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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.

The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting a la Hedberg and Netrusov (2007), which includes weighted anisotropic mixed-norm Besov and Lizorkin–Triebel spaces. In the special case of the classical Lizorkin–Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted L q - L p -maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin–Triebel spaces occur as spaces of boundary data.

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.

Meixner type polynomials ( q n ) n ≥ 0 are defined from the Meixner polynomials by using Casoratian determinants whose entries belong to two given finite sets of polynomials ( S h ) h = 1 m 1 and ( T g ) g = 1 m 2 . They are eigenfunctions of higher order difference operators but only for a careful choice of the polynomials ( S h ) h = 1 m 1 and ( T g ) g = 1 m 2 , the sequence ( q n ) n ≥ 0 is orthogonal with respect to a measure. In this paper, we prove that the Meixner type polynomials ( q n ) n ≥ 0 always satisfy higher order recurrence relations (hence, they are bispectral). We also introduce and characterize the algebra of difference operators associated to these recurrence relations. Our characterization is constructive and surprisingly simple. As a consequence, we determine the unique choice of the polynomials ( S h ) h = 1 m 1 and ( T g ) g = 1 m 2 such that the sequence ( q n ) n ≥ 0 is orthogonal with respect to a measure.

In this paper, we study the distributed learning algorithm and the distribution regression problem of coefficient regularization for Mercer kernels. By utilizing divided-and-conquer approach, we partition a data set into disjoint data subsets for different learning machines, and get the global estimator from local estimators. By using second order decomposition on the difference of operator inverse and properties of trace operator, we show that under some priori conditions of regression function, the result of distributed learning algorithm is as good as that in single batch data algorithm. On the other hand, we give a learning rate of distribution regression problem under the coefficient regularization scheme by using similar operator methods. We find that our learning scheme performs well when the regression function has stronger regularity. And we can see the deep relation of these two different problems.

Information theoretic learning is a learning paradigm that uses concepts of entropies and divergences from information theory. A variety of signal processing and machine learning methods fall into this framework. Minimum error entropy principle is a typical one amongst them. In this paper, we study a kernel version of minimum error entropy methods that can be used to find nonlinear structures in the data. We show that the kernel minimum error entropy can be implemented by kernel based gradient descent algorithms with or without regularization. Convergence rates for both algorithms are deduced.

In this paper, we find strong uniqueness constants for a certain class of operators with a unit norm from a space of dimension 2 n onto its subspace of codimension n , which is formed by using hyperplanes in l ∞ 2 n ( n ≥ 2 ) .

In this paper we will study the harmonic analysis of a class of Cantor–Moran measures μ with three-element digit sets on R . Our results give some sufficient conditions for the maximal orthonormal set of exponential functions to be or not to be a basis for the space L 2 ( μ ) .

In this paper we derive optimal subspaces for Kolmogorov n-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases.

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 the divergence behavior of linear approximation processes in general Banach spaces. We are interested in the structure of the set of vectors creating divergence. The Banach–Steinhaus theory gives some information about this set, however, it cannot be used to answer the question whether this set contains subspaces with linear structure. We give necessary and sufficient conditions for the lineability and the spaceability of the set of vectors creating divergence.

In this paper, we study the sequence of orthogonal polynomials { S n } n = 0 ∞ with respect to the Sobolev-type inner product 〈 f , g 〉 = ∫ − 1 1 f ( x ) g ( x ) d μ ( x ) + ∑ j = 1 N η j f ( d j ) ( c j ) g ( d j ) ( c j ) where μ is a finite positive Borel measure whose support supp μ ⊂ [ − 1 , 1 ] contains an infinite set of points, η j > 0 , N , d j ∈ Z + and { c 1 , … , c N } ⊂ R ∖ [ − 1 , 1 ] . Under some restriction of order in the discrete part of 〈 ⋅ , ⋅ 〉 , we prove that for sufficiently large n the zeros of S n are real, simple, n − N of them lie on ( − 1 , 1 ) and each of the mass points c j “attracts” one of the remaining N zeros. The sequences of associated polynomials { S n [ k ] } n = 0 ∞ are defined for each k ∈ Z + . If μ is in the Nevai class M ( 0 , 1 ) , we prove an analogue of Markov’s Theorem on rational approximation to Markov type functions and prove that convergence takes place with geometric speed.

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.

The D -invariant polynomial subspaces play a crucial role in ideal interpolation. In this paper, we analyze the structure of a second-degree D -invariant polynomial subspace P 2 . As an application for ideal interpolation, we solve the discrete approximation problem for ? z P 2 ( D ) under certain conditions, i.e., we compute pairwise distinct points, such that the limiting space of the evaluation functionals at these points is the given space ? z P 2 ( D ) , as the evaluation sites all coalesce at one site z .

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.

Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data’s structure or on the behavior of the unknown function can make the task well-posed. In this work, we study the problem of learning nonlinear functions from corrupted and weakly dependent data. The learning problem is recast as a sparse robust linear regression problem where we incorporate both the unknown coefficients and the corruptions in a basis pursuit framework. The main contribution of our paper is to provide a reconstruction guarantee for the associated l 1 -optimization problem where the sampling matrix is formed from weakly dependent data. Specifically, we prove that the sampling matrix satisfies the null space property and the stable null space property, provided that the data is compact and satisfies a suitable concentration inequality. We show that our recovery results are applicable to various types of weakly dependent data such as exponentially strongly α -mixing data, geometrically C -mixing data, and uniformly ergodic Markov chain. Our theoretical results are verified via several numerical simulations.

After the appearance of the article “A retrospective on 60 years of approximation theory and associated fields” (J. Approx. Theory 160 (1–2) (2009) 3–18), several readers informed me (PLB) that they would like to see an article based upon research contacts and conference participations of members of the chair “Lehrstuhl A fur Mathematik” at Aachen throughout the world. The present paper is devoted to the research visits of members of the Aachen research group to America and Western Europe. The countries covered are USA and Canada, Austria, Netherlands, Belgium, Switzerland, Sweden, England and Scotland, Italy, Spain, Portugal, Norway, Greece, and Ireland.

In this paper, we discuss various basic properties of moduli of smoothness of functions from L p ( R d ) , 0 p ≤ ∞ . In particular, complete versions of Jackson-, Marchaud-, and Ulyanov-type inequalities are given for the whole range of p . Moreover, equivalences between moduli of smoothness and the corresponding K -functionals and the realization concept are proved.

In this paper we prove that if a multivariate function of a certain smoothness class is represented by a sum of k arbitrarily behaved ridge functions, then it can be represented by a sum of k ridge functions of the same smoothness class and a polynomial of degree at most k − 1 . This solves the problem posed by A. Pinkus in his monograph “Ridge Functions” up to a multivariate polynomial.

We study bracketing covering numbers for spaces of bounded convex functions in the L p norms. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Bracketing number upper bounds in the supremum distance are known for bounded classes that also have a fixed Lipschitz constraint. However, in most settings of interest, the classes that arise do not include Lipschitz constraints, and so standard techniques based on known bracketing numbers cannot be used. In this paper, we find upper bounds for bracketing numbers of classes of convex functions without Lipschitz constraints on arbitrary polytopes. Our results are of particular interest in many multidimensional estimation problems based on convexity shape constraints. Additionally, we show other applications of our proof methods; in particular we define a new class of multivariate functions, the so-called m -monotone functions. Such functions have been considered mathematically and statistically in the univariate case but never in the multivariate case. We show how our proof for convex bracketing upper bounds also applies to the m -monotone case.

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.

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight e - n ( z - log z + t / z ) for t < 0 and z on certain contours in the complex plane. When the parameters n , t and the degree k are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy ? -function of the Painleve III equation. When the degree k = n , n is large and t is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquee solution to the Painleve I equation. Our approach is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

In 1987 the author gave an example of a non convex Chebyshev set S in the incomplete inner product space E consisting of the vectors in l 2 which have at most a finite number of non zero terms. In this paper, we show that the closure of S in the Hilbert space completion l 2 of E is not Chebyshev in l 2 .

The purpose of this paper is to find the relative projection constant of ker { 1 } = { x : ? i = 1 n x i = 0 } in ? p n for an arbitrary n (see Theorem 2.28). This extends a result of L. Skrzypek obtained for n = 3 , 4 . We also improve on the formula for the relative projection constant onto hyperplanes in L p 0 , 1 obtained by C. Franchetti (see Theorem 2.21) and reprove a result of S. Rolewicz (see Corollary 2.11).

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 this paper we investigate the asymptotic behaviour of Weyl numbers of embeddings of tensor product Besov spaces into Lebesgue spaces. These results will be compared with the known behaviour of entropy numbers., 54 pages, 2 figures

The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the Holder spaces H p r , α for all 0 < p ? ∞ and 0 < α ? r . By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria for the precise order of decrease of the best approximation in these spaces. Moreover, we obtained strong converse inequalities for general methods of approximation of periodic functions in H p r , α . Sharp estimates of approximation of periodic functions in Holder spaces.Improvements of the direct and inverse theorems of approximation in Holder spaces.New estimates of the error of the best approximation in Holder spaces.The criteria of precise order of decrease of the best approximation in Holder spaces.Strong converse inequalities for general methods of approximation in Holder spaces.

The paper deals with Haar wavelet bases in function spaces of Besov and Triebel–Lizorkin type with local Muckenhoupt weights. We show that Haar wavelets can be used to characterize such function spaces as far as absolute value of smoothness parameter is small enough and weights fulfill some conditions. The result is based on mapping properties of linear operators involving characteristic functions of dyadic cubes in related spaces and on local means characterization of weighted Besov and Triebel–Lizorkin spaces.

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.

There have been known two generalized Orlicz-Morrey spaces. One is defined earlier by Nakai and the other is by Sugano, the second and third authors. In this paper we investigate differences between these two spaces in some typical cases. The arguments rely upon property of the characteristic function of the Cantor set.

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.

For n ? 1 let A n ? { P : P ( z ) = ? j = 1 n z k j : 0 ? k 1 < k 2 < ? < k n , k j ? Z } , that is, A n is the collection of all sums of n distinct monomials. These polynomials are also called Newman polynomials. If α < β are real numbers then the Mahler measure M 0 ( Q , α , β ] ) is defined for bounded measurable functions Q ( e i t ) on α , β ] as M 0 ( Q , α , β ] ) ? exp ( 1 β - α ? α β log | Q ( e i t ) | d t ) . Let I ? α , β ] . In this paper we examine the quantities L n 0 ( I ) ? sup P ? A n M 0 ( P , I ) n and L 0 ( I ) ? lim inf n ? ∞ L n 0 ( I ) with 0 < | I | ? β - α ? 2 π .

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.

This paper is to construct tight wavelet frame systems containing a set of canonical filters by applying the unitary extension principle of Ron and Shen (1997). A set of filters are canonical if the filters in this set are generated by flipping, adding a conjugation with a proper sign adjusting from one filter. The simplest way to construct wavelets of s -variables is to use the 2 s - 1 canonical filters generated by the refinement mask of a box spline. However, almost all wavelets (except Haar or the tensor product of Haar) defined by the canonical filters associated with box splines do not form a tight wavelet frame system. We consider how to build a filter bank by adding filters to a canonical filter set generated from the refinement mask of a box spline in low dimension, so that the wavelet system generated by this filter bank forms a tight frame system. We first prove that for a given low dimension box spline of s -variables, one needs at least additional 2 s filters to be added to the canonical filters from the refinement mask (that leads to the total number of highpass filters in the filter bank to be 2 s + 1 - 1 ) to have a tight wavelet frame system. We then provide several methods with many interesting examples of constructing tight wavelet systems with the minimal number of framelets that contain canonical filters generated by the refinement masks of box splines. The supports of the resulting framelets are not bigger than that of the corresponding box spline whose refinement mask is used to generate the first 2 s - 1 canonical filters in the filter bank. In many of our examples, the tight frame filter bank has the double-canonical property, meaning it is generated by adding another set of canonical filters generated from a highpass filter to the canonical filters generated by the refinement mask to make a tight frame system.

The collective behavior of the singular values of large Toeplitz matrices is described by the Avram-Parter theorem. In the case of Hermitian matrices, the Avram-Parter theorem is equivalent to Szeg?'s theorem on the eigenvalues. The Avram-Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values.

Spherical tt-design is a finite subset on sphere such that, for any polynomial of degree at most tt, the average value of the integral on sphere can be replaced by the average value at the finite subset. It is well-known that an equivalent condition of spherical design is given in terms of harmonic polynomials. In this paper, we define a spherical design of harmonic index tt from the viewpoint of this equivalent condition, and we give its construction and a Fisher type lower bound on the cardinality. Also we investigate whether there is a spherical design of harmonic index attaining the bound.

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form.

We consider the extremal problem of maximizing a point value � f ( z ) � at a given point z � G by some positive definite and continuous function f on a locally compact Abelian group (LCA group) G , where for a given symmetric open set � � z , f vanishes outside � and is normalized by f ( 0 ) = 1 .This extremal problem was investigated in R and R d and for � a 0-symmetric convex body in a paper of Boas and Kac in 1945. Arestov, Berdysheva and Berens extended the investigation to T d , where T : = R / Z . Kolountzakis and Revesz gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Caratheodory and Fejer.Moreover, following observations of Boas and Kac, Kolountzakis and Revesz showed how the general problem can be reduced to equivalent discrete problems of "Caratheodory-Fejer type" on Z or Z m : = Z / m Z . We extend their results to arbitrary LCA groups.

This paper gives the following generalization of Markov's inequalities ? P ( k ) ? p ? ( C ( p + 1 ) k 2 ) 1 / p ? T n ( k ) ? ∞ ? P ? p for the k th derivative of an algebraic polynomial in L p norms, where p ? 1 . In particular we show that for any k ? 3 the constant C in the Markov inequality satisfies C ? 12 2 3 e 2 .

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.

The generalized Gegenbauer functions of fractional degree (GGF-Fs), denoted by r G ν ( λ ) ( x ) (right GGF-Fs) and l G ν ( λ ) ( x ) (left GGF-Fs) with x ∈ ( − 1 , 1 ) , λ > − 1 ∕ 2 and real ν ≥ 0 , are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for λ > − 1 ∕ 2 and x = cos θ with θ ∈ ( 0 , π ) , ( sin θ ) λ r G ν ( λ ) ( cos θ ) = 2 λ Γ ( λ + 1 ∕ 2 ) π ( ν + λ ) λ cos ( ( ν + λ ) θ − λ π ∕ 2 ) + R ν ( λ ) ( θ ) , and derive the precise expression of the “residual” term R ν ( λ ) ( θ ) for all real ν ≥ ν 0 (with some ν 0 > 0 ). With this at our disposal, we obtain the bounds of GGF-Fs uniform in ν . Under an appropriate weight function, the bounds are uniform for θ ∈ [ 0 , π ] as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree ν . The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.

We correct a mistake in the statements of Corollaries 4.11 and 4.12. The mistake has no bearing to their proofs or other results in the paper.

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] .

Let $X$ be a Banach space and $\mathcal{K}$ be a compact subset in $X$. We consider a greedy algorithm for finding an $n$-dimensional subspace $V_n\subset X$ which can be used to approximate the elements of $\mathcal{K}$. We are interested in how well the space $V_n$ approximates the elements of $\mathcal{K}$. For this purpose we compare the performance of greedy algorithm measured by $\sigma_n(\mathcal{K})_X:=\text{dist}(\mathcal{K},V_n)_X$ with the Kolmogorov width $d_n(\mathcal{K})_X$ which is the best possible error one can achieve when approximating $\mathcal{K}$ by $n$-dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that there exists a constant $C>0$ such that $$ \sigma_n(\mathcal{K})_X\leq C n^{-s+\mu}\big(\log(n+2)\big)^{\min(s,1/2)}, \quad \ n\geq 1\,, $$ if Kolmogorov widths $d_n(\mathcal{K})_X$ decay as $n^{-s}$ and the Banach-Mazur distance between an arbitrary $n$-dimensional subspace $V_n \subset X$ and $\ell_2^n$ satisfies $d(V_n,\ell_2^n)\leq C_1 n^\mu$. In particular, when some additional information about the set $\mathcal{K}$ is given then there is no logarithmic factor in this estimate., Comment: 14 pages

This paper deals with a study of E n ( f ) L w 2 ( S ) , the rate of approximation by polynomials of total degree n in terms of the norm of L w 2 ( S ) where S is the simplex S = { ( x 1 , … , x d ) : x i ≥ 0 , x 0 = 1 − ∑ i = 1 d x i ≥ 0 } and w = w γ = x 0 γ 0 … x d γ d , γ i > − 1 . Direct and converse results are achieved relating E n ( f ) L w 2 ( S ) to the operators φ ξ r ( x ) ( ∂ ∂ ξ ) r f and corresponding K -functionals, where ξ ∈ E ( S ) and E ( S ) are the edges of S . For the special case that S is the triangle somewhat weaker results were recently achieved in Feng et al. (2019).

In this paper we consider a generalization of the Bernstein-Durrmeyer operator where the integrals are taken with respect to measures that may vary from term to term. This construction is more general than the one considered by the first named author and her coauthors earlier, and it includes a number of well-known operators of Bernstein type as particular cases. We give conditions on the collections of measures that guarantee pointwise convergence at a point of continuity of a function.

With respect to generalized trigonometric functions, since the discovery of double-angle formula for a special case by Edmunds, Gurka and Lang in 2012, no double-angle formulas have been found. In this paper, we will establish new double-angle formulas of generalized trigonometric functions in two special cases., Comment: 7 pages

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