We study the continuity and compactness of embeddings for radial Besov and Triebel-Lizorkin spaces with weights in the Muckenhoupt class $A_{\infty }$. The main tool is a discretization in terms of an almost orthogonal wavelet expansion adapted to the radial situation.

Let w be a generalized Jacobi weight on the interval [-1,1] and, for each function f, let Snf denote the n-th partial sum of the Fourier series of f in the orthogonal polynomials associated to w. We prove a result about uniform boundedness of the operators Sn in some weighted Lp spaces. The study of the norms of the kernels Kn related to the operators Sn allows us to obtain a relation between the Fourier series with respect to different generalized Jacobi weights.

We obtain weighted $L^p$ boundedness, with weights of the type $y^{\delta }$, δ > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, ${{ℒ}_k^{\alpha }}_k$, when the parameter α is greater than -1. It is proved that when -1 < α < 0, the maximal operator is of strong type (p,p) if p > 1 and 2(1+δ)/(2+α) < p < 2(1+δ)/(-α), and if α ≥ 0 it is of strong type for 1 < p ≤ ∞ and 2(1+δ)/(2+α) < p. The behavior at the end points of the intervals where there is strong type is studied...

We give a weighted version of the Sobolev-Lieb-Thirring inequality for suborthonormal functions. In the proof of our result we use phi-transform of Frazier-Jawerth.

In this paper, we establish an one-to-one mapping between complex-valued functions defined on $R^{+}\cup \left\{0\right\}$ and complex-valued functions defined on $p$-adic number field $Q_p$, and introduce the definition and method of Weyl-Heisenberg frame on hormonic analysis to $p$-adic anylysis.

We discuss the concept of Sobolev space associated to the Laguerre operator $L_{\alpha }=-yd²/dy²-d/dy+y/4+\alpha ²/4y$, y ∈ (0,∞). We show that the natural definition does not agree with the concept of potential space defined via the potentials ${\left(L_{\alpha }\right)}^{-s}$. An appropriate Laguerre-Sobolev space is defined in order to achieve that coincidence. An application is given to the almost everywhere convergence of solutions of the Schrödinger equation. Other Laguerre operators are also considered.

We give conditions such that the least degree solution of a Bézout identity is nonnegative on the interval [-1,1].

In nonparametric statistics a classical optimality criterion for estimation procedures is provided by the minimax rate of convergence. However this point of view can be subject to controversy as it requires to look for the worst behavior of an estimation procedure in a given space. The purpose of this paper is to introduce a new criterion based on generic behavior of estimators. We are here interested in the rate of convergence obtained with some classical estimators on almost every, in the sense...

Given a probability measure μ with non-polar compact support K, we define the n-th Widom factor W²ₙ(μ) as the ratio of the Hilbert norm of the monic n-th orthogonal polynomial and the n-th power of the logarithmic capacity of K. If μ is regular in the Stahl-Totik sense then the sequence ${\left(W²ₙ\left(\mu \right)\right)}_{n=0}^{\infty }$ has subexponential growth. For measures from the Szegő class on [-1,1] this sequence converges to some proper value. We calculate the corresponding limit for the measure that generates the Jacobi polynomials, analyze...