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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

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.

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 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.

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.

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 ) .

We introduce the concept of D-operators associated to a sequence of polynomials (p"n)"n and an algebra A of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family (p"n)"n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials (q"n)"n by considering a linear combination of two consecutive p"n: q"n=p"n+@b"np"n"-"1, @b"n@?R. Using the concept ofD-operator, we determine the structure of the sequence (@b"n)"n in order that the polynomials (q"n)"n are common eigenfunctions of a higher order difference operator. In addition, we generate sequences (@b"n)"n for which the polynomials (q"n)"n are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.

In this paper, we solve a characterization problem involving a suitable basic-hypergeometric form of a polynomial set. That allows us to introduce new examples of L q -classical d -orthogonal polynomials, generalizing the discrete q -Hermite polynomials in the context of d -orthogonality, and a q -analogous for the d -orthogonal polynomials of Gould–Hopper. For the resulting polynomials, we derive miscellaneous properties. Those turn out to be limit relations, recurrence relations of order ( d + 1 ) , difference formulas, generating functions, inversion formulas, and d -dimensional functional vectors.

It has recently been shown that the Lebesgue constant for Berrut's rational interpolant at equidistant nodes grows logarithmically in the number of interpolation nodes. In this paper we show that the same holds for a very general class of well-spaced nodes and essentially any distribution of nodes that satisfies a certain regularity condition, including Chebyshev-Gauss-Lobatto nodes as well as extended Chebyshev nodes.

We consider Leja sequences of points for polynomial interpolation on the complex unit disk U and the corresponding sequences for polynomial interpolation on the real interval [−1,1] obtained by projection. It was proved by Calvi and Phung in Calvi and Phung (2011, 2012) [3,4] that the Lebesgue constants for such sequences are asymptotically bounded in O(klogk) and O(k3logk) respectively, where k is the number of points. In this paper, we establish the sharper bound 5k2logk in the real interval case. We also give sharper bounds in the complex unit disk case, in particular 2k. Our motivation for producing such sharper bounds is the use of these sequences in the framework of adaptive sparse polynomial interpolation in high dimension.

In this paper, we investigate nonlinear m-term approximation with regard to orthogonal dictionaries. We consider this problem in the periodic multivariate case for generalized Besov classes MB"q","@q^@W under the condition @W(t)=@w(t"1@?...@?t"d) where @w(t)@?@J"l^* is a univariate function. We prove that the well-known dictionary U^d which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthogonal dictionaries. Moreover, it is established that for these classes near-best m-term approximation, with regard to U^d, can be achieved by simple greedy-type algorithms.

In the present paper we find a new interpretation of Narayana polynomials N-n(x) which are the generating polynomials for the Narayana numbers N-n,N-k = 1/nC(n)(k-1)C(n)(k) where C-j(i) stands for the usual binomial coefficient, i.e. C-j(i) = j!/i!(j-i)!. They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1-2) (2002) 311-326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67-82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909-2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147-3160]. Strangely enough Narayana polynomials also occur as limits as n -> infinity of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n - 1)-tuples of polynomials of the form (x +1)(n-1) (x + a) to their Schur-Szego product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N-n(x)}.

This paper presents a framework of iterative algorithms for the variational inequality problem over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings in real Hilbert spaces. Strong convergence theorems are established under a certain contraction assumption with respect to the weighted maximum norm. The proposed framework produces as a simplest example the hybrid steepest descent method, which has been developed for solving the monotone variational inequality problem over the intersection of the fixed point sets of nonexpansive mappings. An application to a generalized power control problem and numerical examples are demonstrated.

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss-Weierstrass operators. We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form [email protected][email protected]^'^'[email protected]^'[email protected] Shape-preserving and regularity properties are also investigated.

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L"~ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper. We show that the L"~ approximation problem in the worst case setting is related to the weighted L"2 approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter @a>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-^(^@a^-^1^)^/^2 and n^-^1^/^2, respectively. We also study tractability of L"~ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L"~ approximation in terms of tractability conditions of the weighted L"2 approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form W ( t ) = t α e - t e At t B t B * e A * t , where A and B are certain (nilpotent and diagonal, respectively) N × N matrices. These weight matrices are the first examples illustrating this new phenomenon which are not reducible to scalar weights.

Starting from a slight modification of the dyadic sets introduced by M. Christ in [A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601–628] on a space of homogeneous type (X,d,μ), an MRA type structure and a Haar system H controlled by the quasi distance d, can be constructed in this general setting in such a way that H is an orthonormal basis for L2(dμ). This paper is devoted to explore under which conditions on the measure ν, the system H is also an unconditional basis for the Lebesgue spaces Lp(dν). As a consequence, we obtain a characterization of these spaces in terms of the H–coefficients.

There is a series of publications which have considered inequalities of Markov–Bernstein–Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159–176; B.D. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (1982) 181–190; P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov–Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1 (1972) 15–25; A. Guessab, G.V. Milovanovic, Weighted L2-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244–249; I.I. Ibragimov, Some inequalities for algebraic polynomials, in: V.I. Smirnov (Ed.), Fizmatgiz, 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570–572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) (1957) 32–35; M.K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1944) 165–209; P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960) 373–378]). In this paper we find an inequality of the same type for algebraic polynomials on (0,∞) with the Laguerre weight function e-xxα (α>-1).

The sets of nodes in the plane for which its nth degree Lagrange polynomials can be factored as a product of first degree polynomials satisfy a geometric characterization: for each node there exists a set of ≤n lines containing the other nodes. Generalized principal lattices are sets of nodes defined by three families of lines. A generalized principal lattice satisfies the geometric characterization and there exist exactly three lines in the plane containing more nodes than the degree. In this paper, we show a converse, valid for degrees n≤7: if a set of nodes satisfy the geometric characterization and there exist exactly three lines containing n+1 nodes, then it is a generalized principal lattice.

The Walsh system will be investigated in the Kaczmarz rearrangement. In an earlier paper we have shown that the maximal operator of the (C, 1)-means of the Walsh-Kaczmarz-Fourier series is bounded from the dyadic Hardy space Hp into Lp for every ½ < p ≤ 1. In the present work, we extend this result to the (C, α) means when 0 < α ≤ 1 and prove their maximal operator σα : Hp → Lp is bounded for all 1/(α + 1) < p ≤ 1. By known results on interpolation we get from this theorem that σα is of weak type (1,1) and bounded from Lq into Lq if 1 < q ≤ ∞. Moreover, the (C, α) means of an integrable function f converge to f a.e.

This paper studies the structure of shift-invariant spaces. A characterization for the univariate shift-invariant spaces of tempered distributions is given. In Lp case, an inclusive relation in terms of Fourier transform is established.

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by Mn,p(G), 1⩽p

S. A. Telyakovskiı (1964, Izv. Akad. Nauk. SSR. Ser. Mat.28, 1209–1236) proved an integrability condition for cosine series. No condition superior to that has been given so far. In this paper we identify the atomic structure of the Hardy type space that can be associated with this condition. As a consequence, we conclude that Telyakovskiı's condition is equivalent to certain Sidon type inequalities. Then on the basis of this equivalence we show how the atomic technique can be used to extend Telyakovskiı's condition to several systems, including Walsh series and integrals, in a uniform way.

In this paper we prove convergence rates for the problem of approximating functions f by neural networks and similar constructions. We show that the rates are the better the smoother the activation functions are, provided that f satisfies an integral representation. We give error bounds not only in Hilbert spaces but also in general Sobolev spaces Wm, r(Ω). Finally, we apply our results to a class of perceptrons and present a sufficient smoothness condition on f guaranteeing the integral representation.

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147-156) a theorem of Borwein and Chen was used to show that for each [email protected]?N the zeros of the hypergeometric polynomials F(-n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z:|z^k(1-z)|=k^k/(k+1)^k^+^1}, with Re{z}>k/(k+1) as n->~. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.

Let l k , 1⩽ k ⩽ n , be the fundamental polynomials of Lagrange interpolation on the nodes x n x n −1 x 1 . The classical Erdős–Turan inequality is l k (x)+l k +1 (x)⩾1, x∈[x k +1 , x k ], 1⩽k⩽n−1. This paper gives an extension for such an inequality to the sum of successive fundamental polynomials of Hermite interpolation.

Let H be a real Hilbert space and let T:H→2H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit v∈T−10. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.

For [email protected]?C[-1,1], let H"m","n(f,x) denote the (0, 1, ...,anbsp;m) Hermite-Fejer (HF) interpolation polynomial of f based on the Chebyshev nodes. That is, H"m","n(f,x) is the polynomial of least degree which interpolates f(x) and has its first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind. In this paper a precise pointwise estimate for the approximation error |H"2"m"," "n(f, x)-f(x)| is developed, and an equiconvergence result for Lagrange and (0, 1, ..., 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0, 1, ..., 2m) HF interpolation methods on the Chebyshev nodes, is convergent for all [email protected]?C[-1, 1].

Let C(B) denote the space of real-valued continuous functions on B. At the conference “Harmonic Analysis and Approximations” held at Nor Amberd in Armenia in September, 1998, the following general problem was posed by Professor George G. Lorentz: Find conditions on finite-dimensional subspaces U and V so that U/V is a uniqueness set in the problem of best uniform approximation to elements of C(B). In this paper we consider this problem.

This paper considers Lagrange interpolation in the rational system {1/(x−a1), 1/(x−a2),…}, which is based on the zeros of the Chebyshev polynomial for the rational system {;1,1/(x−a1), 1/(x−a2),…} with distinct real poles {ak}∞k=1R\[−1,1]. The corresponding Lebesgue constant is estimated, and is shown to be asymptotically of order lnnwhen the poles stay outside an interval which contains [−1,1] in its interior. Some well-known results of classical polynomial interpolation are extended.

Letf∈B2σ, i.e.,f∈L2(R) and its Fourier transformF(s)=∫Rf(t)e−2πistdtvanishes outside of [−σ, σ], then the Shannon sampling theorem says thatfcan be reconstructed by its infinitely many sampling points at {k/(2σ)},k∈Z, i.e.,f(t)=∑k=−∞∞fk2σsinπ(2σt−k)π(2σt−k),∀t∈R.But, in practice, only finitely many samples are available, so one would like to study the truncation errorTN(t)=f(t)−∑k=−NNfk2σsinc(2σt−k),∀f∈B2σ.The error bounds commonly seen in literature are not uniform. In this paper, the author gives uniform bounds for the truncation error forf∈B2σ, when its Fourier transform satisfies some smooth conditions.

This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolation is a linear approximation that makes use of values of f on a well chosen finite set. The approximation is a discrete least-squares approximation constructed with the aid of a high-order quadrature rule: the role of the quadrature rule is to approximate the Fourier coefficients of f with respect to an orthonormal basis of the space of polynomials of degree ≤ n. The principal result is a generalization of the result of Erdos and Turan for classical interpolation at the zeros of orthogonal polynomials: for a rule of suitably high order (namely 2n or greater), the L2 error of the approximation is shown to be within a constant factor of the error of best uniform approximation by polynomials of degree ≤ n. The L2 error therefore converges to zero as the degree of the approximating polynomial approaches ∞. An example discussed in detail is the approximation of continuous functions on the sphere in R s by spherical polynomials. In this case the number of quadrature points must exceed the number of degrees of freedom if n > 2 and s ≥ 3. In such a situation the classical interpolation property cannot hold, whereas satisfactory hyperinterpolation approximations do exist.

The purpose of this paper is to show regularity of (0, 1, ..., r − 2, r) and (0, 1, ..., r − 2, r)* interpolations on the sets obtained by projecting vertically the zeros of (1 − x2)P(α,β)n(x) (− 1) < α,β ≤ 12), (1 − x2)P(α,β)n(x) (− 1 < α ≤ 12, − 1 < β ≤ −12) and (1 + x) P(α,β)n(x) (− 1 < α ≤ −12, − 1 < β ≤ 12) respectively onto the unit circle, where P(α,β)n(x) stands for the nth Jacobi polynomial.

Definite quadrature formulae are used to obtain inclusions for the integral of a function f for which f(r) has no sign change. In this paper, error bounds for optimal definite quadrature formulae for the classes Cr[0, 1] (r ≥ 1) are considered. In the case of odd r (except for special cases), such estimates seem to be unknown.

It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper we show that the case of equally spaced nodes is not an exceptional one in this sense. Namely, we prove that the divergence everywhere in 0 < |x| < 1 of the Lagrange interpolation to |x| takes place for a broad family of nodes, including in particular the Newman nodes, which are known to be very efficient for rational interpolation.

In this paper we study the quantities [formula] which define error bounds for the approximation of functions ƒ ∈ Wm∞[a, b] by the interpolating Lagrange polynomials lm−1, Δ(f) of degree m − 1, constructed on the given mesh of interpolating nodes Δ = Δm = {a ≤ t1 < ··· < tm ≤ b}. Set [formula] It is clear that Lm, k(Δ, x) ≥ 1m!|ω(k)Δ(x)|, Lm, k(Δ) ≥ 1m!∥ω(k)Δ(·)∥. Our main result is THEOREM 1. For all m and k (0 ≤ k ≤ m − 1), and for any mesh Δ of the interpolating nodes {ti}m1Lm, k(Δ) = 1m!∥ω(k)Δ(·)∥.

The notion of mean dimension was introduced in the 1970s by Tikhomirov. It determines the mean number of linear dimensions required to identify an element of a given function class. Tikhomirov then posed the following problem: find the mean dimension of the unit ball B p E of the space of L p -functions on R n with spectra inside a given Lebesgue measurable bounded set E . In the language of signal analysis: determine the amount of linear information carried by generalized band-limited signals. In this paper Tikhomirov′s conjecture on mean dimension is confirmed in certain important cases and yet shown to fail in certain other cases.

This paper gives new admissible values for the constant in Markov inequality in the p -metric. We improve a classical theorem of Hille, Szego, and Tamarkin. For p = 2, sharp numerical values are obtained.

Recently I. J. Schoenberg studied the cardinal splines that interpolate the function λ x at the integers, where λ is a complex number. This paper deals with cardinal splines which together with their successive derivatives interpolate λ x and its successive derivatives at the integers.

This paper contains a brief account on complex planar splines which are complex valued functions defined piecewise on a grid. For noncontinuous (so called nonconforming) splines the problem of the placement of knots at which these splines are required to be continuous is investigated. It is shown that this problem reduces to finding complex Chebyshev polynomials under the additional requirement that the zeros of the polynomials are on the boundary of the corresponding domains. It is proved that the zeros of a generalized Chebyshev polynomial are in the convex hull of the domain on which the Chebyshev polynomials are defined. Some open problems are stated. A numerical and graphical display for the optimal location of three and six points on certain triangles is provided.

In this paper we develop quadrature formulas for splines with equispaced knots. For special classes of splines, we give simple explicit expressions for the weights in the quadrature formulas in terms of the zeros of the Euler-Frobenius polynomials and show that these weights are positive. The zeros of these polynomials of odd degree up to 15, are given by Nilson [2] and by Schoenberg and Silliman [4] to a high degree of accuracy. The general quadrature formulas can also be used to obtain the cubic natural spline quadrature formula given in Ahlberg, Nilson and Walsh [ 1, pp. 44-471 and the semicardinal odd order natural spline formula of Schoenberg and Silliman [4]. Two of the quadrature formulas that we derive can be stated as follows. Let S be a spline of odd degree m with knots at the integers 0, l,..., n. Suppose that S’*“(O+) = S’*“(C) = 0, for 2 < 2i < m 1, where ,S”’ denotes the kth derivative of S. Then

This paper discusses the problem of choosing the Lagrange interpolation points T = ( t 0 , t 1 ,…, t n ) in the interval −1 ⩽ t ⩽ 1 to minimize the norm of the error, considered as an operator from the Hardy space H 2 ( R ) of analytic functions to the space C [−1, 1]. It is shown that such optimal choices converge for fixed n , as R → ∞, to the zeros of a Chebyshev polynomial. Asymptotic estimates are given for the norm of the error for these optimal interpolations, as n → ∞ for fixed R . These results are then related to the problem of choosing optimal interpolation points with respect to the Eberlein integral. This integral is based on a probability measure over certain classes of analytic functions, and is used to provide an average interpolation error over these classes. The Chebyshev points are seen to be limits of optimal choices in this case also.

Orthogonal polynomials satisfy a three term recurrence relation. The purpose of the paper is to give estimates for the orthogonal polynomials and the corresponding weight function provided that the coefficients in the recurrence formula behave in a prescribed manner. (Author)

At present there are only a few approximate identity kernels for the Walsh system, for example, the p N -truncated Dirichlet kernel D p N − 1 ( t ) = ∑ j = 0 p N − 1 w j ( t ) [6]; the Abel-Poisson kernel λ γ ( t ) = ∑ k = 0 ∞ γ k w k ( t ) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p -adic derivative of product type kernels and we estimate this derivative in L 1 -norm.

This paper discusses Sobolev orthogonal polynomials for a class of scalar products that contains the sequentially dominated products introduced by Lagomasino and Pijeira. We prove asymptotics for Markov type functions associated to the Sobolev scalar product and an extension of Widom's Theorem on the location of the zeroes of the orthogonal polynomials. In the case of measures supported in the real line, we obtain results related to the determinacy of the Sobolev moment problem and the completeness of the polynomials in a suitably defined weighted Sobolev space.

We study the rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. Our main interest is in characterizing the slowest type of pointwise convergence possible. This is a continuation of the paper Deutsch and Hundal (2010) [14]. The main result is a “lethargy” theorem (Theorem 3.3) which gives useful conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, we obtain a “dichotomy” theorem, which states the surprising result that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. The dichotomy theorem is applied to generalize and sharpen: (1) the von Neumann–Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.

In this paper we discuss relations between Fourier-Stieltjes transforms of (bounded Borel) measures on (−∞, ∞) and sequences of Fourier-Stieltjes coefficients of measures on [0, 2π). We show (Theorem B) that if \ gm is the Fourier-Stieltjes transform of a measure μ on (−∞, ∞), then { \ gm(n)} n = −∞ ∞ is the sequence of Fourier-Stieltjes coefficients of some measure on [0, 2π). In the other direction we show (Theorem F) that if {an}n = −∞∞ is the sequence of Fourier-Stieltjes coefficients of some measure on [0, 2π), then the function whose graph consists of the line segments successively joining the points (n, an) must be the Fourier-Stieltjes transform of a measure on (−∞, ∞). We also prove a similar theorem for distributions.

The paper is concerned with the calculation of the derived mapping of the metric projection onto a finite dimensional subspace of a space of integrable functions. Abstract results for quotient spaces and for spaces which are l1-direct sums are obtained and are applied to real or complex spaces of integrable functions to obtain improvements of a number of results of Li for real spaces of integrable functions. Metric projections onto one-dimensional subspaces of spaces of integrable functions are considered and it is shown that there is an essential difference between the real and complex cases.

In this paper, we obtain a necessary and sufficient condition for the incompleteness of complex exponential polynomials in Cα, where Cα is a weighted Banach space of complex continuous functions f on the real axis R with f(t)exp(−α(t)) vanishing at infinity, in the uniform norm with respect to the weight α(t). We also prove that, if the above condition of incompleteness holds, then each function in the closure of complex exponential polynomials can be extended to an entire function represented by a Dirichlet series.