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By means of the regularity of the differential operators generated by the Szasz-Mirakjan operators, this paper further investigates the relation between the approximation rate of the linear combinations of the Szasz-Mirakjan operators and the smoothness of the approximated function. We precisely give both the upper and lower bounds, which are of the same order, of the approximation of the linear combinations of these operators. Our results under some conditions include the direct theorem, the inverse theorem and the saturation of the uniform approximation of the linear combinations of these operators, and hence the approximation order of the uniform approximation of the linear combinations of the Szasz-Mirakjan operators are further characterized.

In this paper, for new defined Bernstein type operators, we show that taking derivatives and taking the limit are sequence independent when the Voronovskaja-type expansion is differentiated.

In this paper, we introduce the class of ( β , γ ) -Chebyshev functions and corresponding points, which can be seen as a family of generalized Chebyshev polynomials and points. For the ( β , γ ) -Chebyshev functions, we prove that they are orthogonal in certain subintervals of [ − 1 , 1 ] with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev–Lobatto points are instances of ( β , γ ) -Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters β and γ .

In this paper, we study a class of matrix-valued orthogonal polynomials (MVOPs) that are related to 2-periodic lozenge tilings of a hexagon. The general model depends on many parameters. In the cases of constant and $2$-periodic parameter values we show that the MVOP can be expressed in terms of scalar polynomials with non-Hermitian orthogonality on a closed contour in the complex plane. The 2-periodic hexagon tiling model with a constant parameter has a phase transition in the large size limit. This is reflected in the asymptotic behavior of the MVOP as the degree tends to infinity. The connection with the scalar orthogonal polynomials allows us to find the limiting behavior of the zeros of the determinant of the MVOP. The zeros tend to a curve $\widetilde{\Sigma}_0$ in the complex plane that has a self-intersection. The zeros of the individual entries of the MVOP show a different behavior and we find the limiting zero distribution of the upper right entry under a geometric condition on the curve $\widetilde{\Sigma}_0$ that we were unable to prove, but that is convincingly supported by numerical evidence., Comment: 31 pages, 10 figures

Analysis of functions on combinatorial graphs is an emerging field attracting more and more attention. In this paper, we study the approximation of functions defined on combinatorial graphs by functions in Paley–Wiener spaces. First, we use a family of graph translation operators to define the modulus of smoothness, which has several properties similar to their counterparts in the classical approximation theory. Next, we establish Jackson’s and Bernstein’s inequalities for functions defined on graphs. Finally, we provide an estimation on the decay of graph Fourier coefficients in terms of the modulus of smoothness. These results lead to a theory of approximation of functions on combinatorial graphs and have potential applications to filtering, denoising, data dimension reduction, image processing and learning theory.

In this paper order estimates of the Kolmogorov widths for some weighted Sobolev classes in a weighted Lebesgue space are obtained. The weighted Sobolev classes are defined by a restriction on the weighted L p 1 -norm of the highest order derivative and the weighted L p 0 -norm of the function.

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Turán's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Turán type inequalities and transformations into more convenient orthogonal polynomial systems., Main changes towards first version: The part on associated symmetric Pollaczek polynomials was extended (with more emphasis on Tur\'{a}n's inequality and including a larger parameter region), and the part on little $q$-Legendre polynomials became a separate paper. We added several references and corrected a few typos. Title, abstract and MSC class were changed

In this paper we verify a sharp Bernstein type inequality for multivariate algebraic polynomials in L p norm on general cuspidal domains which is based on a proper measure of the distance to the boundary of the domain. In particular, in case of Lip γ cuspidal graph domains the exact rate is given by n 2 γ − 1 .

The constrained covariance (COCO) has been proposed for measuring dependence between random vectors. Kernel cross-covariance operators on reproducing kernel Hilbert spaces, as one of kernel methods which could extract nonlinear dependence, have attracted considerable attention. This paper establishes learning rates of some estimators associated with kernel cross-covariance. For kernel cross-covariance operators, we bound a weighted summation of squared estimation errors of empirical singular functions by 16 times of the estimation error of empirical cross-covariance. Our method actually applies in general setting, so that a new bound is obtained for perturbation of singular functions of Hilbert–Schmidt operators. It is much tighter than the classical result as the latter only bounds each error of singular function individually. This is interest in its own right. For normalized cross-covariance operator, we propose an estimator and obtain a learning rate.

In this paper, we extend the sharp upper bound of Christiansen et al. (2017) and the sharp lower bound of Schiefermayr (2008) to the case of weighted Chebyshev polynomials on subsets of [ − 1 , 1 ] for the weight w ( x ) = 1 − x 2 . We then analyse the norm of Chebyshev polynomials on a circular arc, prove monotonicity of the corresponding Widom factors, find exact values of their supremum and infimum, and obtain a new proof for their limit.