APPROXIMATION BY SAMPLING TYPE DISCRETE OPERATORS

In this paper, we deal with discrete operators of sampling type. It is known that this type of operators are related to generalized sampling series and they have important applications. In this work, using bounded and uniformly continuous functions we get general estimations under usual supremum norm with the help of summability method. We also study the degree of approximation with respect to suitable Lipschitz class of continuous functions. Finally, we give specific kernels which verify our kernel assumptions.


Introduction
Sampling type discrete operators have signi…cant applications in speech processing, medicine, economic forecasting, geophysics and etc. (see [2,11,12,13,14,15,16,25]). In this paper, we mainly inspired from the paper [1], where Angeloni and Vinti had some convergence results using discrete operators. The authors utilized from convergence in '-variation to get some convergence results in that work. Now, our aim is to get some approximations under usual supremum norm by generalizing them using Bell-type summability method. In this process, we use bounded and uniformly continuous functions on R: Furthermore, we study the rate of approximation for our main theorem using suitable Lipschitz class. Then, taking some appropriate kernels we also get more general case of generalized sampling series. Finally, we illustrate the kernels l k;w which satisfy our kernel assumptions.
Some notations and de…nitions are given below. k k l 1 denotes the l 1 norm, i.e., for a given u k : Z ! R; ku k k l 1 = P k2Z ju k j : By k k ; we mean the usual supremum norm on R: The space of bounded and uniformly continuous functions on R is shown by BU C (R). Let A = fA g 2N = f[a nw ]g 2N (n; w 2 N) be a family of in…nite matrices of real or complex numbers. Then, for a given sequence is called by A-transform of x; if the series is convergent for all n; 2 N: a nw x w = L uniformly in holds, we call "x is A-summable to L" and denote by A lim x = L (see [9]). A is called regular if for any lim k x k = L implies that A lim x = L ([9, 10]). A characterization for the regularity of the given method A is found by Bell in [10] such that A is regular , for each w 2 N; lim n!1 a nw = 0 (uniformly in ); for all n; 2 N; 1 P w=1 ja nw j < 1 and there exist integers N and M such that sup Throughout the paper, we will assume that A is regular with nonnegative real entries. We should note that Bell-type summability method consists many well-known methods such as Cesàro summability [18], almost convergence [23], order summability [19,20] and etc. It also allows us to increase the speed of convergence [21,27,29]. Some applications of Bell-type summability method are given in [3,4,5,6,7,8,17,22,24,28]. Now, we can de…ne our operator as follows: where f : R ! R is bounded and l k;w 2 l 1 (Z) is a family of discrete kernels for all w 2 N: Our aim is to prove the following general convergence result where f; : R ! R and generalized sampling series is a special case of (1.2).

Approximation in Usual Supremum Norm
In this section, we will prove our main approximation theorem. For this, we need the following conditions on the kernel of the corresponding operator.
(l 1 ) There exists a constant A > 0 such that sup n; 2N 1 P w=1 a nw k l k;w k l 1 = A < 1; Here, when A is taken the identity matrix, conditions (l 1 ) (l 3 ) reduce to the approximate identities given in [1].
The following lemma shows that (1.1) is well de…ned for all bounded functions.
1. If f is bounded on R and (l 1 ) holds, then kT n; (f )k < 1 for every n; 2 N: Proof. Since f is bounded, there exists a positive number M such that jf (x)j M for all x 2 R. Considering this with (l 1 ) ; we get and having supremum over x2R; we have for all n; 2 N; which shows that T n; maps from the space of bounded functions into itself. For the second part of the theorem, assume that and from a theorem of integration by series (see [26] for all n; 2 N: Proof. By the previous lemma it is clear that if f is bounded, then T n; (f ) is too. Now, let " > 0 be given and let jx yj < where corresponds to given " and f: Then, for all n; 2 N: The main approximation theorem is given below. Proof. From triangle inequality, it is possible to write that holds. In A 1 , we concentrate on the continuity of f: Since f is uniformly continuous, for every " > 0 we can …nd a > 0 such that whenever jx yj < : Then, for a …xed r it is easy to …nd a number w 1 satisfying r w < for all w > w 1 :Now, if we divide A 1 as follows 1 , from the regularity of A; one can …nd a number n 1 = n 1 (") such that : And from (l 3 ) ; we see that A 3 1 < 2 kf k " for su¢ ciently large n 2 N: Finally, it follows from (l 2 ) A 2 < kf k " yields for su¢ ciently large n 2 N: Hence, having supremum over x2R in the …rst inequality, we complete the proof.

Rate of Convergence
In this section we investigate the rate of approximation, and therefore we need the following Lipschitz class.
For any given > 0; de…ne Lip ( ) as follows: ) as t ! 0 means that, there exist ; N > 0 such that jf (t)j N jg (t)j for jtj < . Let be family of all functions : R + 0 ! R + 0 , such that (0) = 0, (t) > 0 for t > 0 and be continuous at t = 0: Now, for any …xed > 0 and 2 ; consider the following conditions: We obtain the following rates of approximations.  Finally, from (3.1) we conclude that 1=n)) as n ! 1 (uniformly in ).
Notice that, it is possible to …nd regular methods such that (3.4) is satis…ed, for instance, fC 1 g (Cesàro Matrix) and F (almost convergence matrix) which are given in Corollary 4.3.

Conclusions and Applications
In the present section, we give some applications of the operators of type (1.1). Let f : R ! R be given, and suppose that l k;w (k) ; that is, l k;w is not depending on w where : R ! R: Then, (1.1) reduces to which is in some cases equal to A transform of generalized sampling series, namely In this case (l 1 ) and (l 2 ) coincide with the following assumptions where on the other hand, (l 3 ) is clearly not satis…ed. But these two conditions are still enough to verify the following approximations (see also [1]). Proof. Considering (l 0 2 ) ; by the proof of the Theorem 2.3; we obtain the following inequalities Since P k2Z j (k)j < 1 from (l 0 1 ) ; for all " > 0 there exists a number r > 0 such that P jkj r j (k)j < " and hence, for su¢ ciently large n 2 N for su¢ ciently large n 2 N: Since " is arbitrary, the proof is completed.
Although T and S are similar, they are di¤erent in general. However, in some cases, they coincide (see [1]).  It is clear that operator (1.1) can be written as where T w is given by Considering (4.1) and (4.2), we get the following corollary. kT n (f ) f k = 0 i.e., T n (f ) is uniformly convergent to f, where T n (f ) is given in (4.2). Similar corollaries also hold for generalized sampling series Now, we will give a speci…c kernel of l k;w , which satis…es (l 1 ) (l 3 ) respectively: Take A = fC 1 g ; and then de…ne l k;w as follows: It is easy to see that (l 1 ) and (l 2 ) are satis…ed from the following calculations: ; 6) which is symmetric for k. But in the classical sense, l k;w does not satisfy the condition of (A1) since P k2Z l k;w 1 = ( 1) w + 1 is divergent. Therefore, our approximation is not trivial.
Acknowledgement. The author would like to thank to the reviewer(s) for reading the manuscript carefully.