Constructive Approximation最新文献

Abstract

We construct a non-polynomial generalization of the q-Askey scheme. Whereas the elements of the q-Askey scheme are given by q-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars’ hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev’s quantum dilogarithm, Woronowicz’s quantum exponential, or Kurokawa’s double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits.

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an (L^infty ) norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of (overline{{{mathbb {R}}}}). We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szegő–Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau–Widom and DCT conditions.

Abstract

We study the approximation capacity of some variation spaces corresponding to shallow ReLU (^k) neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with less smoothness, the approximation rates in terms of the variation norm are established. Using these results, we are able to prove the optimal approximation rates in terms of the number of neurons for shallow ReLU (^k) neural networks. It is also shown how these results can be used to derive approximation bounds for deep neural networks and convolutional neural networks (CNNs). As applications, we study convergence rates for nonparametric regression using three ReLU neural network models: shallow neural network, over-parameterized neural network, and CNN. In particular, we show that shallow neural networks can achieve the minimax optimal rates for learning Hölder functions, which complements recent results for deep neural networks. It is also proven that over-parameterized (deep or shallow) neural networks can achieve nearly optimal rates for nonparametric regression.

We prove several new families of Bernstein inequalities of two types on the simplex. The first type consists of inequalities in (L^2) norm for the Jacobi weight, some of which are sharp, and they are established via the spectral operator that has orthogonal polynomials as eigenfunctions. The second type consists of inequalities in (L^p) norm for doubling weight on the simplex. The first type is not necessarily a special case of the second type when (d ge 3).

We prove that the Weyl function of the one-dimensional Dirac operator on the half-line ({mathbb {R}}_+) with exponentially decaying entropy extends meromorphically into the horizontal strip ({0geqslant mathop {textrm{Im}}nolimits z > -delta }) for some (delta > 0) depending on the rate of decay. If the entropy decreases very rapidly then the corresponding Weyl function turns out to be meromorphic in the whole complex plane. In this situation we show that poles of the Weyl function (scattering resonances) uniquely determine the operator.

Let (Gamma subset mathbb {C}) be a curve of class (C(1,alpha )). For (z_{0}) in the unbounded component of (mathbb {C}setminus Gamma ), and for (n=1,2,...), let (nu _n) be a probability measure with (mathop {textrm{supp}}nolimits (nu _{n})subset Gamma ) which minimizes the Bergman function (B_{n}(nu ,z):=sum _{k=0}^{n}|q_{k}^{nu }(z)|^{2}) at (z_{0}) among all probability measures (nu ) on (Gamma ) (here, ({q_{0}^{nu },ldots ,q_{n}^{nu }}) are an orthonormal basis in (L^2(nu )) for the holomorphic polynomials of degree at most n). We show that ({nu _{n}}_n) tends weak-* to ({{widehat{delta }}}_{z_{0}}), the balayage of the point mass at (z_0) onto (Gamma ), by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to (Gamma ).

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous branching processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein’s representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish a sufficient condition for families of Bernstein functions, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner–Kufarev differential equation. Most of these results are then applied in the sequel paper [35] to study continuous-state branching processes.

We extent our definition of Euler Gamma function to higher Gamma functions, and we give a unified characterization of Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson q-Gamma function, and Nishizawa higher q-Gamma functions in the space of finite order meromorphic functions. The method extends to more general functional equations and unveils the multiplicative group structure of solutions that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. With the new definition, Barnes–Hurwitz zeta functions are no longer necessary in the definition of Barnes multiple Gamma functions. This simplifies the classical definition, without the analytic preliminaries about the meromorphic extension of Barnes–Hurwitz zeta functions, and defines a larger class of Gamma functions. For some algebraic independence conditions on the parameters, we prove uniqueness of the solutions. Hence, this implies the identification of classical Barnes multiple Gamma functions as a subclass of our multiple Gamma functions.

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on ({mathbb {R}}^n) and X a ball quasi-Banach function space on ({mathbb {R}}^n) satisfying some mild assumptions. Denote by (H_{X,, L}({mathbb {R}}^n)) the Hardy space, associated with both L and X, which is defined via the Lusin area function related to the semigroup generated by L. In this article, the authors establish both the maximal function and the Riesz transform characterizations of (H_{X,, L}({mathbb {R}}^n)). The results obtained in this article have a wide range of generality and can be applied to the weighted Hardy space, the variable Hardy space, the mixed-norm Hardy space, the Orlicz–Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L. In particular, even when L is a second order divergence form elliptic operator, both the maximal function and the Riesz transform characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L, obtained in this article, are completely new.

The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as (n rightarrow infty ) of the minimal logarithmic potential energy of n point charges restricted to move in the interval ([-1,1]) in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion as (N rightarrow infty ) of the logarithmic energy (sum _{jne k} log (1/| x_j - x_k |)) of Fekete points, which, by definition, maximize the product of all mutual distances (prod _{jne k} | x_j - x_k |) of N points in ([-1,1]). The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at (pm 1) of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.