Hyperbolic measures, moments and coefficients. Algebra on hyperbolic functions

With convolutions, we determine the Fourier transform of when n is a positive integer. Studying the expansion and taking the Fourier transform of when n and d are strictly positive integers, we obtain some polynomials and new probability densities related to them.     

The symmetric Meixner-Pollaczek polynomials with real parameter

The Symmetric Meixner-Pollaczek polynomials for λ>0 are well-studied polynomials. These are polynomials orthogonal on the real line with respect to a continuous, positive real measure. For λ?0, are also polynomials, however they are not orthogonal on the real line with respect to any real measure. This paper defines a non-standard inner product with respect to which the polynomials for λ?0, become orthogonal polynomials. It examines the major properties of the polynomials, for λ>0 which are also shared by the polynomials, for λ?0.     

Large degree asymptotics of generalized Bessel polynomials

Asymptotic expansions are given for large values of n of the generalized Bessel polynomials . The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the z-plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points z=±i/n are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed.     

Wavelet expansions and asymptotic behavior of distributions

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S₀(R)⊂S(R) and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.     

Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let be its real form in a formally real Euclidean Jordan algebra J⊂V; is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal-Bargmann transform from a unitary G-space of holomorphic functions on to an L²-space on . We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to of the spherical functions on and find their expansion in terms of the L-spherical polynomials on , which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L²-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on . Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.     

Stable functions and Vietoris' theorem

An analytic function f(z) in the unit disc D is called stable if s_n(f,·)/f?1/f holds for all for . Here s_n stands for the nth partial sum of the Taylor expansion about the origin of f, and ? denotes the subordination of analytic functions in . We prove that (1−z)^λ, λ∈[−1,1], are stable. The stability of turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials.     

Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb(K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of , then the set of all strongly norm attaining elements in is dense. In particular, the set of all points at which the norm of is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to . Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.     

Integer programming, Barvinok's counting algorithm and Gomory relaxations

We propose an algorithm based on Barvinok's counting algorithm for . It runs in time polynomial in the input size of when n is fixed, and under a condition on c, provides the optimal value of . We also relate Barvinok's counting formula and Gomory relaxations.     

Systems of orthogonal polynomials arising from the modular j-function

Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an inner product on for every . Suppose a system of orthogonal polynomials {P_n,ψ(x)}_n=0^∞ with respect to exists. We prove that if n is sufficiently large and ψ(x)P_n,ψ(x) is p-integral, then over . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.     

Semigroups associated to generalized polynomials and some classical formulas

We study operator semigroups associated with a family of generalized orthogonal polynomials with Hermitian matrix entries. For this we consider a Markov generator sequence, and therefore a Markov semigroup, for the family of orthogonal polynomials on related to the generalized polynomials. We give an expression of the infinitesimal generator of this semigroup and under the hypothesis of diffusion we prove that this semigroup is also Markov. We also give expressions for the kernel of this semigroup in terms of the one-dimensional kernels and obtain some classical formulas for the generalized orthogonal polynomials from the correspondent formulas for orthogonal polynomials on .     

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.     

New Pólya-Schoenberg type theorems

In this paper the theory of Hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so-called disk-like domains, i.e. such domains which are the union of disks or half-planes, all containing the origin. In such a domain, say Ω, we define (the class of) generalized prestarlike functions of order α?1 and ask for Hadamard multipliers g analytic at z=0 for which implies . We prove that such a multiplier necessarily has to be analytic in