arXiv subjects

Ana P. Peron

Publications and source records attributed to Ana P. Peron.

6 recordsLinked to original sources

arXiv2019

A family of entire functions connecting the Bessel function $J_1$ and the Lambert $W$ function

Motivated by the problem of determining the values of $α>0$ for which $f_α(x)=e^α- (1+1/x)^{αx},\ x>0$ is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family $φ_α$, $α>0$, of entire functions such that $f_α(x) =\int_0^\infty e^{-sx}φ_α(s)\,ds, \ x>0.$ We show that each function $φ_α$ has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions $φ_α$, which turn out to be related to the well known Bessel function $J_1$ and the Lambert $W$ function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of $φ_α$ as $α$ increases from $0$ to $\infty$ and to obtain a very precise approximation of the largest $α>0$ such that $φ_α(s)\geq0,\, s>0$, or equivalently, such that $f_α$ is completely monotonic.

math.CA↗

arXiv2017

Schoenberg's theorem for real and complex Hilbert spheres revisited

Schoenberg's theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let \overline{\D} denote the closed unit disc in the complex plane. Continuous functions f:\overline{\D}\times L\to \C such that f(ξ\cdot η,u^{-1}v) is a positive definite kernel on the product of the unit sphere in \ell_2(\C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=\sum_{m,n=0}^\infty φ_{m,n}(u)z^m\overline{z}^n, where φ_{m,n} is a double sequence of continuous positive definite functions on L such that \sumφ_{m,n}(e_L)<\infty (e_L is the neutral element of L). It is shown how the coefficient functions φ_{m,n} are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials. Similar results are obtained for the real Hilbert sphere.

math.CA↗

arXiv2016

Orthogonal expansions related to compact Gelfand pairs

Given a compact Gelfand pair (G,K) and a locally compact group L, we characterize the class P_K^\sharp(G,L) of continuous positive definite functions f:G\times L\to \C which are bi-invariant in the G-variable with respect to K. The functions of this class are the functions having a uniformly convergent expansion \sum_{φ\in Z} B(φ)(u)φ(x) for x\in G,u\in L, where the sum is over the space Z of positive definite spherical functions φ:G\to\C for the Gelfand pair, and (B(φ))_{φ\in Z} is a family of continuous positive definite functions on L such that \sum_{φ\in Z}B(φ)(e_L)<\infty. Here e_L is the neutral element of the group L. For a compact abelian group G considered as a Gelfand pair (G,K) with trivial K=\{e_G\}, we obtain a characterization of P(G\times L) in terms of Fourier expansions on the dual group \widehat{G}. The result is described in detail for the case of the Gelfand pairs (O(d+1),O(d)) and (U(q),U(q-1)) as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg-Porcu (2016) and Guella-Menegatto (2016)

math.CA↗

arXiv2016

Strictly Positive Definite Kernels on a Product of Spheres II

We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695].

math.CA↗

arXiv2015

An extension of a theorem of Schoenberg to products of spheres

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.

math.CA↗

arXiv2012

Traceability of positive integral operators in the absence of a metric

We investigate the traceability of positive integral operators on $L^2(X,μ)$ when $X$ is a Hausdorff locally compact second countable space and $μ$ is a non-degenerate, $σ$-finite and locally finite Borel measure. This setting includes other cases proved in the literature, for instance the one in which $X$ is a compact metric space and $μ$ is a special finite measure. The results apply to spheres, tori and other relevant subsets of the usual space $\mathbb{R}^m$.

math.FA↗