DETERMINATION OF ORDER OF MAGNITUDE OF MULTIPLE FOURIER COEFFICIENTS OF FUNCTIONS OF BOUNDED ? -VARIATION HAVING LACUNARY FOURIER SERIES USING JENSEN'S INEQUALITY

摘要

For a Lebesgue integrable complex-valued function ? defined over the m-dimensional torus T~m:= [0,2π)~m, let ?(n) denote the Fourier coefficient of ?, where n = (n~((1)),...,n~((m))) ∈?~m. Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded ? -variation for a complex-valued function on a rectangle [a_1,b_1] ×... × [a_m，b_m] 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 ? -variation from [0,2π]~m to ? 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.