Approximation by Shannon sampling operators in terms of an averaged modulus of smoothness Andi Kivinukk Dept. of Mathematics Tallinn University Narva mnt 25 Tallinn 10120, Estonia Email: andik@tlu.ee Gert Tamberg Dept. of Mathematics Tallinn University of Technology Ehitajate tee 5 Tallinn 19086, Estonia Email: gtamberg@staff.ttu.ee Abstract—The aim of this paper is to study the approximation properties of generalized sampling operators in L p (R)-space in terms of an averaged modulus of smoothness. I. I NTRODUCTION For the uniformly continuous and bounded functions f ∈ C(R) the generalized sampling series are given by (t ∈ R; w> 0) (S w f )(t) := ∞ X k=-∞ f ( k w )s(wt - k), (1) where the condition for the operator S w : C(R) → C(R) to be well-defined is ∞ X k=-∞ |s(u - k)| < ∞ (u ∈ R), (2) the absolute convergence being uniform on compact intervals of R. If the kernel function is s(t) = sinc(t) := sin πt πt , we get the classical (Whittaker-Kotel’nikov-)Shannon opera- tor, (S sinc w f )(t) := ∞ X k=-∞ f ( k w ) sinc(wt - k). A systematic study of sampling operators (1) for arbitrary kernel functions s with (2) was initiated at RWTH Aachen by P. L. Butzer and his students since 1977 (see [1], [2], [3] and references cited there). Since in practice signals are however often discontinuous, this paper is concerned with the convergence of S w f to f in the L p (R)-norm for 1 6 p< ∞, the classical modu- lus of continuity being replaced by the averaged modulus of smoothness τ k (f ;1/w)p. For the classical (Whittaker- Kotel’nikov-Shannon) operator this approach was introduced by P. L. Butzer, C. Bardaro, R. Stens and G. Vinti (2006) in [4] (see also [5]) for 1 <p< ∞. For time-limited kernels s this approach was applied for 1 6 p< ∞ in [6] and [7]. In this paper we use this approach for band-limited kernels for 1 6 p< ∞. In this paper we study an even band-limited kernel s, defined by an even window function λ ∈ C [-1,1] , λ(0) = 1, λ(u)=0 (|u| > 1) by the equality s(t) := s(λ; t) := 1 Z 0 λ(u) cos(πtu) du. (3) We first used the band-limited kernel in general form (3) in [8], see also [9], [10]. We studied the generalized sampling operators S W : C(R) → C(R) with the kernels in form (3) in [11]-[12]. We computed exact values of operator norms kS w k := sup kf k C 61 kS w f k C = sup u∈R ∞ X k=-∞ |s(u - k)| (4) and estimated the order of approximation in terms of the classical modulus of smoothness. In this paper we give similar results for L p (R) norm in terms of the averaged modulus of smoothness. The main result of this paper, Theorem 2, was proved for f ∈ C(R) in [11]. II. PRELIMINARY RESULTS A. Averaged modulus of smoothness In this section we follow the approach of Butzer et al [4] of convergence problems of Shannon sampling series in a suitable subspace of L p (R). Let f ∈ M (R) be measurable and bounded on R, and δ > 0. The k-th averaged τ -modulus of smoothness for 1 6 p 6 ∞ is defined as ([4], Def. 1) τ k (f ; δ)p := kω k (f ; ·; δ)k p , (5) where ω k (f ; t; δ) is a local modulus of smoothness of order k ∈ N at t ∈ R, ω k (f ; t; δ) := := sup{|Δ k h f (x)|; x, x + kh ∈ [t - kδ 2 ,t + kδ 2 ]}, where the classical finite forward difference is given by Δ k h f (x)= k X ‘=0 (-1) k-‘  k ‘  f (x + ‘)h). (6) Proceedings of the 10th International Conference on Sampling Theory and Applications 540