub.n](f).

The plan of the paper goes as follows : in Section 2 we deal with the saturation order for [T.sup.(M).sub.n], Section 3 deals with the saturation order for [W.sup.(M).sub.n], in Section 4 a local inverse saturation result for [T.sup.(M).sub.n] is presented, Section 5 contains a localization result and a local direct saturation result for [T.sup.(M).sub.n], while in Section 6 a localization result and local inverse and local direct saturation results for [W.sup.(M).sub.n] are presented.

Everywhere, by convention we take sin(O)/O = 1, which means that for every x = k[pi]/n, k [member of] {0, 1, ..., n}, we take sin(nx-k[pi])/nx-k[pi]] = 1.

2 The Saturation Order for the [T.sup.(M).sub.n] Operator

Firstly, we need some auxiliary results.

Lemma 2.1 Denoting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. See the proof of Lemma 4.3 in [11].

Lemma 2.2. For any function f : [0, [pi]] [right arrow] [R.sub.+], where [R.sub.+] : {x [member of] R; x [greater than or equal to]0}, and for all n [member of] N, n [greater than or equal to] 1, and j [member of] {0, 1,...,n}, j [less than or equal to] n/2, we have:

(i) [T.sup.(M).sub.n](f)(j[pi]/(n + 1)) [greater than or equal to] f(j[pi]/n);

(ii) [T.sup.(M).sub.n](f)(j[pi]/(n/1)) [greater than or equal to] f(j[pi]/n+1)).

Proof. (i) Firstly, by Lemma 2.1 we observe that for x [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, if j [less than or equal to] n/2 then it is easy to check that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1)). This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) Since j [less than or equal to] n/2, one can easily prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Therefore, by Lemma 2.1 we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since j [greater than or equal to] n/2 easily implies that (j + 1)[pi]/(n + 1) [member of] [j[pi]/n, (j + 1/2)[pi]/n] and j[pi]/n [member of] [(j + 1/2)[pi]/(n + 1), (j + 1)[pi]/(n + 1)], by similar reasonings with those for Lemma 2.2, we also get the following.

Lemma 2.3. For any function f : [0, [pi]] [right arrow] [R.sub.+] and for all n [member of] N, n [greater than or equal to] 1, and j [member of] {0, 1, ..., n}, j [greater than or equal to] n/2, we have:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We are now in position to determine the saturation order and the associated special class of functions for the truncated max-product operator T(M). Theorem 2.4. Denote C+[0, [pi]] = {f: [0, [pi]] [right arrow] [R.sub.+]; f continuous on [0, [pi]]} and [parallel]f[parallel] = sup{|f(x)]; x [member] [0, [pi]]}. Then for the max-product [T.sup.(M).sub.n] operator, the saturation order in C+[0, [pi]] is 1/n, that is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], implies that f is a constant function on [0, [pi]].

Proof. By hypothesis, there exists [a.sub.n] [member of] R, n [member of] N with the property [a.sub.n] [??] 0 as n [right arrow] + [infinity], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us choose arbitrary [epsilon] > 0. Since [a.sub.n] [??] 0 as n [right arrow] + [infinity], it follows that there exists [n.sub.0] [member of] N such that [a.sub.n] < [epsilon] for all n [member of] N, n [greater than or equal to] [n.sub.0]. Noting the above relation we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Then, from the uniform continuity of f it results the existence of [n.sub.1] [member of] N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We will obtain the desired conclusion by following the following steps: (A) we prove that f is constant on any interval [a, b] with 0 < a < b < [pi]/2; (B) we prove that f is constant on any interval [a, b] with [pi]/2 < a < b < 1. Indeed, if (A) holds then thanks to the continuity of f we easily obtain that f is constant on [0, [pi]/2]. Similarly, if (B) holds then we obtain that f is constant on [[pi]/2, 1]. Then, from the continuity of f it easily follows that f is constant on [0, [pi]]. So, we start by proving that (A) and (B) hold.

(A) Let us choose arbitrary a, b [member of] R such that 0 < a < b < [pi]/2. Further one, let [x.sub.0] and [y.sub.0] be the points where f attaints its minimum and maximum respectively on the interval [a, b]. Without any loss of generality we may suppose that [x.sub.0] [not equal to] [y.sub.0] (contrariwise it follows that f is constant on [a, b] and there is nothing to prove). We have two subcases: [A.sub.1]) [x.sub.0] < [y.sub.0] and [A.sub.2]) [x.sub.0] > [y.sub.0].

Subcase [A.sub.1]) Let us choose arbitrary n [member of] N with n > max{[n.sub.0], [n.sub.1], 2[pi]/([y.sub.0]- [x.sub.0])}. By relation (1) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, combining the inequality in Lemma 2.2 (i) with the above inequality, we get

f(j[pi]/n) - f(j[pi]/(n + 1)) [less than or equal to] [epsilon]/n for all j [member of] {0, 1, ..., n}, j [less than or equal to] n/2. (3)

Further one, let us choose [j.sub.1] [member of] {0, 1, ..., n-l} such that [j.sub.i][pi]/n [less than or equal to] [y.sub.0] [less than or equal to] ([j.sub.1]+l)[pi]/n and [x.sub.0] [less than or equal to] [j.sub.1][pi]. Note that there exists such an index [j.sub.1], because the previous inequalities are equivalent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] while the condition n 2[pi]/([y.sub.0] - [x.sub.0]) is equivalent to the condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Also, from [j.sub.1][pi]/n [less than or equal to] [y.sub.0] b < [pi]/2 it easily follows that [j.sub.1] [less than or equal to] n/2.

As a first consequence, from the relation (2) we obtain

|f([j.sub.1][pi]/n) - f([y.sub.0])| < [epsilon]. (4)

Then, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by [x.sub.0] > 0 and [x.sub.0] [less than or equal to] [j.sub.1][pi]/n it follows that there exists [l.sub.0] [member of] N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is worth noting here that indeed, the above lo cannot be equal to O, because if we would have [l.sub.0] = O, then we would obtain [j.sub.1][pi]/(n + 1) [less than or equal to] [x.sub.0] [less than or equal to] [x.sub.0] < [y.sub.0] [less than or equal to] ([j.sub.1] +l)[pi]/n [less than or equal to] ([j.sub.1] +2)[pi]/(n+1), which would imply [y.sub.0]-[x.sub.0] [less than ore qual to] 2[pi]/(n+ 1) < 2[pi]/n, in contradiction with the supposition that n > 2[pi]/([y.sub.0] - [x.sub.0]).

The inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and (2) also implies that

|f(([j.sub.1][pi]/(n + [l.sub.0])) - f([x.sub.0])| < [epsilon]. (5)

Since [j.sub.1] [less than or equal to] n/2, applying successively relation (3) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking the sum of all these inequalities we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, by relations (4)-(5) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and since 0 [less than or equal to] f([Y.sub.0]) - f([x.sub.0]), we obtain

0 [less than or equal to] f([Y.sub.0]) - f([x.sub.0]) [less than or equal to] 2[epsilon] + [l.sub.0][epsilon] / n. (6)

On the other hand, since 0 < [x.sub.0] [less than or equal to] [j.sub.1][pi]/(n+[l.sub.0]), after some simple calculations we get (note that [J.sub.1] [less than or equal to] n/2)

[l.sub.0] [less than or equal to] [j.sub.1][pi]/[X.sub.0] - n [less than or equal to] n([pi]/(2[X.sub.0]) - 1)

Using this information in relation (6) we obtain

0 [less than or equal to] f([Y.sub.0]) - f([x.sub.0]) [less than or equal to] [epsilon](2 + [pi]/(2[x.sub.0]) - 1)

where [epsilon] > 0 was chosen arbitrary. Therefore, passing to the limit for [epsilon] [??] 0 in the previous inequality, we obtain f([x.sub.0]) = f([Y.sub.0]) (here, it is important that [x.sub.0] > 0 ). Since on the interval [a, b] the maximum value and the minimum value of the function f coincide, we obtain that f is a constant function on the interval [a, b] and hence we obtained the desired conclusion for this case.

The proof of Subcase [A.sub.2]) is similar to that in the case of Subcase [A.sub.1]), which proves the Case (A).

(B) Let us choose arbitrary a, b [member of] N such that [pi]/2 < a < b < [pi] and further one, let [x.sub.0] and [y.sub.0] be the points where f attaints its minimum and maximum respectively on the interval [a, b]. Again we have two subcases [B.sub.1]) [x.sub.0] < [y.sub.0] and [B.sub.2]) [x.sub.0] > [y.sub.0], which by similar reasonings with those in the Case (A) implies that (B) holds.

Now, by the discussion just before the beginning of the case (A), we conclude that f is constant on the whole interval [0, [pi]].

Remark. Because it is easy to check that [T.sup.(M).sub.n] reproduces the constant functions in [C.sub.+] [0, [pi]], it follows that the saturation class in [C.sub.+] [0, [pi]] for [T.sup.(M).sub.n] is exactly the class of constant functions.

3 The Saturation Order for the [W.sup.(M).sub.n] Operator

Firstly, we need some auxiliary results.

Lemma 3.1 Denoting now [s.sub.n,k](x) sin(nx - [pi]k)/nx - [pi]k we, for any J [member of] {0, ... n} we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. See the proof of Lemma 3.3 in [11].

Lemma 3.2. For any function f : [0, [pi]] [right arrow] [R.sub.+] and for all n [member of] N, n [greater than or equal to] 1, and j [member of] {0, 1,...,n}, j [less than or equal to] n/2, we have:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The proof is identical with the proof of Lemma 2.2.

Lemma 3.3. For any function f: [0, [pi]] [right arrow] [R.sub.+] and for all n [member of] N, n [greater than or equal to] 1, and j [member of] {0, 1, ..., n}, j [greater than or equal to] n/2, we have:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The proof is identical with the proof of Lemma 2.3.

Since Theorem 2.4 is a direct consequence of the Lemmas 2.1-2.3 and in its proof does not intervene at all the form of the kernel [s.sub.n,k](x), as a consequence of the Lemmas 3.1-3.3, analogously we can prove the following saturation result for the truncated max-product operator [W.sup.(M).sub.n].

Theorem 3.4. For the max-product [W.sup.(M).sub.n] operator, the saturation order in [C.sub.+] [0,[pi]] is 1/n, that is ||[W.sup.(M).sub.n] (f) - f|| = o (1/n), implies that f is a constant function on [0, [pi]].

Remark. Because it is easy to see that [W.sup.(M).sub.n] reproduces the constant functions in [C.sub.+] [0, [pi]], it follows that the saturation class for [W.sup.(M).sub.n] in [C.sub.+] [0, [pi]] is exactly the class of constant functions.

4 Local Inverse Saturation Result for the [T.sup.(M).sub.n] Operator

In this section we present the following local inverse saturation result for the operator [T.sup.(M).sub.n].

Theorem 4.1. Let f: [0,[pi]] [right arrow] [R.sub.+] and 0 < [alpha] < [beta] < [pi] be such that f is continuous on [[alpha],[beta]]. If there exists a constant M > 0 (independent of n but depending on f, [alpha] and [beta]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [f|.sub.[[alpha],[beta]]] [member of] Lip[[alpha],[beta]] , that is f is a Lipschitz function on [[alpha],[beta]]. Here [||f|||.sub.[[alpha],[beta]]] = sup {|f(x)|;x[member of] [[alpha],beta]]} and

Lip [alpha], beta]] = {g:[alpha],[beta]] [right arrow] R; |g(x) - g(y)| [less than or equal to] C|x - y|, for all x,y [member of] [[alpha], [beta]]}.

The proof of Theorem 4.1 requires the following three lemmas.

Lemma 4.2. Let f: [0, [pi]] [right arrow] R, n [member of] N and 0 < [alpha] < [beta] [less than or equal to] [pi]/2 be fixed, such that f is continuous on [[alpha], [beta]]. Also, denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. We prove only the direct implication since the converse one is immediate. Since f is continuous on the interval [[alpha],[beta]], it easily follows that for each n [member of] N, n [greater than or equal to] 2, [pi]/n [less than or equal to] [beta] - [alpha], there exist [x.sub.n], [y.sub.n] [member of] [[alpha],[beta]] satisfying [absolute value of [x.sub.n] - [y.sub.n]] [less than or equal to] [pi]/n and [w.sub.1] [(f, [pi]/n).sub.[[alpha],[beta]]] = [absolute value of f([x.sub.n]) - f([y.sub.n])]]. Without any loss of generality, we may suppose that [x.sub.n] [not equal to] [y.sub.n] and [x.sub.n] < [y.sub.n], for all n [member of] N.

Let us consider the sequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us fix n [member of] N. Since f is uniformly continuous on [[alpha], [beta]], it follows that there exists m [member of] N such that for all x, y [member of] [[alpha],[beta]] satisfying [absolute value of x - y] [less than or equal to] [pi]/m we have [absolute value of f(x) - f(y)] [less than or equal to] 1/n. In addition, we may choose sufficiently large m [member of] N such that [y.sub.n] - [x.sub.n] > 2[pi]/m, that is m > 2[pi]/([y.sub.n] - [X.sub.n]).

Since 0 < [alpha] < [y.sub.n] [less than or equal to] [beta] < [pi]/2, clearly there exists j [member of] {1,..., m-1} (depending on m and n) such that j[pi]/m [less than or equal to] [y.sub.n] [less than or equal to] (j + 1) [pi]/m.

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and since [x.sub.n] [greater than or equal to] [alpha] > 0, it results the existence of [l.sub.0] [member of] N (depending on j and m) such that j[pi]/(m + [l.sub.0] + 1) [less than or equal to] [x.sub.n] [less than or equal to] j[pi]/(m + [l.sub.0]).

By the inequalities [x.sub.n] [less than or equal to] j[pi]/(m + [l.sub.0]) < j[pi]/m [less than or equal to] [y.sub.n], we get

[absolute value of f([x.sub.n]) - f([y.sub.n])]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where p [member of] {0, 1, ..., [l.sub.0] - 1} is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, we observe that max {[absolute value of j[pi]/(m + [l.sub.0]) - [x.sub.n]], [absolute value of j[pi]/(m - [y.sub.n]]} [less than or equal to] [pi]/m, which implies |f([x.sub.n]) - f(j[pi]/ (m = [l.sub.0]))| [less than or equal to 1/n and |f([j[pi]/m) - f([y.sub.n]| [less than or equal to] 1/n. We thus obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

By the inequalities [x.sub.n] [less than or equal to] j[pi]/ ([m + [l.sub.0]) [less than or equal to] j[pi]m [less than or equal to] [y.sub.n] we get j[pi]/m - j[pi]/(m + [l.sub.0] [less than or equal to] and this implies j[pi][l.sub.0])) [less than or equal to] [pi]/n and then [l.sub.0] [less than or equal to] m/j * (m + [l.sub.0])/n [less than or equal to][pi]/[alpha]* (m + [l.sub.0])/n. (Here we used that [alpha] [less than or equal to] [x.sub.n] < j[pi]/m).

Then, by the inequalities 0 < [alpha] [less than or equal to] [x.sub.n] [less than or equal to] A : = j[pi]/(m + [l.sub.0]) [less than or equal to] B := j[pi]/m [less than or equal to] [y.sub.n] [less than or equal to] [beta] we easily get B/A [less than or equal to] [beta]/[alpha], which immediately implies j[pi]/(m + [l.sub.0]) [greater than or equal to] j[pi]/m x [alpha]/[beta]. From here we get m + [l.sub.0] [less than or equal to] m ([beta]/[alpha]), that is [l.sub.0] [less than or equal to]_ m([beta]/[alpha] - 1). Replacing this last inequality in the inequality [l.sub.0] [less than or equal to] [pi]/[alpha] x (m + [l.sub.0])/n just proved above, we conclude that [l.sub.0] [less than or equal to] [pi][beta]/[[alpha].sup.2] * m/n.

Replacing now in relation (7) and then multiplying with n, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and clearly this implies that [a.sub.n] [less than or equal to] 2 + [pi][beta]/[[alpha].sup.2] x [M.sub.m+p] ([alpha], [beta]). Summarizing, for any n [member of] N there exist m + p [member of] N such that [a.sub.n] [less than or equal to] [pi][beta]//[[alpha].sup.2] * [b.sub.m+p] + 2. Since m > 2[pi]/([y.sub.n] - [x.sub.n]) and [y.sub.n] - [x.sub.n] < [pi]/n, we get m > 2n. Therefore, by lim sup [a.sub.n] = [infinity], it easily follows that lim sup [b.sub.n] = [infinity] and the lemma is proved. n [right arrow][infinity]

By similar reasoning with those for the Lemma 4.2, we also get he following

Lemma 4.3. Let f: [0, [pi]] [right arrow] R, n [member of] N and [pi]/2 [less than or equal to] [alpha] < [beta] < [pi] be fixed, such that f is continuous on [[alpha],[beta]]. Also, denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 4.4. Let f: [0, [pi]] [right arrow] [R.sub.+] and 0 < [alpha] < [beta] < [pi] be such that f continuous on [[alpha],[beta]]. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [parallel]f[parallel] = sup{[absolute value of f(x)];x [member of] [[alpha],[beta]]}.

Proof. If [alpha] < [pi]/2 < [beta] then by the hypothesis it is elementary to prove that either [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Therefore, without any loss of generality we may suppose that we have only two cases: (i) 0 < [alpha] < [beta] [less than or equal to] [pi]/2 and (ii) [pi]/2 [less than or equal to] [alpha] < [beta] < [pi].

Case (i) For fixed n [member of] N with n [greater than or equal to] ([pi] + [alpha])/([beta] - [alpha]), let us choose k(n) [member of] {1, ..., n} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that such an index k(n) exists, because the inequalities [alpha] [less than or equal to] k(n)[pi]/(n+1) [less than or equal to] k(n)[pi]/n [less than or equal to] [beta] imply [alpha](n + 1)/[pi] [less than or equal to] k(n) [less than or equal to] [beta]n/[pi], where [beta]n/[pi] - [alpha](n + 1)/[pi] [greater than or equal to] 1.

Since [beta] [less than or equal to] [pi]/2, it results that k(n) [less than or equal to] n/2 and hence we can use the conclusion of Lemma 2.2. This means that we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If f(k(n)[pi]/n) [greater than or equal to] f(k(n)[pi]/(n + 1)) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this implies

If f(k(n)[pi]/n) < f(k(n)[pi]/(n + 1)) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In conclusion, for any n [member of] N with n [greater than or equal to] [pi] ([beta] - [alpha]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since by Lemma 4.3 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it easily follows now that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof of the Case (if) is absolutely similar, which proves the lemma.

Now, we are in position to prove Theorem 4.1.

Proof of Theorem 4.1. Firstly. it is immediate that f is a Lipschitz function on [a, 2] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, by the hypothesis it follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all n [member of] N. Supposing that f is not a Lipschitz function on [[alpha], [beta]], by the above considerations it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. But then, by Lemma 4.4 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a contradiction. The theorem is proved.

5 Localization and Local Direct Saturation Result for the [T.sup.(M).sub.n] Operator

In this section we obtain a local direct saturation result as a consequence of the following strong localization theorem, of independent importance.

Theorem 5.1. Let f, g: [0, [pi]] [right arrow] [R.sub.+] be both bounded on [0, [pi]] with strictly positive lower bounds and suppose that there exists a, b [member of] [0, [pi]], 0 < a < b < [pi] such that f(x) = g(x) for all x [member of] [a, b]. Then for all c, d [member of] [a, b] satisfying a < c < d < b there exists [??] [member of] N which depends only on f, g, a, b, e, d such that [T.sup.(M).sub.n] (f)(x) = [T.sup.(M).sub.n] (g)(x) for all x [member of] [c, d] and n [member of] N, n [greater than or equal to] [??].

Proof. Let us choose arbitrary x [member of] [c, d] and for each n [member of] N let [j.sub.x] [member of] {0, 1, ..., n} ([j.sub.x] depends on n too, but there is no need at all to complicate on the notations) be such that x [member of] [[j.sub.x][pi]/n, ([j.sub.x] + 1)[pi]/n]. Then we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where for k [member of] {0,1, ..., n} we have [s.sub.n,k](x) - [sin.sup.2](nx-k[pi])/[(nx-k[pi]).sup.2] Since x [member of] [c, d] [intersection] [jx[pi]/n, ([j.sub.x] + 1)[pi]/n] and since a < e < d < b it is immediate that for n [greater than or equal to] [n.sub.0] where [n.sub.0] is chosen such that [pi]/[n.sub.0] < min{c - a, d - b}, we obtain a < [j.sub.x][pi]/n < b which gives na/[pi] < [j.sub.x] < nb/[pi] for all n [greater than or equal to] [n.sub.0] (indeed, if we would suppose that there exists n > [n.sub.0] which does not satisfy the previous double inequalities, then we would easily get a contradiction).

It is important to notice here that no does not depend on x. From the previous inequality it follows that if n [greater than or equal to] [n.sub.0] then for any x [member of] [c, d] there exists [[alpha].sub.x] [member of] [a, b] such that [j.sub.x] = n[[alpha].sub.x]/[pi].

In what follows, it will serve to our purpose to use the sequence (an)n>1, [a.sub.n] = [square root of n]. For this sequence there exists [n.sub.1] [member of] N such that na/[pi] - [a.sub.n] > 0 for all n [greater than or equal to] [n.sub.1].

Our intention is to prove as an intermediate result, that there exits an absolute constant [N.sub.0] [member of] N independent of x [member of] [c, d] such that for any n [greater than or equal to] [N.sub.0] and x [member of] [c, d] we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In order to obtain this conclusion, for n [greater than or equal to] max {[n.sub.0], [n.sub.1]} let us choose k [member of] {0, 1, ..., n}\In,x. We have two cases: i) k + [a.sub.n] < [j.sub.x] and ii) [j.sub.x] + [a.sub.n] < k.

Case i) We have two subcases: [i.sub.a]) x [member of] [[j.sub.x][pi]/n, ([j.sub.x] + 1/2)[pi]/n] and [i.sub.b]) x [member of] [([j.sub.x] + 1/2)[pi]/n, ([j.sub.x] + 1)[pi]/n].

Subcase [i.sub.a]) Since x [member of] [[j.sub.x][pi]/n, ([j.sub.x] + 1/2)[pi]/n], we observe that (nx-[j.sub.x][pi]) [member of] [0, [pi]/2] and by the well-known property sin(x) [greater than or equal to] 2x/[pi], x [member of] [0, [pi]/2], it results that [S.sub.n,jx] (x) [greater than or equal to] 4/[[pi].sup.2]. This implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since nx [greater than or equal to] [j.sub.x][pi] [greater than or equal to] k[pi] it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, denoting the infimum and the supremum of f on [0, 1] with mf and MI, respectively (according to the hypotheses they are strictly positive real numbers), we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it follows that there exists [n.sub.2] [member of] N, [n.sub.2] > max{[n.sub.0], [n.sub.1]} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x [member of] [c, d], n [greater than or equal to] [n.sub.2] and k [member of] {0,1, ..., n}, k < [j.sub.x] - [a.sub.n] (as k [not member of] [I.sub.n,x]). In addition, it is important to notice that [n.sub.2] does not depend on x E [c, d], but of course it depends on f.

Subcase [i.sub.b]) It is easy to check that x [member of] [([j.sub.x] + 1/2)[pi])/n, ([j.sub.x] + 1)[pi]/n] implies (nx - ([j.sub.x] + 1)[pi]) [member of] [-[pi]/2, 0]. Therefore, reasoning as in case [i.sub.a]) (because of [sin.sup.2]) we obtain that for sufficiently large n we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Summarizing, we conclude that there exists an absolute constant N1 E N which depends only on a, b, c, d, f such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] [c, d], n [greater than or equal to] [N.sub.1] and k E {0, 1,...,n}, k < [j.sub.x] - [a.sub.n].

Case ii) The proof is identical with the proof of the above Case i) and therefore we conclude that there exists an absolute constant [N.sub.2] [member of] N which depends only on a, b, c, d, f such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] [c, d], n [greater than or equal to] [N.sub.2] and k [member of] {0, 1, ...,n}, k > [j.sub.x] + [a.sub.n].

Analyzing the results obtained in cases i)-ii), it results that for all x [member of] [c, d], n [greater than or equal to] [N.sub.0], [N.sub.0] = max{[N.sub.1], [N.sub.2]} and k [member of] {0, 1, ..., n}, with k < [j.sub.x] - [a.sub.n] or k > [j.sub.x] + [a.sub.n], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In conclusion, we obtain our preliminary result, that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [I.sub.n,x] = {k [member of] {0, 1, ...,n}: [j.sub.x] - [a.sub.n] [less than or equal to] k [less than or equal to] [j.sub.x] + [a.sub.n]}.

Next, let us choose arbitrary x [member of] [c, d] and n [member of] N so that n [greater than or equal to] [N.sub.0]. If there exists k [member of] [I.sub.n,x] such that k[pi]/n [not member of] [c, d] then we distinguish two cases. Either k[pi]/n < c or k[pi]/n > d. In the first case we observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it results that for sufficiently large n we necessarily have ([a.sub.n]+1)[pi]/n < c - a which clearly implies that k[pi]/n [member of] [a, c]. In the same manner, when k[pi]/n > d, for sufficiently large n we necessarily have k[pi]/n [member of] [d, b].

Summarizing, there exists N1 E N independent of any x E [c, d], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and in addition for any x [member of] [c, d], n [greater than or equal to] [??]1 and k [member of] [I.sub.n,x], we have k[pi]/n [member of] [a, b].

Also, it is easy to check that N1 depends only on a, b, c, d, f. We thus obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and in addition for any x [member of] [c, d], n [greater than or equal to] [[??].sub.1] and k [member of] [I.sub.n,x], we have k[pi]/n [member of] [a, b].

Reasoning for the function g exactly as in the case of the function f, it follows that there exists [[??].sub.2] [member of] N which depends only on a, b, c, d, g such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and in addition for any x Z [c, d], n [greater than or equal to] [[??].sub.2] and k [member of] [I.sub.n,x], we have k[pi]/n [member of] [a, b]. Taking [??] = max{[[??].sub.1], [[??].sub.2]}, we easily obtain the desired conclusion.

As a consequence of the localization result in Theorem 5.1 we present a local shape preserving property.

Corollary 5.2. Let f: [0, [pi]] [right arrow] [R.sub.+] be bounded on [0, [pi]] with strictly positive lower bound and suppose that there exists a, b [member of] [0, [pi]], 0 < a < b < [pi] such that f is constant on [a, b] with the constant value [alpha] > O. Then for any c, d E [a, b] with a < c < d < b, there exists [??] [member of] N which depends only on a, b, c, d and f such that [T.sup.(M).sub.n] (f)(x) = [alpha] for all x C [c,d] and n E N, n > [??].

Proof. Let g : [0,1] [right arrow] [R.sub.+] be given by g(x) = [alpha] > 0 for all x [member of] [0, [pi]]. Since f(x) = g(x) for all x [member of] In, b] and since obviously [T.sup.(M).sub.n](g)(x) = [alpha] for all x [member of] [0, [pi]], by Theorem 5.1 we easily obtain the desired conclusion.

At the end of this section, as a consequence of the localization result in Theorem 5.1 we present a direct result, as follows.

Corollary 5.3. Let f: [0, [pi]] [right arrow] [R.sub.+] be bounded on [0, [pi]] with the lower bound strictly positive and suppose that there exists a, b [member of] [0, [pi]], 0 < a < b < [pi] and the constant Co which depends only on a and b, such that

|f(x) - f(Y)| [less than or equal to] [C.sub.0]|x - Y| for all x [member of] [a, b], (8)

that is [f|.sub.[a,b]] [member of] Lip[a, b]. Then, for any c, d [member of] [0,1] satisfying a < e < d < b, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the constant C depends only on f and the values a, b, c, d.

Proof. Let us define the function F : [0, [pi]] [right arrow] R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The hypothesis imply that F is continuous strictly positive and, according to Theorem 4.4 in [11] it results that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since by the definition of F we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and since by the relation (8) it easily follows [[omega].sub.1][(f, 1/n).sub.[a,b]] [less than or equal to] [C.sub.0]/n, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let us choose arbitrary c, d E [a, b] such that a < c < d < b. Then, by Theorem 5.1 (applicable to F and f) it results the existence of [??] [member of] N which depends only on a, b, c, d, f, F such that [T.sup.(M).sub.n] (F)(x) = [T.sup.(M).sub.n] (f)(x) for all x [member of] [c, d]. But since actually the function F depends on the function f, it is clear that in fact [??] depends only on a, b, e, d and f.

Therefore, for arbitrary x [member of] [c, d] and n [member of] N with n [greater than or equal to] [??] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.0] and [??] depend only on a, b, c, d and f.

Now, denoting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we finally obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with C = max{[C.sub.0], [C.sub.1]} depending only on a,b,c,d and f. This proves the corollary.

Remark. The results in Theorem 5.1 and Corollary 5.2 show the nice property of the truncated max-product operator [T.sup.(M).sub.n] to reproduce locally with great accuracy the graph of non-smooth strictly positive, bounded and locally continuous function f, with important implications in the signals' representation. For example, if on a subinterval [c, d] the signal f is zero, we may consider there to be constant equal to an arbitrary small [epsilon] > 0 and then, the above Corollary 5.2 proves that for sufficiently large n, [T.sup.(M).sub.n] (f)(x) = [epsilon] for all x [member of] [c, d]. Note that the truncated linear sampling operator [T.sub.n] does not have these properties.

6 Localization, Local Inverse and Local Direct Saturation Results for the [W.sup.(M).sub.n] Operator

In this section we present a local inverse saturation result, a strong localization result and a local direct saturation result for the [W.sup.(M).sub.n] operator.

In this sense, let us make the following remarks. Firstly, note that the proof of Theorem 4.1 follows from Lemma 4.4 and the proof of Lemma 4.4 follows from the Lemmas 4.2-4.3 (which are independent of any kind of operator) and Lemmas 2.2-2.3. Since the Lemmas 2.2-2.3 are identical with Lemmas 3.2-3.3 (making abstraction of the kernels), in the case of the truncated max-product operator based on Whittaker-type kernel we easily obtain corresponding results for Theorems 4.1 and 5.1 and for Corollaries 5.2 and 5.3, which are the main results of the previous two sections.

In conclusion, for [W.sup.(M).sub.n] we can state the following local inverse saturation result, localization result and local direct saturation result, respectively.

Theorem 6.1. Let f : [0, [pi]] [right arrow] [R.sub.+] and 0 < [alpha] < [beta]< [pi]be such that f is continuous on [[alpha], [beta]]. If there exists a constant M > 0 (independent of n but depending on f, [alpha] and [beta]) such that

[[parallel][W.sup.(M).sub.n](f) - f[[parallel].sub.[[alpha],[beta]]] [less than or equal to] M/n, for all n [member of]

then f|[[alpha],[beta]] [member of] N, Lip [[alpha],[beta]] , that is f is a Lipschitz function on [[alpha],[beta]].

Theorem 6.2. Let f, g : [0,[pi]] [right arrow] [R.sub.+] be both bounded on [0,[pi]] with strictly positive lower bounds and suppose that there exists a,b [member of] [0,[pi]], 0 < a < b < [pi] such that f(x) = g{x) for all x [member of] [a,b]. Then for all c,d [member of] [a,b] satisfying a < c < d < b there exists n [member of] N which depends only on f, g, a, b, c, d such that [W.sup.(M).sub.n] (f)(x) = [W.sup.(M).sub.n] (g)(x) for all x [member of] [c,d] and n [member of] N, n [greater than or equal to] n.

Corollary 6.3. Let f : [0,[pi]] [right arrow] . R+ be bounded on [0,[pi]] with strictly positive lower bound and suppose that there exists a, b [member of] [0, [pi]], 0 < a < b < [pi] and the constant Co which depends only on a and b, such that

|(f)(x) - f(y)| [less than or equal to] [C.sub.0] |x - y| for all x [member of] [a,b],

that is [f|.sub.[a,b]] [member of] Lip([a, b]). Then, for any c, d [member of] [0, 1] satisfying a < c < d < b, we have

[w.sup.(M).sub.n] (f)(x) - f(x) [less than or equal to] C/n for all n [member of] N and [pi] [member of] [c,d],

where the constant C depends only on f and the values a, b, c, d.

Corollary 6.4. Let f : [0,[pi]] [right arrow] [R.sub.+] be bounded on [0,[pi]] with strictly positive lower bound and suppose that there exists a, b [member of] [0,[pi]], 0 < a < b < [pi] such that f is constant on [a, b] with the constant value a [Greater than] O. Then for any c, d [member of] [a, b] with a < c < d < b, there exists [??] [member of] N which depends only on a,b,c,d and f such that [W.sup.(M).sub.n](f)](x)= [alpha] for all x [member of] [c,d] and n [member] N, n [greater than or equal to] [??].

Remarks. 1) Similar to the case of the operator [T.sub.(M).sub.n], the results in the Theorem 6.2 and Corollary 6.4 show the nice properties of the truncated max-product operator [W.sup.(M).sub.n] to reproduce locally with great accuracy the graph of a non-smooth strictly positive, bounded and locally continuous function f, with important potential applications in the signals' representation. Note that the truncated linear sampling operator [W.sub.n] does not have these properties.

2) Although the max-product operators [W.sub.(M).sub.n] and [T.sub.(M).sub.n] studied by this paper are well-defined only for bounded functions with values in [R.sub.+], they can easily be used to generate max-product type operators for the approximation of functions of arbitrary sign f : [0,[pi]] [right arrow]s R, as follows : if a < min{f(x); x [member of] [0,[pi]]}, then the new operators [p.sup.(M).sub.n](f)](x) = W(nM) (f - a)(x) + a and [Q.sup.(M).sub.n](f)(x) = [T.sup.(M).sub.n] (f - a)(x) + a, approximate f with the same Jackson type order [[omega].sub.1] [(f; 1/n).sub.[0,[pi]] and keep the interpolation properties [P.sup.(M).sub.n](f)](j[pi]r/n)] = [Q.sup.(M)sub.n](j[pi]r/n) = f(j[pi]/n), j = O, 1, ..., n.

ACKNOWLEDGEMENT

The work of both authors was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-IDPCE-2011-3-0861.

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Lucian Coroianu and Sorin G. Gal

Department of Mathematics and Computer Science,

The University of Oradea

Universitatii 1, 410087

Oradea, Romania

lcoroianu@uoradea.ro and galso@uoradea.ro
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Author:Coroianu, Lucian; Gal, Sorin G.
Publication:Sampling Theory in Signal and Image Processing
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Date:Jan 1, 2012
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