TY - JOUR

T1 - On a Sturm Liouville periodic boundary values problem

AU - Fabbri, I. M.

AU - Lucianetti, A.

AU - Krasikov, I.

PY - 2009/5

Y1 - 2009/5

N2 - Consider the operator, acting in the Hilbert space, where v is a constant, with the periodic boundary values conditions, related to the maximum and minimum of the even Hermite Orthogonal polynomial(s) (OP). The problem is of interest in the field of optics when spiral geometries are involved. Specifically, the solutions represent the azimuthal component of the propagating modes into spiral optical fibres, previously studied. We establish two-sided asymptotic bounds for the relative extrema of the even Hermite OP and relate them to the values of ν = νk,j. The ratio between the upper and the lower bounds does not exceed, √3 near to the least and the largest maxima, and is very close to one in the bulk of. We also apply the same method to prove the inequality, for the Bessel function, and show that it is sharp outside monotonicity region.

AB - Consider the operator, acting in the Hilbert space, where v is a constant, with the periodic boundary values conditions, related to the maximum and minimum of the even Hermite Orthogonal polynomial(s) (OP). The problem is of interest in the field of optics when spiral geometries are involved. Specifically, the solutions represent the azimuthal component of the propagating modes into spiral optical fibres, previously studied. We establish two-sided asymptotic bounds for the relative extrema of the even Hermite OP and relate them to the values of ν = νk,j. The ratio between the upper and the lower bounds does not exceed, √3 near to the least and the largest maxima, and is very close to one in the bulk of. We also apply the same method to prove the inequality, for the Bessel function, and show that it is sharp outside monotonicity region.

KW - Asymptotic

KW - Bessel function

KW - Extremum

KW - Hermite polynomial

KW - Sharp inequalities

KW - Sturm Liouville problem

UR - http://www.scopus.com/inward/record.url?scp=68949132663&partnerID=8YFLogxK

U2 - 10.1080/10652460802522751

DO - 10.1080/10652460802522751

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AN - SCOPUS:68949132663

SN - 1065-2469

VL - 20

SP - 353

EP - 364

JO - Integral Transforms and Special Functions

JF - Integral Transforms and Special Functions

IS - 5

ER -