On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces

Abstract

We denote by Λ_β(λ), β > 0, the Lorentz space equipped with the (quasi) norm

|f|_Λβ(λ) := ( ∫₀¹ ( f^*(t)tλ ( 1 / t )^β ) dt / t )^1/β

for a function f on [0,1] and with λ positive and equipped with some additional growth properties. Some estimates of this quantity and some corresponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system.