Constructive Approximation最新文献

This article examines the asymptotic behavior of the Widom factors, denoted ({mathcal {W}}_n), for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom’s proposal in Widom (Adv Math 3:127–232, 1969), when dealing with a single smooth Jordan arc, ({mathcal {W}}_n) converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval ([-2,2]), and we provide a complete description of the asymptotic behavior of ({mathcal {W}}_n) for symmetric star graphs and quadratic preimages of ([-2,2]). We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential extension of the conjecture posed in Christiansen et al. (Oper Theory Adv Appl 289:301–319, 2022). Lastly, we propose a possible connection between the S-property and Widom factors converging to 2.

We study the approximation of non-negative multi-variate couplings in the uniform norm while matching given single-variable marginal constraints.

We study the characteristic polynomial (p_{n}(x)=prod _{j=1}^{n}(|z_{j}|-x)) where the (z_{j}) are drawn from the Mittag–Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large n asymptotics for the moment generating function (mathbb {E}[e^{frac{u}{pi } , text {Im,}ln p_{n}(r)}e^{a , text {Re,}ln p_{n}(r)}]), in the case where r is in the bulk, (u in mathbb {R}) and (a in mathbb {N}). This expectation involves an (n times n) determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump- and root-type singularities along the circle centered at 0 of radius r. This “circular" root-type singularity differs from earlier works on Fisher–Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.

We approximate d-variate periodic functions in weighted Korobov spaces with general weight parameters using n function values at lattice points. We do not limit n to be a prime number, as in currently available literature, but allow any number of points, including powers of 2, thus providing the fundamental theory for construction of embedded lattice sequences. Our results are constructive in that we provide a component-by-component algorithm which constructs a suitable generating vector for a given number of points or even a range of numbers of points. It does so without needing to construct the index set on which the functions will be represented. The resulting generating vector can then be used to approximate functions in the underlying weighted Korobov space. We analyse the approximation error in the worst-case setting under both the (L_2) and (L_{infty }) norms. Our component-by-component construction under the (L_2) norm achieves the best possible rate of convergence for lattice-based algorithms, and the theory can be applied to lattice-based kernel methods and splines. Depending on the value of the smoothness parameter (alpha ), we propose two variants of the search criterion in the construction under the (L_{infty }) norm, extending previous results which hold only for product-type weight parameters and prime n. We also provide a theoretical upper bound showing that embedded lattice sequences are essentially as good as lattice rules with a fixed value of n. Under some standard assumptions on the weight parameters, the worst-case error bound is independent of d.

Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann–Hilbert framework. We then focus on the special case, defined by a constant weight function, and use the Riemann–Hilbert problem to derive recurrence relations and differential equations for the orthogonal polynomials. We further show that the sub-class of even polynomials is associated to the elliptic form of Painlevé VI, with the tau function given by the Hankel determinant of even moments, up to a scaling factor. The first iteration of these even polynomials relates to the special case of Painlevé VI studied by Hitchin in relation to self-dual Einstein metrics.

In 2011, Armentano, Beltrán and Shub obtained a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.

Complex valued measures of finite total variation are a powerful signal model in many applications. Restricting to the d-dimensional torus, finitely supported measures can be exactly recovered from their trigonometric moments up to some order if this order is large enough. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the 1-Wasserstein distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the indicator function on the support of the measure and to converge to zero outside.

Abstract

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for P (_{nu }^{-mu }left( tanh left( alpha +beta right) right) ) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.

We begin our analysis with the study of two collections of lattice paths in the plane, denoted ({mathcal {D}}_{[n,i,j]}) and ({mathcal {P}}_{[n,i,j]}). These paths consist of sequences of n steps, where each step allows movement in three directions: upward (with a maximum displacement of q units), rightward (exactly one unit), or downward (with a maximum displacement of p units). The paths start from the point (0, i) and end at the point (n, j). In the collection ({mathcal {D}}_{[n,i,j]}), it is a crucial constraint that paths never go below the x-axis, while in the collection ({mathcal {P}}_{[n,i,j]}), paths have no such restriction. We assign weights to each path in both collections and introduce weight polynomials and generating series for them. Our main results demonstrate that certain matrices of size (qtimes p) associated with these generating series can be expressed as matrix continued fractions. These results extend the notable contributions previously made by Flajolet (Discrete Math 32:125–161, 1980) and Viennot (Une Théorie Combinatoire des Polynômes Orthogonaux Généraux. University of Quebec at Montreal, Lecture Notes, 1983) in the scalar case (p=q=1). The generating series can also be interpreted as resolvents of one-sided or two-sided difference operators of finite order. Additionally, we analyze a class of random banded matrices H, which have (p+q+1) diagonals with entries that are independent and bounded random variables. These random variables have identical distributions along diagonals. We investigate the asymptotic behavior of the expected values of eigenvalue moments for the principal (ntimes n) truncation of H as n tends to infinity.

We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of ({mathbb {R}}^d). All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a SVF F, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to F. At points of discontinuity of F, we derive estimates, which yield the convergence to a certain set described in terms of the metric selections of F. To obtain these estimates we refine and extend known results on approximation of real-valued functions by integral operators. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in (L^1) provides our global estimates. The theory is applied to concrete operators: the Bernstein–Durrmeyer operator and the Kantorovich operator.