Relations between functions from some Lorentz type spaces and summability of their Fourier coefficients

Abstract: This Licentiate Thesis is devoted to the study of summability of the Fourier coefficients for functions from some Lorentz type spaces and contains three papers (papers A - C) together with an introduction, which put these papers into a general frame.Let $\Lambda_p(\omega),\;\; p>0,$ denote the Lorentz spaces equipped with the (quasi) norm$$\|f\|_{\Lambda_p(\omega)}:=\left(\int_0^1\left(f^'(t)\omega(t)\right)^p\frac{dt}{t}\right)^{\frac1p}$$for a function $f$ on [0,1] and with $\omega$ positive and equipped with some additional growth properties.In paper A some relations between this quantity and some corresponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system. Under certain circumstances even two-sided estimates are obtained.In paper B we study relations between summability of Fourier coefficients and integrability of the corresponding functions for the generalized spaces $\Lambda_p(\omega)$ in the case of a regular system. For example, all trigonometrical systems, the Walsh system and Prise's system are special cases of regular systems. Some new inequalities of Hardy-Littlewood-P\'{o}lya type with respect to a regular system for the generalized Lorentz spaces $\Lambda_p(\omega)$ are obtained. It is also proved that the obtained results are in a sense sharp.The following inequalities are well-known:\begin{equation}\label{f--}c_1\left\|\overline{f}\right\|_{L_p\left[0,1\right]}^p\leq \sum_{k=1}^{\infty}k^{p-2}|a_k|^{p}\leqc_2\left\|tf'\right\|_{L_p\left[0,1\right]}^p,\;\;\;\text{for}\;1\end{equation}where $\overline{f(t)}=\frac1t\left|\int_0^tf(s)ds\right|$ and $f'(t)$ is the derivative of the function $f(t).$ (Here $\{a_k\}_{k=1}^\infty $ are the Fourier coefficients of the function $f$). In paper C we prove some analogues Hardy-Littlewood-P\'{o}lya type inequalities \eqref{f--} with respect to the regular system for generalized Lorentz spaces $\Lambda_{p}(\omega).$