On splitting functions in Paley-Wiener space

Abstract

We consider the problem on splitting functions in the Paley-Wiener space into the sum of two ones, each being "large" only in their domain.

Hilbert transform on $W_{\sigma}^1$

“…The main result. The decomposition problem for functions in Paley-Wiener space W 1 σ into the sum of two functions each of them characterized by the module which is "big" only in the upper or lower half-planes was investigated by B. V. Vynnytskyi and his followers (see [7], [8]). R. S. Yulmukhametov in [9] solved the problem of decomposition (splitting) into the product of two functions for some subspace of the Paley-Wiener space.…”

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

“…Especially important are the problems of splitting functions into the product or sum of two functions separated by the corresponding domains of the complex plane. For instance, the decomposition problem of holomorphic functions from the Paley-Wiener and Hardy spaces into the sum of two functions, specified by their growth degree, was actively studied in [1][2][3], solving the important filter identification problem [4][5][6][7][8] subject to electrical, optical, and acoustical signals. Especially, mathematically challenging the factorization problem of holomorphic functions in Bergman space with assigned growth degree properties proved to have many applications [9][10][11][12], inspired by recent advances in wavelet, co-orbit, control, and coherent state representation theories.…”

“…In our work, we are mainly interested both in the additive decomposition of holomorphic functions of Bergman spaces on connected domains, subject to the related growth degree problem on suitably separated subdomains, inspired by recent studies in [2,3], and in the Bergman space direct sum invariant decompositions [13][14][15][16] with respect to isometry operators, defined by means of some invariant measurable mappings, especially ergodic.…”

“…the upper and down intersections of the region D with the half-planes +  and −  of the complex plane , the representation (5) and (6) on the domains D + and Dcan be naturally analytically continued from D + and Drespectively, on the whole region D, giving rise to the functions 2 and satisfying, by construction, the expected[2,3] conditions: the reduced function | D ϕ − is "small" on Dsubject to the wage function needs to remark that the functions : D…”