On weighted $$\ell ^p$$- convergence of Fourier series: a variant of theorems of Wiener and Lévy

Abstract

Let $$1<p<\infty$$ , let q be the conjugate index of p, and let $$\omega$$ be an almost monotone algebra weight on $$\mathbb Z$$ . Let f be a complex valued continuous function on the unit circle such that Fourier series of f is p-th power $$\omega$$ -absolutely convergent, i.e., $$\sum _{n\in {\mathbb {Z}}}|{\widehat{f}}(n)|^p \omega (n)^p<\infty$$ . If f is nowhere vanishing, then there is an almost monotone algebra weight $$\nu$$ on $${\mathbb {Z}}$$ such that $$\nu \le K\omega$$ for some $$K>0$$ and the Fourier series of $$\frac{1}{f}$$ is p-th power $$\nu$$ -absolutely convergent. If $$\varphi$$ is holomorphic on the range of f, then there exists an almost monotone algebra weight $$\chi$$ on $${\mathbb {Z}}$$ such that $$\chi \le K\omega$$ for some constant $$K>0$$ and the Fourier series of $$\varphi \circ f$$ is p-th power $$\chi$$ -absolutely convergent. This gives p-th power analogue of a Theorem by Bhatt and Dedania; and rectifies Theorems of Kinani and Bouchikhi.