We study weighted $L_p$-integrability (1 ≤ p < ∞) of trigonometric series. It is shown how the integrability of a function with weight $x^{-\alpha }$ depends on some regularity conditions on Fourier coefficients. Criteria for the uniform convergence of trigonometric series in terms of their coefficients are also studied.

Integrability and $L^1-$convergence of modified cosine sums introduced by Rees and Stanojević under a class of generalized semi-convex null coefficients are studied by using Cesàro means of non-integral orders.

We show that, if the coefficients (an) in a series $a_0/2+{\sum }_{n=1}^{\infty }{a}_{n}cos\left(nt\right)$ tend to 0 as n → ∞ and satisfy the regularity condition that ${\sum }_{m=0}^{\infty }{{\sum }_{j=1}^{\infty }[{\sum }_{n=j{2}^m}^{(j+1)2^m-1}|a_n-a_{n+1}\left|\right]²}^{1/2}<\infty$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series ${\sum }_{n=1}^{\infty }{b}_{n}sin\left(nt\right)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if ${\sum }_{n=1}^{\infty }\left|b_n\right|/n<\infty$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences...

In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi$, where $\phi$ is a function of positive lower type and upper type less than $1$, and show that $I_{\phi }$ and $D_{\phi }$ are bounded from Lipschitz spaces ${\Lambda }^{\xi }$ to ${\Lambda }^{\xi \phi }$ and ${\Lambda }^{\xi /\phi }$ respectively, with suitable restrictions on the quasi-increasing function $\xi$ in each case. We also prove that $I_{\phi }$ and $D_{\phi }$ are bounded from the generalized Besov ${\dot{B}}_p^{\psi ,q}$, with $1\le p,q<\infty$, and Triebel-Lizorkin spaces ${\dot{F}}_p^{\psi ,q}$, with $1<p,q<\infty$, of order $\psi$ to those of order $\phi \psi$ and $\psi /\phi$ respectively,...

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ${\ell }^{q̅}\left(L_v^{p̅}\right)$ into ${\ell }^q\left(L_u^p\right)$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^p$-spaces. Amalgams of the form ${\ell }^q\left(L_w^p\right)$, 1 < p,q < ∞ , q ≠ p, $w\in A_p$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

We analyze some aspects of Mercer's theory when the integral operators act on L²(X,σ), where X is a first countable topological space and σ is a non-degenerate measure. We obtain results akin to the well-known Mercer's theorem and, under a positive definiteness assumption on the generating kernel of the operator, we also deduce series representations for the kernel, traceability of the operator and an integration formula to compute the trace. In this way, we upgrade considerably similar results...

We investigate approximation properties of de la Vallee Poussin right-angled sums on the classes of periodic functions of several variables with a high smoothness. We obtain integral presentations of deviations of de la Vallee Poussin sums on the classes $C_{\beta ,\infty }^{m\alpha }$.

We give integral representations for multiple Hermite and multiple Laguerre polynomials of both type I and II. We also show how these are connected with double integral representations of certain kernels from random matrix theory.

Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative.