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

In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments cn(?) ? cn = ?10tnd?(t), n = 0, 1, 2, ..., of some singular distributions ? (with support ? 0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

For modified Szasz operators, S. M. Mazhar and V. Totik gave a direct approximation theorem for continuous functions. In this paper we extend this direct result to combinations of these operators. An inverse theorem to this direct estimate is given. An equivalent relation between the derivatives of these operators and smoothness of functions is also presented.

The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.

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.

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .

In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .

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.

The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderon products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderon complex interpolation spaces.

Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m -term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m -term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.

In this paper we introduce Laguerre expansions to approximate vector-valued functions. We apply this result to approximate C 0 -semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions in order to estimate the rate of convergence of these expansions. Finally, we illustrate the main results of this paper with some examples: shift, convolution and holomorphic semigroups, where the rate of convergence is improved.

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 generalize the notion of Kantorovich-type sampling operators using the Fejer-type singular integral. By means of these operators we are able to reconstruct signals (functions) which are not necessarily continuous. Moreover, our generalization allows us to take the measurement error into account. The goal of this paper is to estimate the rate of approximation by the above operators via high-order modulus of smoothness.

We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded ( k , p ) -variation ( V p k functions). It includes assertions on discontinuity sets and pointwise differentiability of V p k functions and their Luzin type and C ∞ approximations. The second part presents results on Banach structure of V p k spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing V p k spaces as second duals of their separable subspaces consisting of functions of “vanishing variation”.

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.

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painleve III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method.

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

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

In this paper, we investigate the smoothness of multivariate refinable functions with infinitely supported masks and an isotropic dilation matrix. Using some methods as in [R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089–4112], we characterize the optimal smoothness of multivariate refinable functions with polynomially decaying masks and an isotropic dilation matrix. Our characterizations extend some of the main results of the above mentioned paper with finitely supported masks to the case in which masks are infinitely supported.

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.

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.

In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest.

In the paper [Y. Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996) 220–238], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [-1,1]2, and derived a compact form of the corresponding Lagrange interpolation formula. In [L. Bos, M. Caliari, S. De Marchi, M. Vianello, A numerical study of the Xu polynomial interpolation formula in two variables, Computing 76(3–4) (2005) 311–324], we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows like O((logn)2), n being the degree. The aim of the present paper is to provide an analytic proof to show that the Lebesgue constant does have this order of growth.

This paper treats in detail the life and work of Otto Blumenthal, one of the most tragic figures of the 188 emigré mathematicians from Germany and the Nazi-occupied continent. Blumenthal, the first doctoral student of David Hilbert, was crucial in the publication and communication system of German mathematics between the two World Wars. There has been an unusual revival of interest in his mathematical work in the last three decades. Thus his work on orthogonal polynomials whose zeros are dense in intervals, called the Blumenthal theorem by T.S. Chihara (1972), lead to over two dozen recent papers in the field. The Blumenthal–Nevai theorem, with applications to scattering theory in physics, is one example. In modern work on Hilbert modular forms, increasingly being called Hilbert–Blumenthal modular forms, many recent papers even contain the word Blumenthal in their titles. This paper contains 212 references.

We investigate the solutions of vector refinement equations of the form ϕ= ∑ α ∈ Zs a(α)ϕ(M ċ - α), where the vector of functions ϕ = (ϕ1.....ϕr)T is in (Lp(Rs))r, 1 ≤ p ≤ ∞, a =: (a(α))α ∈ Zs is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn → ∞ M-n = 0. Associated with the mask a and M is a linear operator Qa defined on (Lp(Rs))r by Qaψ := ∑β ∈ Zsa(β)ψ(M ċ-β). The iteration scheme (Qanψ)n = 1.2,... is called a cascade algorithm (see [D.R. Chen, R.Q. Jia, S.D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002) 128-149; B. Han, The initial functions in a cascade algorithm, in: D.X. Zhou (Ed.), Proceeding of International Conference of Computational Harmonic Analysis in Hong Kong, 2002; B. Han, R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177-1199; R.Q. Jia, Subdivision schemes in Lp spaces, Adv. Comput. Math. 3 (1995) 309-341; R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1363; S. Li, Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms associated with nonhomogeneous refinement equations, Adv. Comput. Math. 20 (2004) 311-331; Q. Sun, Convergence and boundedness of cascade algorithm in Besov space and Triebel-Lizorkin space I, Adv. Math. (China) 29 (2000) 507-526]). Cascade algorithm is an important issue to wavelets analysis and computer graphics. Main results of this paper are related to the convergence and convergence rates of vector cascade algorithm in (Lp(Rs))r (1 ≤ p ≤ ∞). We give some characterizations on convergence of cascade algorithm and also give estimates on convergence rates of this cascade algorithm with M being isotropic dilation matrix. It is well known that smoothness is a very important property of a multiple refinable function. A characterization of Lp(1 ≤ p ≤ ∞) smoothness of multiple refinable functions is also presented when M = qIs × s, where Is×s is the s×s identity matrix, and q ≥ 2 is an integer. In particular, the smoothness results given in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999) 1-28] is a special case of this paper.

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.

Orthogonality of polynomials in several variables with respect to a positive Borel measure supported on an algebraic set is the main theme of this paper. As a step towards this goal quasi-orthogonality with respect to a non-zero Hermitian linear functional is studied in detail; this occupies a substantial part of the paper. Therefore necessary and sufficient conditions for quasi-orthogonality in terms of the three term recurrence relation modulo a polynomial ideal are accompanied with a thorough discussion. All this enables us to consider orthogonality in full generality. Consequently, a class of simple objects missing so far, like spheres, is included. This makes it important to search for results on existence of measures representing orthogonality on algebraic sets; a general approach to this problem fills up the three final sections.

The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.

Suppose that X- = (X 0 , X 1 ) and Y-=(Y 0 ,Y 1 ) are Banach couples and suppose that T 0 : X 0 → Y 0 and T 1 : X 1 → Y 1 are bounded and linear. Also assume that Γ ∈ (Δ(X-))' and that T 0 and T 1 agree as maps from Δ(X-) ∩ ker Γ to Σ(Y-). If the maps do not agree as maps from all of Δ(X-) we cannot interpolate T 0 and T 1 to a map T : J 0,p (X- → J 0,p (Y-), where J 0,p denotes the classical J-method. This situation can for example be found in an article on interpolation of Hardy-type inequalities by Krugljak, Maligranda and Persson. We will in this paper define functors J 0,p;Γ such that T 0 and T 1 interpolate to a map T: J 0,p:Γ (X-) → J 0,p (Y-). The main purpose of this paper is to make the definition of the J 0,p;Γ (X-) spaces and build a theory for them. We will also do this for more general real parameters. If Γ is bounded on X 0 it holds that J 0,p:Γ (X-)=J 0,p (X 0 ∩ ker Γ, X 1 ). These spaces have been studied by Kalton, Ivanov and Lofstrom. Their results will follow as corollaries to the more general results of this article and our new theory can be thought of as a theory for generalized subcouples of codimension one. In the last section, we apply our theory to a situation considered by Krugljak, Maligranda and Persson in connection with Hardy-type inequalities. We prove new results and provide a new way of understanding that kind of problems.

In his fundamental paper (RAIRO Anal. Numer. 12 (1978) 325) Duchon presented a strategy for analysing the accuracy of surface spline interpolants to sufficiently smooth target functions. In the mid-1990s Duchon's strategy was revisited by Light and Wayne (J. Approx. Theory 92 (1992) 245) and Wendland (in: A. Le Méhauté, C. Rabut, L.L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, Nashville, 1997, pp. 337–344), who successfully used it to provide useful error estimates for radial basis function interpolation in Euclidean space. A relatively new and closely related area of interest is to investigate how well radial basis functions interpolate data which are restricted to the surface of a unit sphere. In this paper we present a modified version Duchon's strategy for the sphere; this is used in our follow up paper (Lp-error estimates for radial basis function interpolation on the sphere, preprint, 2002) to provide new Lp error estimates (p∈[1,∞]) for radial basis function interpolation on the sphere.

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejer (or (C, 1)) means of functions on unbounded Vilenkin groups. There was no known positive result before the author's paper appeared in 1999 (J. Approx. Theory 101(1) (1999) 1) with respect to the a.e. convergence of the one-dimensional (C, 1) means of Lp (p > 1) functions. This paper is concerned with the almost everywhere convergence of a subsequence of the two-dimensional Fejer means of functions in L log- L. Namely, we prove the a.e. relation limn,k → ∞ σMnċM-k f = f (for the indices the condition |n - k| > α is provided, where α > 0 is some constant).

Let Q: R → R be even, nonnegative and continuous, Q' be continuous, Q' > 0 in (0, ∞), and let Q'' be continuous in (0, ∞). Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight WrQ2(x) = |x|2r exp(-2Q(x)). In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials {Pn(WrQ2; x)}n=0x with the generalized Freud-type weight WrQ2(x) on R. In this paper we treat three themes. Firstly, we give an estimate of Pn(WrQ2; x) in the Lp-space, 0 < p ≤ ∞. Secondly, we obtain the Markov inequalities, and third we study the higher-order Hermite Fejer interpolation polynomials based at the zeros {xkn}k=1n of Pn(WrQ2; x). In Section 5 we show that our results are applicable to the study of approximation for continuous functions by the higher-order Hermite-Fejer interpolation polynomials.

The paper considers properties of compactly supported, locally linearly independent refinable function vectors Φ = (φ 1 , ..., φ r ) T , r ∈ N. In the first part of the paper, we show that the interval endpoints of the global support of φ v , v = 1,..., r , are special rational numbers. Moreover, in contrast with the scalar case r = 1. we show that components φ v of a locally linearly independent refinable function vector Φ can have holes. In the second part of the paper we investigate the problem whether any shift-invariant space generated by a refinable function vector Φ possesses a basis which is linearly independent over (0, 1). We show that this is not the case. Hence the result of Jia, that each finitely generated shift-invariant space possesses a globally linearly independent basis, is in a certain sense the strongest result which can be obtained.

Consider the Hardy-type integral operator T : Lp(a,b) → Lp(a,b), -∞ ≤ a < b ≤ ∞, which is defined by (Tf)(x) = v(x) ∫ax u(t)f(t)dt.In the papers by Edmunds et al. (J. London Math. Soc. (2) 37 (1988) 471) and Evans et al. (Studia Math. 130 (2) (1998) 171) upper and lower estimates and asymptotic results were obtained for the approximation numbers an(T) of T. In case p = 2 for "nice" u and v these results were improved in Edmunds et al. (J. Anal. Math. 85 (2001) 225). In this paper, we extend these results for 1 < p < ∞ by using a new technique. We will show that under suitable conditions on u and v, lim supn → ∞ n1/2 |λp-1/p ∫ab |u(t)v(t)|dt - nan(T)| ≤ c(||u'||p'/(p'+1) + ||v'||p/(p+1))(||u||p' + ||v||p) + 3αp||uv||1, where ||w||p = (∫ab |w(t)|p dt)1/p and λp is the first eigenvalue of the p-Laplacian eigenvalue problem on (0, 1).

The topic of this paper is the study of four real, linear, possibly constrained minimum norm approximation problems, which arise in connection with the design of linear-phase nonrecursive digital filters and are distinguished by the type of used trigonometric approximation functions. In the case of unconstrained minimax designs these problems are normally solved by the Parks–McClellan algorithm, which is an application of the second algorithm of Remez to these problems and which is one of the most popular tools in filter design. In this paper the four types of approximation problems are investigated for all Lp and lp norms, respectively. It is especially proved that the assumptions for the Remez algorithm are satisfied in all four cases, which has been claimed, but is not obvious for three of them. Furthermore, results on the existence and uniqueness of solutions and on the convergence and the rate of convergence of the approximation errors are derived.

We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved Lp error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper.

In the present paper we explore an approximation theoretic approach to some classical convergence theorems of real analysis. The background of this paper is the intuition that some of the usual compactness theorems on various modes of convergence in classical analysis are based on suitable ways of obtaining good decompositions of functions to exploit rates of approximation, cancellations, or appropriate control of sizes that can be controlled by the basic functionals of real interpolation.

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

The left Bernstein quasi-interpolant operator introduced by Sablonniere is a kind of modified Bernstein operator that has good stability and convergence rate properties. However, we recently found that it is not very convenient for practical applications. Fortunately, we showed in a previous paper that there exist many operators having stability and convergence rate properties similar to those of Sablonniere's operator. In this paper, we introduce another specific class of such operators generated from the operator introduced by Stancu. We present detailed results about this class, some of which can be applied to numerical quadrature. Finally, we clarify its advantages and assert that it is more natural and more convenient, both theoretically and practically, than that of Sablonniere. Our paper, at the same time, provides several new results regarding Stancu's operator. (C) 1999 academic Press.

It is known that if f ∈ W k p , then ω m ( f , t ) p ⩽ tω m −1 ( f ′, t ) p ⩽…. Its inverse with any constants independent of f is not true in general. Hu and Yu proved that the inverse holds true for splines S with equally spaced knots, thus ω m ( S , t ) p ∼ tω m −1 ( S ′, t ) p ∼ t 2 ω m −2 ( S ″, t ) p …. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.

LetD?R2be the open unit disk. We consider best harmonic approximation to functions continuous onD. In a basic paper, Haymanet al.characterized best harmonic approximants which are themselves continuous onD. In this paper we give sufficient conditions and many simple examples of functions continuous onDwhich have no best harmonic approximants which are continuous onD.

The aim of this paper is to outline the life and work of Chebyshev, creator in St. Petersburg of the largest prerevolutionary school of mathematics in Russia, who permitted himself to be equated only with Archimedes. Chebyshev, who was regularly in Paris, at the latest by 1852, if not already by 1842, a friend of Liouville and Hermite, was the author of ca 80 publications, covering approximation theory, probability theory, number theory, theory of mechanisms, as well as many problems of analysis and practical mathematics. He was also proud to be a constructor of various mechanisms, including an arithmometre. Although the paper is intended for an approximation theorist readership, an attempt is made to give proportionate coverage of the broad spectrum of Chebyshev's achievements, emphasis being placed upon their background. The paper is based in part upon the authors' studies during 1985?1991

This paper is mainly concerned with the study of recurrences defined by Möbius-transformations, whose solutions are the orbits of points on the Riemann-sphere under a sequence of Möbius-transformations. We study the asymptotic behaviour of such solutions in relation to the asymptotic behaviour of the coefficients of the Möbius-transformations. Most of the theorems give sufficient conditions in order that there exist converging solutions, but a section of examples is added where examples are given of recurrences whose solutions do not converge because one or several of the conditions of the theorems are violated. One of the most important results of this paper is that if the fixpoints of the Möbius-transformations are of bounded variation and converge to distinct limits, then the behaviour of the solutions depends entirely on the products of the derivatives in the fixpoints. Several methods will be proposed to deal with the case that the fixpoints converge to one single limit. The paper starts with a few results onnth order recurrences and matrix recurrences and concludes with an investigation of the asymptotic behaviour of the solutions of linear second-order recurrences having coefficients that are asymptotic expressions in fractional powers of the indexn. A number of examples are added in order to show how some of the theorems can be applied.