An application of Jensen's inequality in determining the order of magnitude of multiple Fourier coefficients of functions of bounded φ-variation

Abstract

For a Lebesgue integrable complex-valued function f defined over the n -dimensional torus Tn := [0,2π)n , n∈N , let f (k) denote the Fourier coefficient of f , where k = (k1, . . . ,kn) ∈ Zn . The Riemann-Lebesgue lemma shows that f (k) = o(1) as |k| → 0 for any f ∈ L1(Tn) . However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive results are due to V. Fulop and F. Moricz for functions of bounded variation, and due to B. L. Ghodadra for functions of bounded p -variation. In this paper, defining the notion of bounded φ -variation for a function from [0,2π]n to C in two different ways, we prove that this is the case for Fourier coefficients of such functions also. Interestingly, in proving our main results we use the famous Jensen’s inequality for integrals. Our new results with φ(x) = xp (p 1) gives our earlier results [Acta Math. Hungar, 128 (4) (2010), 328–343]. Mathematics subject classification (2010): 42B05, 26D15, 26B30.