The Fomin Classes F p

Abstract

For p > l, G.A. Fomin introduced a class F p of null-sequences a defined by the property that Fp (ą) < ∞, where Δan = n −an+1 and $$ {F_p}(\mathop a\limits_ ): = \mathop \sum \limits_{n = 1}^\infty {(\frac{1}{n}\mathop \sum \limits_{r = n}^\infty |\Delta {a_r}{|^p})^{1/p}} $$ [G.A. Fomin, A class of trigonometric series. Mat. Zametki 2 3 (1978), 213–222]. Another writer claimed that “the class F p is wider when p is closer to 1”, which is equivalent to claiming that Fp(ą) is an increasing function of p for each null-sequence ą [C.V. Stanojević, Classes of L1 -convergence of Fourier and Fourier-Stieltjes series. Proc. Amer. Math. Soc. 82 (1981), 209–215].