Asymptotics of rearranged trigonometric and Walsh-Fourier coefficients of smooth functions

AbstractFor a 2π-periodic function f ϵ Lp[0, 2π] (1 ⩽ p ⩽ 2) there exists A(p) > 0 such that \̂tf∗(n) ⩽ n−1 + 1pω(n−1; p; f), where \̂tf∗ is the nonincreasing rearrangement of the moduli of the sequence of Fourier coefficients {¦\̂tf(n)¦}. This inequality accounts for virtually all “absolute convergence” theorems expressible in terms of Lebesgue or Lorentz sequence spaces. A similar inequality, with the (smaller) dyadic modulus of continuity holds for Walsh-Fourier coefficients.These inequalities lead to extensions, for both trigonometric and Walsh-Fourier series of, the Wiener-Lozinski theorems, characterizing continuous functions of bounded p-variation (p < 2) in terms of the moduli of their Fourier coefficients. In turn these imply seemingly new estimates for the rate of convergence of Fourier series of functions in Lip(p−1, p) which have slowly varying moduli of continuity