Order of magnitude of multiple Walsh-Fourier coefficients of functions of bounded φ-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus In:=[0,1)n(n is an element of N), let f<^></mml:mover>(k) denote the multiple Walsh-Fourier coefficient of f, where k=(k1,,kn)is an element of (Z+)n, Z+=N{0}. The Riemann-Lebesgue lemma shows that f<^></mml:mover>(k)=o(1) as |k|-> infinity for any f is an element of L1<mml:mo stretchy="false">(In<mml:mo stretchy="false">). 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.