Wiener's lemma

• Main Author: Fredriksson, Henrik
• Format: Others
• Language: English
• Published: Linnéuniversitetet, Institutionen för matematik (MA) 2013
• Subjects: Wiener's lemma; Banach algebra; quasi-norm; p-norm; submultiplicative weight function
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-27270

In this thesis we study Wiener’s lemma. The classical version of the lemma, whose realm is a Banach algebra, asserts that the pointwise inverse of a nonzero function with absolutely convergent Fourier expansion, also possesses an absolutely convergent Fourier expansion. The main purpose of this thesis is to investigate the validity inalgebras endowed with a quasi-norm or a p-norm.As a warmup, we prove the classical version of Wiener’s lemma using elemen-tary analysis. Furthermore, we establish results in Banach algebras concerning spectral theory, maximal ideals and multiplicative linear functionals and present a proof Wiener’s lemma using Banach algebra techniques. Let ν be a submultiplicative weight function satisfying the Gelfand-Raikov-Shilov condition. We show that if a nonzero function f has a ν-weighted absolutely convergent Fourier series in a p-normed algebra A. Then 1/f also has a ν-weightedabsolutely convergent Fourier series in A.