Müntz-type theorems on the half-line with weights

We consider the linear span S   of the functions t^_ak$t^{a_k}$ (with some _ak>0$a_k>0$) in weighted L²$L^2$ spaces, with rather general weights. We give one necessary and one sufficient condition for S to be dense. Some comparisons are also made between the new results and those that can be deduced from older ones in the literature.     

Essential norms of weighted composition operators on Müntz spaces

In this paper, we discuss the problem of compactness for weighted composition operators, defined on a Müntz space . We compute the essential norm of such operators on the Müntz spaces. As a corollary, we obtain the exact values of essential norms of composition and multiplication operators.     

Generalized Christoffel functions and error of positive quadrature

The author gives some upper and lower bounds for the generalized Christoffel functions related to a Ditzian-Totik generalized weight. As an application, an error estimate of Gauss quadrature formula inL ₁-weighted norm is derived.Dedicated to Prof. Luigi Gatteschi on the occasion of his 70th birthdayWork sponsored by MURST 40%.     

Local approximation of functions with several variables

This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave-convex functions are obtained as well.     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.     

The version of Newman's Inequality for lacunary polynomials

The principal result of this paper is the establishment of the essentially sharp Markov-type inequality

for every with distinct real exponents greater than and for every . A remarkable corollary of the above is the Nikolskii-type inequality

for every with distinct real exponents greater than and for every . Some related results are also discussed.

     

Markov-Type Inequalities for Products of Müntz Polynomials

Let Λ(λ_j)^∞_j=0 be a sequence of distinct real numbers. The span of {x^λ₀, x^λ₁, …, x^λ_n} over is denoted by M_n(Λ)span{x^λ₀, x^λ₁, …, x^λ_n}. Elements of M_n(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λ_j)^∞_j=0andΓ(γ_j)^∞_j=0be increasing sequences of nonnegative real numbers. Let

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Then

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18(n+m+1)(λ_n+γ_m).In particular ,

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Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b](0, ∞).     

Generalizations of Müntz's Theorem via a Remez-type inequality for Müntz spaces

The principal result of this paper is a Remez-type inequality for Müntz polynomials:

or equivalently for Dirichlet sums:

where . The most useful form of this inequality states that for every sequence satisfying , there is a constant depending only on and (and not on , , or ) so that

for every Müntz polynomial , as above, associated with , and for every set of Lebesgue measure at least . Here denotes the supremum norm on . This Remez-type inequality allows us to resolve two reasonably long-standing conjectures.

The first conjecture it lets us resolve is due to D. J. Newman and dates from 1978. It asserts that if , then the set of products is not dense in .

The second is a complete extension of Müntz's classical theorem on the denseness of Müntz spaces in to denseness in , where
is an arbitrary compact set with positive Lebesgue measure. That is, for an
arbitrary compact set with positive Lebesgue measure,
is dense in if and only if .

Several other interesting consequences are also presented.

     

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

This paper is devoted to the testing and comparison of numerical solutions obtained from higher-order accurate finite difference schemes for the two-dimensional Burgers' equation having moderate to severe internal gradients. The fourth-order accurate two-point compact scheme, and the fourth-order accurate Du Fort Frankel scheme are derived. The numerical stability and convergence are presented. The cases of shock waves of severe gradient are solved and checked with the fourth-order accurate Du Fort Frankel scheme solutions. The present study shows that the fourth-order two-point compact scheme is highly stable and efficient in comparison with the fourth-order accurate Du Fort Frankel scheme.     

Convergence Rates of the Hilbert Uniqueness Method via Tikhonov Regularization

In this paper, we identify the Hilbert uniqueness method for a boundary control problem with the calculation of the pseudo inverse. Because of its ill-posedness, we approximate it by a regularized Hilbert uniqueness method, which we prove to be identical with Tikhonov regularization. By this equivalence, we can find sufficient conditions for convergence and convergence rates, which require approximation rates in Müntz spaces. We show that these conditions are fulfilled by an a priori bound in Sobolev norms on the exact solution.     

Asymptotic behavior of roots of random polynomial equations

We derive several new results on the asymptotic behavior of the roots of random polynomial equations, including conditions under which the distributions of the zeros of certain random polynomials tend to the uniform distribution on the circumference of a circle centered at the origin. We also derive a probabilistic analog of the Cauchy-Hadamand theorem that enables us to obtain the radius of convergence of a random power series.     

On Multivariate Quasipolynomials of the Minimal Deviation from Zero

We generalize to several variables both the upper and the lower Gelfond bounds for the least uniform deviation from zero of the quasipolynomials (or Müntz–Legendre polynomials) on the segment [0, 1]. Orthonormal quasipolynomials are also considered.     

Asymptotic behavior of generalized convolutions

We study the behavior of a class of convolution-type nonlinear transformations. Under some smallness conditions we prove the existence of fixed points and analyze the spectrum of the associated linearized operator.      

Series identities and associated families of generating functions

The authors investigate several families of double-series identities as well as their (known or new) consequences involving various hypergeometric functions in one and two variables. A number of associated generating-function relationships, involving certain classes of hypergeometric polynomials, are also considered.     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

Askey-Wilson type functions with bound states

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of q-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Grünbaum. 2000 Mathematics Subject Classification Primary—33D45, 37K10, 14H70, 39A70, 39A13     

Zero distribution of Müntz extremal polynomials in

Let be a sequence of distinct positive numbers. Let and let denote the extremal Müntz polynomial in with exponents . We investigate the zero distribution of . In particular, we show that if

then the normalized zero counting measure of converges weakly as to

while if or , the limiting measure is a Dirac delta at or , respectively.

     

On the multidimensional zeta functions associated with theta functions,and the multidimensional Appell polynomials

We define and study the multidimensional Appell polynomials associated with theta functions. For the trivial theta functions, we obtain the various well-known Appell polynomials. Many other interesting examples are given. To push our study, by Mellin transform, we introduce and investigate the multidimensional zeta functions associated with thetas functions and prove that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions. Using zeta functions techniques, among others, we prove an induction formula for multidimensional Appell polynomials. The last part of this paper is devoted to spectral zeta functions and its generalization associated with Laplacians on compact Riemannian manifolds. From this generalization, we construct new Appell polynomials associated with Riemannan manifolds of finite dimensions.