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We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two well-studied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of computational complexity. However, the WRGA has limited applicability. It converges only for elements of the closure of the convex hull of a dictionary. In this paper we study algorithms that combine good features of both algorithms, the WRGA and the WCGA. In the construction of such algorithms we use different forms of relaxation. First results on such algorithms have been obtained in a Hilbert space by A. Barron, A. Cohen, W. Dahmen, and R. DeVore. Their paper was a motivation for the research reported here.

Meixner polynomials m n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are $j(x;\beta,c) = \frac{c^x(\beta)_x}{x!} \qquad \mbox{at}\quad x=0,1,2\ldots.$ Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m n (nα;β,c) as $n\to\infty$ . One holds uniformly for $0 < \epsilon\le \alpha\le 1+a$ , and the other holds uniformly for $1-b\le \alpha\le M < \infty$ , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases.

We derive an asymptotic approximation of Plancherel—Rotach type for the Charlier polynomials on the positive real line.

Smoothness of the Green functions for the complement of rarefied Cantor-type sets is described in terms of the function $\varphi (\delta)=(1/\log\frac{1}{\delta})$ that gives the logarithmic measure of sets. Markov’s constants of the corresponding sets are evaluated.

It is shown that products of polynomials introduced by Heine and Stieltjes form orthogonal bases in suitable function spaces. A theorem on the expansion of analytic function in these bases is proved.

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. Approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studied.

We analyze the accuracy of the discrete least-squares approximation of a function $$u$$ in multivariate polynomial spaces $$\mathbb {P}_\varLambda :=\mathrm{span} \{y\mapsto y^\nu : \nu \in \varLambda \}$$ with $$\varLambda \subset \mathbb {N}_0^d$$ over the domain $$\varGamma :=[-1,1]^d$$ , based on the sampling of this function at points $$y^1,\ldots ,y^m \in \varGamma $$ . The samples are independently drawn according to a given probability density $$\rho $$ belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. Motivated by recent results on high-dimensional parametric and stochastic PDEs, we restrict our attention to polynomial spaces associated with downward closed sets $$\varLambda $$ of prescribed cardinality n, and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in $$L^2(\varGamma ,d\rho )$$ with the best achievable polynomial approximation error when using downward closed sets of cardinality n. We establish conditions between the dimension n and the size m of the sample, under which these two errors are proved to be comparable. Our main finding is that the dimension d enters only moderately in the resulting trade-off between m and n, in terms of a logarithmic factor $$\ln (d)$$ , and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named anchored sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon (Chkifa et al. in ESAIM Math Model Numer Anal 49(3):815–837, 2015), which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of efficient numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs, but are not limited to this area.

We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted.

We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We study the dual spaces of the classical and nonclassical Besov and Triebel-Lizorkin spaces on this setting. Our results generalize those on Euclidean spaces and are new on several setups of independent interest; the sphere, the ball, more general Riemannian manifolds and other settings.