Wiener's Tauberian Theorem Research Articles

A Remark on Wiener's Tauberian Theorem

A Remark on Wiener's Tauberian Theorem

Two notes on spectral synthesis for discrete Abelian groups

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

Distribution Proof of Wiener's Tauberian Theorem

Distribution Proof of Wiener's Tauberian Theorem

Extensions of Wiener's Tauberian theorem for positive measures

Extensions of Wiener's Tauberian theorem for positive measures

Comments on Wiener's Tauberian Theorems

Comments on Wiener's Tauberian Theorems

Notes on Fourier analysis(XXXVI): On certain applications of Wiener's Tauberian theorems

Notes on Fourier analysis(XXXVI): On certain applications of Wiener's Tauberian theorems

An Extension of Wiener's General Tauberian Theorem

An Extension of Wiener's General Tauberian Theorem