Journal of Approximation Theory最新文献

The concept of $H$-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and $H$-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, $H$-sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of $H$-sets to Linear Programming is studied. But on the downside, it is explained why $H$-sets have a very limited range of applicability in the times of large-scale computing.

We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in $L_p$ and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in $L_p$, find the exact order of decay of the corresponding $L_p$-errors for particular classes of functions, and introduce a special $K$-functional and its realization suitable for studying smoothness properties of sampling operators.

This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed and convex cones. Furthermore, the inverse images of the generalized metric projection are neither a convex set nor a cone. We also prove that the distance from a point to its projection on a convex set is a weakly lower semicontinuous function for all three notions of projections. We provide illustrating examples to highlight the different behavior of the three projections in Banach spaces.

Let $w$ be a weight on the unit disk $D$ having the form $w\left(z\right)=|v\left(z\right){|}^2\prod _{k=1}^s{\left(\frac{z-a_k}{1-z{\overline{a}}_k}\right)}^{m_k},m_k>-2,|a_k|<1,$where $v$ is analytic and free of zeros in $\overline{D}$, and let ${\left(p_n\right)}_{n=0}^{\infty }$ be the sequence of polynomials ($p_n$ of degree $n$) orthonormal over $D$ with respect to $w$. We give an integral representation for $p_n$ from which it is in principle possible to derive its asymptotic behavior as $n\to \infty$ at every point $z$ of the complex plane, the asymptotic analysis of the integral being primarily dependent on the nature of the first singularities encountered by the function $v{\left(z\right)}^{-1}{\prod }_{k=1}^s{(1-z{\overline{a}}_k)}^{-1}$.

The aim of this paper is to use the analytic theory of linear $q$-difference equations for the study of the functional-differential equation $y^{\prime }\left(x\right)=ay\left(qx\right)+by\left(x\right)$, where $a$ and $b$ are two non-zero real or complex numbers. When $0<q<1$ and $y\left(0\right)=1$, the associated Cauchy problem admits a unique power series solution, ${\sum }_{n\ge 0}\frac{{(-a/b;q)}_n}{n!}{\left(bx\right)}^n$, that converges in the whole complex $x$-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.

We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

An upper bound for the Lebesgue constant, i.e., the supremum norm of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to $n$ is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to $n$ for an arbitrary $d$-dimensional simplex was established by the author. The upper bound proved in this article refines this result for $d=2$.

In this paper we prove that under some conditions on the parameters the univariate Triebel–Lizorkin spaces $F_{r,q}^s\left(R\right)$ can be characterized in terms of quarklets. So for functions from Triebel–Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed by means of biorthogonal compactly supported Cohen–Daubechies–Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel–Lizorkin–Morrey spaces $E_{u,r,q}^s\left(R\right)$.