Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case

Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions II: On functions orthogonal with respect to their own zeros, J. London Math. Soc. 14 (1939) 37-44] on functions orthogonal with respect to their real zeros λn, , we will consider, under the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0,1), that is, the functions f(z)=zνF(z), ν∈R, where F is entire and

     

On a conjecture regarding the extrema of Bessel functions and its generalization

It was conjectured by Á. Elbert in J. Comput. Appl. Math. 133 (2001) 65-83 that, given two consecutive real zeros of a Bessel function of order ν, j_ν,κ and j_ν,κ+1, the zero of the derivative between such two zeros j_ν,κ′ satisfies . We prove that this inequality holds for any Bessel function of any real order. In addition to these lower bounds, upper bounds are obtained. In this way we bracket the zeros of the derivative. It is discussed how similar relations can be obtained for other special functions which are solutions of a second order ODE; in particular, the case of the zeros of is considered.     

The asymptotics of the generalised Hermite-Bell polynomials

The Hermite-Bell polynomials are defined by for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for as n→∞ using the method of steepest descents. For a certain value of x, two saddle points coalesce and a uniform approximation in terms of Airy functions is given to cover this situation. An asymptotic approximation for the largest positive zeros of is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.     

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.     

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.     

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.     

A Pochhammer type sampling reconstruction of analytic functions

We give a characterization of the functions f analytic on the half-plane Re(z)>σ that can be expressed in the form for some βn∈C. It will follow that the βn depend on the values of f on the integers so this gives a sampling reconstruction of these functions. Several examples of this representation are given.     

On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

In this paper we study the distribution of zeros of each entire function of the sequence , which approaches the Riemann zeta function for Rez<−1, and is closely related to the solutions of the functional equations f(z)+f(2z)+?+f(nz)=0. We determine the density of the zeros of Gn(z) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of Gn(z) inside certain rectangles in the critical strip.     

Functional inequalities involving special functions

Let up(x) be the generalized and normalized Bessel function depending on parameters b,c,p and let σ(r)=up(1−r²)/up(r²), r∈(0,1). Motivated by an open problem of Anderson, Vamanamurthy, and Vuorinen we prove that for all r₁,r₂∈(0,1) for certain conditions on the parameters b,c,p.     

On k-regular functions with values in a universal Clifford algebra

In this paper, we mainly study properties of nullsolutions of the operator Dk (k∈N^∗=N?{0}), so-called k-regular functions. Firstly, we study the set of all homogeneous polynomials of degree p in x₁,…,xn which are k-regular in the whole Rn, clearly is a right module over C(Vn,n), we construct a basis for the right module . Secondly, we study the k-regular and analytic functions, and we give the Taylor expansions for these functions. At last, the corresponding Taylor expansions for k-regular functions are given since each k-regular function is a real analytic function.     

Spectral synthesis of diagonal operators and representing systems for the space of entire functions

In this paper, we study continuous linear operators on spaces of functions analytic on disks in the complex plane having as eigenvectors the monomials zn whose associated eigenvalues λn are distinct. In particular, we show that under mild conditions, such a diagonal operator has non-spectral invariant subspaces (that is, closed invariant subspaces which are not the closed linear span of collections of monomials) if and only if every entire function of a particular growth rate is representable as a generalized Dirichlet series .     

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 .     

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.     

About a condition for starlikeness

A condition for starlikeness will be improved given by the inequality Re(f^′(x)+αzf^″(z))>0, z∈U, concerning analytic functions of the form f(z)=z+a₂z²+? which are defined on the unit disk .