Determination of order of magnitude of multiple Fourier coefficients of functions of bounded ϕ-variation having lacunary Fourier series using Jensen's inequality

Abstract

For a Lebesgue integrable complex-valued function f defined over the m -dimensional torus Tm := [0,2π)m , let f (n) denote the Fourier coefficient of f , where n = (n(1), . . . ,n(m)) ∈ Zm . Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded φ -variation for a complex-valued function on a rectangle [a1,b1 ]× . . .× [am,bm ] and studied the order of magnitude of Fourier coefficients of such functions on [0,2π]m . In this paper, the order of magnitude of Fourier coefficients of a function of bounded φ -variation from [0,2π]m to C and having lacunary Fourier series with certain gaps is studied and a generalization of our earlier result (Theorem in [Acta Sci. Math. (Szeged), 78, (2012), 97–109]) is proved. Interestingly, the Jensen’s inequality for integrals is used to prove the main result. Mathematics subject classification (2010): 42B05, 26B30, 26D15.