Order of magnitude of multiple Fourier coefficients of functions of bounded p-variation

Abstract

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \( \mathbb{T}^n \):= [0, 2π)n, let \( \hat f \)(k) denote the Fourier coefficient of f, where k = (k1, … kn) ∈ ℤn. In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π]n to ℜ in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Moricz [5] and Fulop and Moricz [3].