pinecone/core-2020-05-10-deduplication huggingface dataset:
Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

AbstractLet τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Qn(x)}∞n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Qn(x)}∞n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: LN[y](x)=∑Ni=1ℓi(x)y(i)(x)=λny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Qn(x)}∞n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions
Discrete Approximation of Unbounded Operators and Approximation of their Spectra

AbstractLet E be a Banach space over C and let the densely defined closed linear operator A:D(A)⊂E→E be discretely approximated by the sequence ((An,D(An)))n∈N of operators An where each An is densely defined in the Banach space Fn. Let σa(A) be the approximate point spectrum of A and let σε(An) denote the ε-pseudospectrum of An. Generalizing our own result, we show that σa(A)⊂liminfσε(An)=∪n∈N ∩k⩾nσε(Ak) holds for every ε>0. We deduce that then for every compact set K⊂Climndist(σa(A)∩K, σa(An))=0 provided there exists M>0 such that ‖(λ−An)−1‖⩽Mdist(λ, σ(An))−1 holds for every n and every λ in the resolvent set ρ(An) of An. We finally treat the problem under which conditions σa(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩ε>0∩n∈N∪k⩾nσε, a(Ak)⊂σa(A) hold where σε, a(A) denotes the ε-approximate pseudospectrum
Rational and Polynomial Interpolation of Analytic Functions with Restricted Growth

AbstractLet f be an analytic function on a domain D⊂C∪{∞} and rn the rational function of degree n with poles at the points Bn={bni}ni=1, interpolating to f at the points An={ani}ni=0⊂D. A fundamental question is whether it is possible to choose the points An and Bn so that rn converges locally uniformly to f on D for every analytic function f on D. In some situations the interpolation points must be allowed to approach the boundary of D as n tends to infinity and then we cannot obtain convergence for every analytic f on D. If we restrict the growth of f(z) when z goes to the boundary of D, we still have some positive convergence results that we prove here
Some Orthogonal Polynomials Related to Elliptic Functions

AbstractWe characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates
Entropy Numbers of Embeddings of Besov Spaces in Generalized Lipschitz Spaces

AbstractWe establish two-sided estimates for entropy numbers of embeddings between certain weighted Banach sequence spaces with mixed norms. These estimates are “almost” sharp, in the sense that upper and lower bounds differ only by logarithmic terms and improve previous results by D. E. Edmunds and D. Haroske (1999, Dissertationes Math.380, 1–43; 2000, J. Approx. Theory104, 226–271). As an application we obtain also new upper entropy estimates for embeddings of Besov spaces in generalized Lipschitz spaces
Uniform Asymptotic Expansions for Charlier Polynomials

AbstractThe asymptotic behaviour of the Charlier polynomials C(a)n(x) as n→∞ is examined. These polynomials satisfy a discrete orthogonality relation and, unlike classical orthogonal polynomials, do not satisfy a second-order linear differential equation with respect to the independent variable x. As such, previous results on their asymptotic behaviour have been restricted to integral methods and consequently have been quite limited in their scope. In this paper a new approach is used, where the polynomials C(a)n(x) are not regarded as a functions of x with a as a parameter, but rather with the roles reversed via a second-order linear differential equation in which a is the (real or complex-valued) independent variable and x is a parameter. This equation has two turning points in the a plane which depend on x, and are either positive or complex conjugates, according to the values of x. Moreover, the turning points can coalesce with one another, or one with a singularity of the equation, for certain critical values of x. By using two general asymptotic theories of differential equations, one for intervals free of turning points and the other for intervals containing a double pole and a coalescing turning point, expansions are derived for C(a)n(x) involving either elementary functions or Bessel functions. Taken together, the results are uniformly valid for −∞<x<∞. In addition, in each case the expansions are uniformly valid for a lying in certain unbounded intervals, each of which contain [−{1−δ}n, {1−δ}n], where δ∈(0, 1) is an arbitrary constant
Approximations from Subspaces of C0(X)

AbstractWe show among other things that if B is a linear space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, for which there is a continuous function h defined in a neighbourhood of 0 in the real line which is non-affine in every neighbourhood of 0 and satisfies |h(t)|⩽k|t| for all t, such that hb is in B whenever b is in B and the composite function is defined, then every function in C0(X) which can be approximated on every pair of points in X by functions in B can be approximated uniformly by functions in B
Error Bounds for Approximation with Neural Networks

AbstractIn 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
Hardy Spaces Generated by an Integrability Condition

AbstractS. 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
Asymptotic Zero Distribution of Orthogonal Polynomials with Discontinuously Varying Recurrence Coefficients

AbstractThe zero distribution of orthogonal polynomials pn, N,n=0, 1,… generated by recurrence coefficients an, N and bn, N depending on a parameter N has been recently considered by Kuijlaars and Van Assche under the assumption that an, N and bn, N behave like a(n/N) and b(n/N), respectively, where a(·) and b(·) are continuous functions. Here, we extend this result by allowing a(·) and b(·) to be measurable functions so that the presence of possible jumps is included. The novelty is also in the sense of the mathematical tools since, instead of applying complex analysis arguments, we use recently developed results from asymptotic matrix theory due to Tyrtyshnikov, Serra Capizzano, and Tilli
Subdivision Schemes and Refinement Equations with Nonnegative Masks

AbstractWe consider the two-scale refinement equation f(x)=∑Nn=0cnf(2x−n) with ∑nc2n=∑nc2n+1=1 where c0, cN≠0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all cn⩾0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 0<c0, cN<1 and the greatest common divisor of S={n:cn>0} is 1. We prove the conjecture for a large class of refinement equations
Curvelets and Curvilinear Integrals

AbstractLet C(t):I↦R2 be a simple closed unit-speed C2 curve in R2 with normal [formula](t). The curve C generates a distribution Γ which acts on vector fields [formula](x1, x2):R2↦R2 by line integration according to[formula] We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame Φ=[formula] with dual Φ*=[formula]; then an m-term approximation to Γ can be formed by selecting m terms (μi:1⩽i⩽m) and taking[formula] Here the μi can be chosen adaptively based on the curve C. We are interested in finding a vector basis or frame for which the above scheme yields the highest-quality m-term approximations. Here performance is measured by considering worst-case error on vector fields which are smooth in an L2 Sobolev sense:[formula] We establish an isometry between this problem and the problem of approximating objects with edges in L2 norm. Starting from the recently-introduced tight frames of scalar curvelets, we construct a vector frame of curvelets for this problem. Invoking results on the near-optimality of scalar curvelets in representing objects with edges, we argue that vector curvelets provide near-optimal quality m-term approximations. We show that they dramatically outperform both wavelet and Fourier-based representations in terms of m-term approximation error. The m-term approximations to Γ are built from terms with support approaching more and more closely the curve C with increasing m; the terms have support obeying the scaling law width≈length2. Comparable results can be developed, with additional work, for scalar curvelet approximation in the case of scalar integrandsI(f)=∫f(C(t))dt
Shape Preserving Interpolatory Subdivision Schemes for Nonuniform Data

AbstractThis article is concerned with a class of shape preserving four-point subdivision schemes which are stationary and which interpolate nonuniform univariate data {(xi, fi)}. These data are functional data, i.e., xi≠xj if i≠j. Subdivision for the strictly monotone x-values is performed by a subdivision scheme that makes the grid locally uniform. This article is concerned with constructing suitable subdivision methods for the f-data which preserve convexity; i.e., the data at the kth level, {x(k)i, fi(k)} is a convex data set for all k provided the initial data are convex. First, a sufficient condition for preservation of convexity is presented. Additional conditions on the subdivision methods for convergence to a C1 limit function are given. This leads to explicit rational convexity preserving subdivision schemes which generate continuously differentiable limit functions from initial convex data. The class of schemes is further restricted to schemes that reproduce quadratic polynomials. It is proved that these schemes are third order accurate. In addition, nonuniform linear schemes are examined which extend the well-known linear four-point scheme to the case of nonuniform data. Smoothness of the limit function generated by these linear schemes is proved by using the well-known smoothness criteria of the uniform linear four-point scheme
On Polynomial Approximation of Square Integrable Functions on a Subarc of the Unit Circle

AbstractLet E be an arc on the unit circle and let L2(E) be the space of all square integrable functions on E. Using the Banach–Steinhaus Theorem and the weak* compactness of the unit ball in the Hardy space, we study the L2 approximation of functions in L2(E) by polynomials. In particular, we will investigate the size of the L2 norms of the approximating polynomials in the complementary arc E of E. The key theme of this work is to highlight the fact that the benefit of achieving good approximation for a function over the arc E by polynomials is more than offset by the large norms of such approximating polynomials on the complementary arc E
Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

AbstractWe 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
Antiproximinal Norms in Banach Spaces

AbstractWe prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal norms in Banach spaces with the convex point of continuity property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed
Approximation in Sobolev Spaces by Kernel Expansions

AbstractFor interpolation of smooth functions by smooth kernels having an expansion into eigenfunctions (e.g., on the circle, the sphere, and the torus), good results including error bounds are known, provided that the smoothness of the function is closely related to that of the kernel. The latter fact is usually quantified by the requirement that the function should lie in the “native” Hilbert space of the kernel, but this assumption rules out the treatment of less smooth functions by smooth kernels. For the approximation of functions from “large” Sobolev spaces W by functions generated by smooth kernels, this paper shows that one gets at least the known order for interpolation with a less smooth kernel that has W as its native space
On Approximate ℓ 1 Systems in Banach Spaces

AbstractLet X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (xi)∞i=1 in X which satisfy ‖∑i∈A±xi‖⩾|A|−f(|A|), for all finite A⊂N and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [xi] is H.I., and we exhibit systems in all isomorphs of ℓ1 which are not equivalent to the unit vector basis of ℓ1. We also prove that certain lacunary Haar systems in L1 are quasi-greedy basic sequences
Zeros of Sobolev Orthogonal Polynomials of Gegenbauer Type

AbstractLet {Sn}n denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product〈f, g〉S=∫f(x)g(x)dψ0 (x)+λ∫f(x)g(x)dψ1 (x), where λ>0 and {dψ0, dψ1} is a so-called symmetrically coherent pair, with dψ0 or dψ1 the classical Gegenbauer measure (x2−1)αdx, α>−1. If dψ1 is the Gegenbauer measure, then Sn has n different, real zeros. If dψ0 is the Gegenbauer measure, then Sn has at least n−2 different, real zeros. Under certain conditions Sn has complex zeros. Also the location of the zeros of Sn with respect to Gegenbauer polynomials, is studied
Inverse Operators, q-Fractional Integrals, and q-Bernoulli Polynomials

AbstractWe introduce operators of q-fractional integration through inverses of the Askey–Wilson operator and use them to introduce a q-fractional calculus. We establish the semigroup property for fractional integrals and fractional derivatives. We study properties of the kernel of q-fractional integral and show how they give rise to a q-analogue of Bernoulli polynomials, which are now polynomials of two variables, x and y. As q→1 the polynomials become polynomials in x−y, a convolution kernel in one variable. We also evaluate explicitly a related kernel of a right inverse of the Askey–Wilson operator on an L2 space weighted by the weight function of the Askey–Wilson polynomials
An Expansion Formula for the Askey–Wilson Function

AbstractThe Askey–Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey–Wilson second order q-difference operator. The kernel is called the Askey–Wilson function. In this paper an explicit expansion formula for the Askey–Wilson function in terms of Askey–Wilson polynomials is proven. With this expansion formula at hand, the image under the Askey–Wilson function transform of an Askey–Wilson polynomial multiplied by an analogue of the Gaussian is computed explicitly. As a special case of these formulas a q-analogue (in one variable) of the Macdonald–Mehta integral is obtained, for which also two alternative, direct proofs are presented
Convergence Rates of Regularized Approximation Processes

AbstractWe define the concept of an A-regularized approximation process and prove for it uniform convergence theorems and strong convergence theorems with optimal and non-optimal rates. The sharpness of non-optimal convergence is also established. The general results provide a unified approach to dealing with convergence rates of various approximation processes, and also of local ergodic limits as well. As applications, approximation theorems, and local Abelian and Cesáro ergodic theorems with rates are deduced for n-times integrated solution families for Volterra integral equations, which include n-times integrated semigroups and cosine functions as special cases. Applications to (Y)-semigroups and tensor product semigroups are also discussed
Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points

AbstractThe relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or −1, then the generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc.39, 233–238)
Best One-Sided L1-Approximation by Blending Functions of Order (2,2)

AbstractLet f∈C2, 2([−1, 1]2) be a real function satisfying ∂4f/∂x2∂y2⩾0 on [−1, 1]2. We study the problem of best one-sided L1-approximation to f from the linear space {h∈C2, 2([−1, 1]2):∂4h/∂x2∂y2=0} of all blending functions of order (2, 2). The unique best one-sided L1-approximant to f from above is characterized by transfinite Hermite interpolation on the canonical grid {(x, y)∈[−1, 1]2:|x|=|y|}. For f even with respect to one of its variables we characterize the unique best one-sided L1-approximant to f from below by transfinite Hermite interpolation on the canonical grid {(x, y)∈[−1, 1]2:|x|+|y|=1}. There is no canonical grid for the entire cone class of functions f with ∂4f/∂x2∂y2⩾0 on [−1, 1]2 when we approximate from below. The best one-sided L1-approximant from above has the smoothness of f. The best one-sided L1-approximant to f from below is a blending-spline function with two line segment knots {(x, 0):−1⩽x⩽1} and {(0, y):−1⩽y⩽1}; i.e., the best one-sided approximation to f from below possesses a saturation effect with respect to the smoothness of f
Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

AbstractThe convergence in L2(T) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with limn→∞1n∑n−1k=0|ak|=0, {an}n⩾0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|ϕn|2dσ}n⩾0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szegő class which consists of measures with ∑∞n=0(1−|an|2)1/2<∞ and describe it in terms of Hessenberg matrices
Sharp Integral Inequalities of the Hermite–Hadamard Type

AbstractWe consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanović and Pecarić, and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning Lp estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds
On the Divergence of the Two-Dimensional Dyadic Difference of Dyadic Integrals

AbstractIn 1989 F. Schipp and W. R. Wade (Appl. Anal.34, 203–218) proved for functions in L(I2)log+L(I2) (I2 is the unit square) that the dyadic difference of the dyadic integral dn(If) converges to f a.e. in the Pringsheim sense (that is, min(n1, n2)→∞, n=(n1, n2)∈P2). We prove that this result cannot be sharpened. Namely, we prove that for all measurable functions δ:[0, +∞)→[0, +∞), limt→∞δ(t)=0 we have a function f∈Llog+Lδ(L) such as dn(If) does not converge to f a.e. (in the Pringsheim sense)
On a Bivariate Interpolation Problem

AbstractIn this paper we solve the poisedness problem for a bivariate interpolation introduced by B. Bojanov and Y. Xu. The authors were informed the problem was solved earlier, by a different method, also by B. Bojanov and Y. Xu (to appear, On a Hermite interpolation by polynomials of two variables, SIAM J. Numer. Anal.). Parameters of the original interpolation from Π2k are the values of a function and its radial derivatives up to some order k at 2k+1 equidistant nodes of the unit circumference. We also consider other closely related polynomial interpolations and prove their poisedness. Meanwhile the poisedness of several general univariate Birkhoff interpolation problems is proved to which the above problem is reduced. At the end we consider the corresponding useful cubature formulas
A Vector Generalisation of de Montessus' Theorem for the Case of Polar Singularities on the Boundary

AbstractIn the context of vector Padé approximants we present an extension of de Montessus' theorem to vector-valued meromorphic functions with poles on the boundary of interest—thus strengthening previous results for these approximants. The proofs are framed in Clifford algebras which provide a natural language for discussing vector rational approximants. We also present results for the asymptotic behaviour of the constituent parts of the Clifford denominator—namely, its scalar and bivector parts. In particular, for the case of vector-valued rational functions in which the principal parts are orthogonal to each other for different poles, we demonstrate that the rate of convergence is doubled for the scalar part of the denominator. Finally, we derive consequences of the convergence theorems for the approximation of poles, using either the complete Clifford denominator or its scalar part