Some recent results on the zeros of Bessel functions and orthogonal polynomials

Abstract. In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J 0 at multiples of π, i.e., at the zeros of J 1/2 . This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.

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“…The lower bound becomes exact for ν = − [12], also reported in [3]. More precise results are given in [7].)…”

Section: Preliminary Remarks On Zeros Of Bessel Functionssupporting

confidence: 53%

“…Elbert and A. Laforgia [5] (see also [3]) have shown how to define a zero j νκ of continuous rank κ of C ν (x, α) by:…”

Section: Preliminary Remarks On Zeros Of Bessel Functionsmentioning

confidence: 99%

Monotonic sequences related to zeros of Bessel functions

Lorch

Muldoon

2008

Numer Algor

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“…There have been many contributions in this direction and we refer to Chapter 6 of Szegő's classical reading [26] and to a recent survey of Elbert [7] for an exhaustive number of inequalities. Generally speaking, when λ ∈ [0, 1], the use of Sturm's comparison theorem provides very precise bounds.…”

Section: Introductionmentioning

confidence: 99%

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

et al. 2004

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“…It can be observed that, in all the cases studied, the explained functional variability exceeds the residual functional variability. The truncation order, the number of components of β, and the functional shape of such components do not substantially affect the goodness of fit of the transformed Hilbert-valued fixed effect model (24). …”

Section: Rectangular Domainmentioning

confidence: 99%

“…The statistics (28) is now computed (see Table 9), as an empirical approximation of the relative magnitude between the explained functional variability and the residual variability, after fitting the transformed Hilbert-valued fixed effect model (24). It can be observed that the values of close to one in all the scenarios analyzed.…”

Section: Ef M Sementioning

confidence: 99%

The effect of the spatial domain in FANOVA models with ARH(1) error term

Álvarez-Liébana

Ruiz-Medina

2017

Statistics and Its Interface

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SummaryFunctional Analysis of Variance (FANOVA) from Hilbert-valued correlated data with spatial rectangular or circular supports is analyzed, when Dirichlet conditions are assumed on the boundary. Specifically, a Hilbertvalued fixed effect model with error term defined from an Autoregressive Hilbertian process of order one (ARH (1) process) is considered, extending the formulation given in [51]. A new statistical test is also derived to contrast the significance of the functional fixed effect parameters. The Dirichlet conditions established at the boundary affect the dependence range of the correlated error term. While the rate of convergence to zero of the eigenvalues of the covariance kernels, characterizing the Gaussian functional error components, directly affects the stability of the generalized least-squares parameter estimation problem. A simulation study and a real-data application related to fMRI analysis are undertaken to illustrate the performance of the parameter estimator and statistical test derived.

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Some recent results on the zeros of Bessel functions and orthogonal polynomials

Cited by 94 publications

References 46 publications

Monotonic sequences related to zeros of Bessel functions

Monotonic sequences related to zeros of Bessel functions

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

The effect of the spatial domain in FANOVA models with ARH(1) error term

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