Multivariate Vector Valued Beurling Algebra Analogues Of Theorems Of Wiener And Ε»elazko
Authors :- PA Dabhi, KB Solanki
Publication :- Communications of the Korean Mathematical Society, 2025
If 0 < p β€ 1, π is a weight on β€2, π is a unital complex Banach algebra and if f is a continuous π valued function on π2 such that ${\sum{_{{(m,n)}{\in}{\mathbb{Z}}^2}}\,{\parallel}{\hat{f}}(m,n){\parallel}^p{\omega}(m,n)\,<\,{\infty}$, where ${\hat{f}}(m,n)$ are Fourier coefficients of f, and f(z, w) is left invertible in π for all (z, w) β π2, then it is shown that there is a weight π on β€2 and a continuous function g : π2 β π such that 1 β€ π β€ π, π is constant if and only if π is constant, ${\sum{_{{(m,n)}{\in}{\mathbb{Z}}^2}}\,{\parallel}{\hat{g}}(m,n){\parallel}^p{\omega}(m,n)\,<\,{\infty}$ and g is a left inverse of f. A similar result is obtained for a continuous function f from πβ (the countable copies of π) to π.