Order of magnitude of multiple Walsh–Fourier coefficients of functions of bounded $$\phi$$-variation

Abstract

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus $$\mathbb {I}^n:=[0,1)^n$$ $$(n\in \mathbb N)$$ , let $$\hat{f}(\mathbf{k})$$ denote the multiple Walsh–Fourier coefficient of f, where $$\mathbf{k}=(k_1,\dots ,k_n)\in (\mathbb {Z}^+)^n$$ , $$\mathbb {Z^+}=\mathbb {N}\cup \{0\}$$ . The Riemann–Lebesgue lemma shows that $$\hat{f}(\mathbf{k})=o(1)$$ as $$|\mathbf{k}|\rightarrow \infty$$ for any $$f\in L^1(\mathbb I^n)$$ . However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive result is due to Ghodadra for functions of bounded p-variation. In this paper, similar definite results are proved for functions of bounded $$\phi$$ -variation which generalize the known results for functions of bounded p-variation. The techniques used are some key inequalities for convex functions (including the Jensen’s inequality) and the Taibleson-like technique for Walsh-Fourier coefficients.