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We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.

This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.

In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation (previously called baker’s transform) was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z 2 by Cristea et al. (2007) and Goda (2015). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b . For this purpose, we introduce the b -adic tent transformation for an arbitrary positive integer b greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer b greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over Z b which are randomly digitally shifted and then folded using the b -adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime b , we prove the existence of good higher order polynomial lattice rules over Z b among a smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S= +\lambda , $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time., Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theory

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the ''convex feasibility'' problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the ''angles'' between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the ''norm'' of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the ''linear regularity property'' of Bauschke and Borwein, the ''normal property'' of Jameson (as well as Bakan, Deutsch, and Li's generalization of Jameson's normal property), the ''strong conical hull intersection property'' of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.

The purpose of this paper is to derive product representations for generalizations of the Rogers–Ramanujan series. Special cases of the results presented here were first stated by Ramanujan in the “Lost Notebook” and proved by George Andrews. The analysis used in this paper is based upon the work of Andrews and the broad contributions made by Mourad Ismail and Walter Hayman. Each series considered is related to an extension of the Rogers–Ramanujan continued fraction and corresponds to an orthogonal polynomial sequence generalizing classical orthogonal sequences. Using Ramanujan's differential equations for Eisenstein series and corresponding analogues derived by V. Ramamani, the coefficients in the series representations of each zero are expressed in terms of certain Eisenstein series.

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

This paper gives the conditions necessary for weighted convergence of Hermite–Fejér interpolation for a general class of even weights which are of exponential decay on the real line or at the end points of (-1,1). The results of this paper guarantee that the conditions of Theorem 2.3 in [11] are optimal.

We discuss and examine Weierstrass’ main contributions to approximation theory. §1. Weierstrass This is a story about Karl Wilhelm Theodor Weierstrass (Weierstras), what he contributed to approximation theory (and why), and some of the consequences thereof. We start this story by relating a little about the man and his life. Karl Wilhelm Theodor Weierstrass was born on October 31, 1815 at Ostenfelde near Munster into a liberal (in the political sense) Catholic family. He was the eldest of four children, none of whom married. Weierstrass was a very successful gymnasium student and was subsequently sent by his father to the University of Bonn to study commerce and law. His father seems to have had in mind a government post for his son. However neither commerce nor law was to his liking, and he “wasted” four years there, not graduating. Beer and fencing seem to have been fairly high on his priority list at the time. The young Weierstrass returned home, and after a period of “rest”, was sent to the Academy at Munster where he obtained a teacher’s certificate. At the Academy he fortuitously came under the tutelage and personal guidance of C. Gudermann who was professor of mathematics at Munster and whose basic mathematical love and interest was the subject of elliptic functions and power series. This interest he was successful in conveying to Weierstrass. In 1841 Weierstrass received his teacher’s certificate, and then spent the next 13 years as a teacher (for 6 years he was a teacher in a pregymnasium in the town of Deutsch-Krone (West Prussia), then for another 7 years in a gymnasium in Braunsberg (East Prussia)). During this period he continued learning mathematics, mainly by studying the work of Abel. He also published some mathematical papers. However these appeared in school journals and were quite naturally not discovered at that time by any who could understand or appreciate them. (Weierstrass’ collected works contain 7 papers from before 1854, the first of which On the development of modular functions (49 pp.) was written in 1840.) In 1854 Weierstrass published the paper On the theory of Abelian functions in Crelle’s Journal fur die Reine und Angewandte Mathematik (the first mathematical research journal, founded in 1826, and now referred to without Crelle’s name in the formal title). It created a sensation within the mathematical community. Here was a 39 year old school teacher whom no one within the mathematical community had heard of. And he had written a masterpiece, not only in its depth, but also in its mastery of an area. Recognition was

In this paper we study sequences of linear operators which are "almost positive" outside sets of small Jordan measure. For them, we prove Korovkin-type theorems in terms of a modification of the R-convergence used previously by W. Dickmeis, K. Mevissen, R. J. Nessel, and E. Van Wickeren and the test families of functions which the author introduced in a previous paper.

In previous papers the first author extended the classical equiconvergence theorem of Walsh by the application of summability methods in order to enlarge the disk of equiconvergence to regions of equisummability. The aim of this paper is to study equisummability of sequences which arise from Hadamard products of a fixed power series with Lagrange polynomial interpolants.

Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to\infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigenvectors of the covariance matrices of the Gaussian limits and study applications to CLTs for the intermediate particles for $k\to\infty$ and then $N\to\infty$., Comment: 20 pages

We prove a partial fraction decomposition of a quotient of two functions E α ( i t x ) and I α ( i t ) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ ( α + 1 ) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α ( x ) of degree n are defined by considering E α ( t x ) ∕ I α ( t ) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α ( 1 ) . For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 ( x ) is the exponential function and E n , 1 ∕ 2 ( x ) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ] .

The study of the restricted isometry property (RIP) of corrupted random matrices is particularly important in the field of compressed sensing (CS) with corruptions. If a matrix still satisfies the RIP after that a certain portion of rows are erased, then we say that this matrix has the strong restricted isometry property (SRIP). In the field of compressed sensing, random matrices which satisfy certain moment conditions are of particular interest. Among these matrices, those with entries generated from i.i.d. Gaussian or symmetric Bernoulli random variables are often typically considered. Recent studies have shown that matrices with entries generated from i.i.d. Gaussian random variables satisfy the SRIP under arbitrary erasure of rows with high probability. In this paper, we study the erasure robustness property of Bernoulli random matrices. Our main result shows that with overwhelming probability, the SRIP holds for Bernoulli random matrices. Moreover, our analysis leads to a robust version of the famous Johnson–Lindenstrauss lemma for Bernoulli random matrices.

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1, Comment: 37 pages, to appear in Journal of Approximation Theory

Let ζ ( s ) be the Riemann zeta function. In 1911, Bohr showed that the set { ζ ( σ + i τ ) : σ > 1 , τ ∈ R } is dense in ℂ . By Voronin’s denseness theorems in 1972, the sets { ( ζ ( σ + i λ 1 + i τ ) , … , ζ ( σ + i λ n + i τ ) ) : σ ≥ 1 , τ ∈ R } with distinct λ 1 , … , λ n ∈ R and { ( ζ ( σ + i τ ) , ζ ′ ( σ + i τ ) , … , ζ ( n − 1 ) ( σ + i τ ) ) : σ ≥ 1 , τ ∈ R } are dense in ℂ n . By Voronin’s universality theorem, for any fixed 1 ∕ 2 σ 1 and any non-negative integer k , the set { ζ σ , τ ( k ) : τ ∈ R } is dense in C [ a , b ] , where ζ σ , τ ( k ) ( t ) ≔ ζ ( k ) ( σ + i t + i τ ) , t ∈ [ a , b ] . In the present paper, we prove that the set { ζ σ , τ ( k ) : σ ≥ 1 , τ ∈ R } ∩ C [ a , b ] is not dense in C [ a , b ] .

In this paper, we study the problem of whether the best approximation to any x in a real Hilbert space X from the closed convex set K ≔ C ∩ D can be characterized by the best approximation to a perturbation x − l of x from the set C for some l in a certain cone in X . The set C is a closed convex subset of X and D ≔ { x ∈ X : g j ( x ) ≤ 0 , ∀ j = 1 , 2 , … , m } , where the functions g j : X ⟶ R ( j = 1 , 2 , … , m ) are continuously Frechet differentiable that are not necessarily convex. We show under suitable conditions that this “perturbation property” is characterized by the strong conical hull intersection property of C and D at the point x 0 ∈ K . We prove this by first establishing a dual cone characterization of a nearly convex set. Our result shows that the convex geometry of K is critical for the characterization rather than the representation of D by convex inequalities, which is commonly assumed for the problems of best approximation from a convex set. In the special case where the set D is convex, we show that the Lagrange multiplier characterization of best approximation holds under the standard Slater’s constraint qualification together with a non-degeneracy condition. The lack of representation of D by convex inequalities is supplemented by the non-degeneracy condition, but the characterization, even in this special case, allows applications to problems with quasi-convex functions g j , j = 1 , 2 , … , m , as they guarantee the convexity of D . Simple numerical examples illustrate the nature of our assumptions.

In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these polynomials., 40 pages, 1 figure

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 .

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 find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems., Accepted for publication in the Journal of Approximation Theory

In this paper, we study the rate of approximation of the combination of some generalized multivariate average on Hardy spaces and obtain its equivalent relation to the K -functionals. The result is an extension of a result in Dai and Ditzian (2004). We also extend and improve Theorem 6.2 in Belinsky et al. (2003).

In this paper the structure of the set of functions for which the peak value of the Shannon sampling series is strongly divergent is analyzed. Strong divergence is closely linked to the non-existence of adaptive reconstruction methods. Signals in the Paley–Wiener space PW π 1 of bandlimited functions with absolutely integrable Fourier transform are considered, and it is shown that the set of strong divergence is spaceable and dense-lineable, i.e., that there exist an infinite dimensional closed subspace and an infinite dimensional dense subspace, such that we have strong divergence of the peak value of the Shannon sampling series for all functions from these sets, except the zero function. Further, it is proved that this result is not restricted to the Shannon sampling series, but rather holds for an entire class of reconstruction processes.

In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem., 21 pages, 1 figure. Revised version, minor changes and references added

Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer’s G function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions, connection with radial positive definite functions and generalizations of Luke’s inequalities for the Kummer and Gauss type functions.

In this paper we study approximation theorems for L 2 -space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K -functional and modulus of smoothness for Damek–Ricci spaces.

In this paper, we discuss the compressed data separation problem. In order to reconstruct the distinct subcomponents, which are sparse in morphologically different dictionaries D 1 ∈ R n × d 1 and D 2 ∈ R n × d 2 , we present a general class of convex optimization decoder. It can deal with signal separation under the corruption of different kinds of noises, including Gaussian noise ( p = 2 ), Laplacian noise ( p = 1 ), and uniformly bounded noise ( p = ∞ ). Although the restricted isometry property adapted to frames is a commonly used tool, the measurement number is suboptimal when p > 2 . Furthermore, the l p robust nullspace property adapted to Ψ , which is constructed by D 1 and D 2 , may fail to work on data separation problem. Here we introduce the modified l p robust nullspace property adapted to Ψ (abbreviated as the modified ( l p , Ψ )-RNSP). First of all, we show the robust recovery of signals based on the modified ( l p , Ψ )-RNSP and the mutual coherence between D 1 and D 2 . Besides, we find that Gaussian measurements meet the modified ( l p , Ψ )-RNSP for any 1 ≤ p ≤ ∞ , provided with the optimal number of measurements O ( s log ( d ∕ s ) ) , where s is the sparsity level and d = d 1 + d 2 . Furthermore, we introduce another properly constrained l 1 -analysis optimization model, called the Split Dantzig Selector. It can recover signals which are approximately sparse in terms of different frame representations, when the measurement matrix satisfies the modified ( l p , Ψ )-RNSP. As a special case, when considering Gaussian white noise, the recovery error by the Split Dantzig Selector is O s log d m . It outperforms the l 2 -constrained model, whose recovery error is O ( log m ) , if the sparsity level is small.

In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets—a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2 . For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2 -weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion.

This paper is devoted to studying an interpolation problem on the circle, which can be considered an intermediate problem between Lagrange and Hermite interpolation. The difference as well as the novelty is that we prescribe Lagrange values at the 2 n roots of a complex number with modulus one and we prescribe values for the first derivative only on half of the nodes. We obtain two types of expressions for the interpolation polynomials: the barycentric expressions and another one given in terms of an orthogonal basis of the corresponding subspace of Laurent polynomials. These expressions are very suitable for numerical computation. Moreover, we give sufficient conditions in order to obtain convergence in case of continuous functions and we obtain the rate of convergence for smooth functions. Finally we present some numerical experiments to highlight the results obtained.

In this paper we study several properties of Chebyshev sets in geodesic spaces. We focus on analyzing if some well-known results that characterize convexity of such sets in Hilbert spaces are also valid in the setting of geodesic spaces with bounded curvature.

In this paper the problem of density in the space C ( X ) , for a compact set X ? C , of polynomial modules of the type { p + z ? d q : p , q ? C z } for integer d 1 , as well as several related problems are studied. We obtain approximability criteria for Caratheodory compact sets using the concept of a d -Nevanlinna domain, which is a new special analytic characteristic of planar simply connected domains. In connection with this concept we study the problem of taking roots in the model spaces, that is, in the subspaces of the Hardy space H 2 which are invariant under the backward shift operator.

Let B R ? ( h ? 2 ; f ) , ( R 0 ) denote the Bochner-Riesz means of order ? - 1 for the Dunkl transform of f ? L 1 ( R d ; h ? 2 d x ) associated with the weight function h ? 2 ( x ) : = ? j = 1 d | x j | 2 ? j on R d , where ? : = ( ? 1 , ? , ? d ) ? 0 , ∞ ) d . This paper shows that if ? ? 0 , then the Bochner-Riesz mean B R ? ( h ? 2 ; f ) ( x ) of each function f ? L 1 ( R d ; h ? 2 d x ) converges almost everywhere to f ( x ) on R d at the critical index ? = λ ? : = d - 1 2 + ? j = 1 d ? j as R ? ∞ . As is well-known in classical analysis, this result is no longer true in the unweighted case where ? = 0 , h ? ( x ) ? 1 , and B R ? ( h ? 2 ; f ) is the Bochner-Riesz mean of the Fourier transform.

The Bochner Classification Theorem (1929) characterizes the polynomial sequences { p n } n = 0 ∞ , with deg p n = n that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials, with certain restrictions on the polynomial parameters, satisfy these conditions. In 2009, Gomez-Ullate, Kamran, and Milson found that for sequences { p n } n = 1 ∞ , deg p n = n (without the constant polynomial), the only such sequences satisfying these conditions are the exceptional X 1 -Laguerre and X 1 -Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials { p n } n ? N 0 ? A , where A is a finite subset of the non-negative integers N 0 and where deg p n = n for all n ? N 0 ? A . We call such a sequence an exceptional polynomial sequence of codimension | A | , where the latter denotes the cardinality of A . All exceptional sequences with a non singular weight, found to date, have the remarkable feature that they form a complete orthogonal set in their natural Hilbert space setting.Among the exceptional sets already known are two types of exceptional Laguerre polynomials, called the Type I and Type II exceptional Laguerre polynomials, each omitting m polynomials. In this paper, we briefly discuss these polynomials and construct the self-adjoint operators generated by their corresponding second-order differential expressions in the appropriate Hilbert spaces. In addition, we present a novel derivation of the Type III family of exceptional Laguerre polynomials along with a detailed disquisition of its properties. We include several representations of these polynomials, orthogonality, norms, completeness, the location of their local extrema and roots, root asymptotics, as well as a complete spectral study of the second-order Type III exceptional Laguerre differential expression.

The Hamburger moment problem for the $q$-Lommel polynomials which are related to the Hahn-Exton $q$-Bessel function is known to be indeterminate for a certain range of parameters. In this paper, the Nevanlinna parametrization for the indeterminate case is provided in an explicit form. This makes it possible to describe all N-extremal measures of orthogonality. Moreover, a linear and quadratic recurrence relation are derived for the moment sequence, and the asymptotic behavior of the moments for large powers is obtained with the aid of appropriate estimates., Comment: 28 pages

In the last 20 years a whole hierarchy of notions of tractability was proposed and analyzed by several authors. These notions are used to classify the computational hardness of continuous numerical problems S = ( S d ) d ? N in terms of the behavior of their information complexity n ( e , S d ) as a function of the accuracy e and the dimension d . By now a lot of effort was spent on either proving quantitative positive results (such as, e.g., the concrete dependence on e and d within the well-established framework of polynomial tractability), or on qualitative negative results (which, e.g., state that a given problem suffers from the so-called curse of dimensionality). Although several weaker types of tractability were introduced recently, the theory of information-based complexity still lacks a notion which allows to quantify the exact (sub-/super-) exponential dependence of n ( e , S d ) on both parameters e and d . In this paper we present the notion of ( s , t ) -weak tractability which attempts to fill this gap. Within this new framework the parameters s and t are used to quantitatively refine the huge class of polynomially intractable problems. For linear, compact operators between Hilbert spaces we provide characterizations of ( s , t ) -weak tractability w.r.t.?the worst case setting in terms of singular values. In addition, our new notion is illustrated by classical examples which recently attracted some attention. In detail, we study approximation problems between periodic Sobolev spaces and integration problems for classes of smooth functions.

Let r 2 ( n ) denote the number of representations of n as a sum of two squares. Finding the precise order of magnitude for the error term in the asymptotic formula for ? n ? x r 2 ( n ) is known as the circle problem. Next, let d ( n ) denote the number of positive divisors of n . Determining the exact order of magnitude of the error term associated with the asymptotic formula for ? n ? x d ( n ) is the divisor problem. In his lost notebook, Ramanujan states without proof two identities that are associated with these two famous unsolved problems. It is natural to ask if identities exist for certain weighted sums, called Riesz sums, that generalize Ramanujan's identities. In this paper, we establish a Riesz sum identity that generalizes Ramanujan's identity linked to the divisor problem.

In this paper, a common generalization of the Rogers-Ramanujan series and the generating function for basis partitions is studied. This leads naturally to a sequence of polynomials, called BsP-polynomials. In turn, the BsP-polynomials provide simultaneously a proof of the Rogers-Ramanujan identities and a new, more rapidly converging series expansion for the basis partition generating function. Finally the basis partitions are identified with a natural set of overpartitions.

Let { p k } k = 0 ∞ be the orthogonal polynomials with certain exponential weights. In this paper, we prove that under certain mild conditions on exponential weights class, a series of the form Â? b k p k converges uniformly and absolutely on compact subsets of an open strip in the complex plane, and diverges at every point outside the closure of this strip.

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums S n + ( f , ζ ) = ∑ k = 0 n f ˆ ( k ) ζ k of f , which converges to g uniformly on E . This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C ∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2] .

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + \lambda_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained., Comment: 18 pages

The purpose of this paper is to develop a unified approach to the characterization of solutions of constrained and unconstrained approximation problems. Several papers have been written on the characterization of solutions of special approximation problems with particular types of constraints or without constraints. For uniform approximation a general theory has been obtained by using generalized weight functions. Recently a new approach via optimization theory has been presented in [I]. The idea is to show, first, that the local Kolmogoroff condition is satisfied. Assuming a convexity condition, it can be shown that the local Kolmogoroff condition implies the Kolmogoroff criterion. Hence best approximations are characterized by the local Kolmogoroff condition. An essential restriction in [I] is the assumption of linear equality constraints For uniform approximation problems with nonlinear equality constraints, the local Kolmogoroff condition has been deduced in [2] under the assumption cd a regularity condition that does not seem to be practical. By deleting inequality constraints a more satisfactory regularity condition has been studied in [3]. Our aim is to treat approximation problems with nonlinear equality and inequality constraints in a normed linear space and to present a new and satisfactory regularity condition. As in [I], we consider the problem as a particular type of optimization problem. Applying new kinds of differentiability, a new approach to optimization problems has been developed in [4]. A generalization of the well-known Lagrange multiplier theorem has been obtained that can be applied to convex optimization problems as well as to differentiable optimization problems. Here we shall apply this theorem to approximation problems with constraints. In particular we obtain new characterization theorems for constrained &-approximation problems of continuous functions.

G. Freud had wide research interests which included Fourier series. In this survey paper we intend to show the great influence of his only paper [2] on strong approximation of Fourier series. This article has been the origin of a new subject called nowadays “converse-type results for strong approximation of Fourier series.” Before stating his initial result we outline briefly the background of the subject. After the classical result of Fejer in 1904 on the convergence of the arithmetical mean of the partial sums of a Fourier series of 2x-periodic functions, Hardy and Littlewood [3] began to investigate the problem of so-called strong summability. It turned out that under certain conditions not only the means

In 1934, Walsh noted that the Taylor polynomial of degree n can be obtained by taking the limit as ϵ → 0 + of the net of n th degree polynomials which best approximate f in the closed discs ¦ z ¦ ⩽ ϵ . Later, this result was generalized to rational approximation. In a recent paper, Shisha and the first two authors generalized this idea to the idea of best local approximation . In this paper, using a different technique, we study this problem in the L 2 setting. Consequently, better results follow under weaker hypotheses.

For 1 d p < co, let L, denote the Banach space of pth power Lebesgue integrable functions on the interval [0, l] with /I f IID = (lh 1 f / p)“p. Let M, EL, denote the set of non-decreasing functions. Then M, is a closed convex lattice. For 1 < p < co, each f E L, has a unique best approximation from M,, while, for p = 1, existence of a best approximation from M, follows from Proposition 4 of [6]. Recently, there has been interest in characterizing best L, approximations from M, [ 1, 2, 3, 41. For example, in [ 1 ] it is shown that iff E L, and if each point in [0, l] is a Lebesgue point off [7], then the best L, approximation to f from M, is unique and continuous. In each of the papers mentioned above, the approach taken was measure theoretic, and the arguments were necessarily complicated. The purpose of this paper is to approach the best approximation problem from a duality viewpoint. This leads to considerable simplification in the derivation of the results, and allows for the omission of the assumption that f E L ~.

In a previous paper ( Numer. Math. 39 (1982), 1–14), M. Gasca and J. I. Maeztu used a geometrical method for the construction of the solutions of certain Hermite and Lagrange interpolation problems in R k . In the present paper, the method is generalized in two different ways: first, the interpolant is not assumed to be a polynomial, and second, a parameter is introduced in order to render the method more versatile.

In this paper we discuss the existence and uniqueness of the interpolating M-splines defined by Lucas [2]. Lucas has given two sets of sufficient conditions for the existence of M-splines. We obtain both necessary and sufficient conditions for the existence and uniqueness of such splines. Schaback [3] has generalized the concept of M-splines and studied the problem in the general setting of vector spaces. We restrict our attention to M-splines in Hilbert spaces. Throughout this paper we will adhere to the following notations: If X is a real Hilbert space and it4 is a closed subspace of X, then M’ denotes the orthogonal complement of A4 in X and PM stands for the projection operator taking X onto M. For x E X, the set @(x; M) is defined by

We prove a few interesting inequalities for Lorentz polynomials. A highlight of this paper states that the Markov-type inequality max x ? - 1 , 1 ] | f ' ( x ) | ? n max x ? - 1 , 1 ] | f ( x ) | holds for all polynomials f of degree at most n with real coefficients for which f ' has all its zeros outside the open unit disk. Equality holds only for f ( x ) : = c ( ( 1 ? x ) n - 2 n - 1 ) with a constant 0 ? c ? R . This should be compared with Erd?s's classical result stating that max x ? - 1 , 1 ] | f ' ( x ) | ? n 2 ( n n - 1 ) n - 1 max x ? - 1 , 1 ] | f ( x ) | for all polynomials f of degree at most n having all their zeros in R ? ( - 1 , 1 ) .