MULTIPLIER ExTENSION AND SAMPLIN G THEOREM ON HARDY SPACES

Extension by integer translates of compactly supported function for multiplier spaces on periodic Hardy spaces to multiplier spaces on Hardy spaces is given . Shannon sampling theorem is extende d to Hardy spaces. 1 . Introduction and statement of results The purpose of this paper is to establish a natural extension from multiplier spaces M(p) on periodic Hardy spaces HP (T) to multiplie r spaces M(p) on Hardy spaces HP (R) by integer translates of a function ~ and to extend Shannon sampling theorem to Hardy spaces. It is the continuation of [13] on stability of integer translates of a function bu t with different interest . In [13], the following stability problem of intege r transiates of 0 ( 1 ) c-l llf IIHP(Z) E f(n)O(x — n ) nEZ Cllf11HP(Z ) HP (R ) was considered which arises in the interpolation of sequences by functions and plays an important role in multiresolution analysis, where o C p < oo, f = {f(n)}Ez is a tempered sequence, HP (Z) and HP (R) denote s Hardy spaces on Z and R respectively, and Z is the set of integers . A natural replacement of the norm in (1) when p = oo is the norm as multiplier operator on HP(T) and HP (R) respectively, which is an This project is partially supported by Postdoctoral Fellowship Foundation of China and Natural Sciences Foundation of China .