Naslov
Probabilistic and analytical aspects of the symmetric and generalized Kaiser-Bessel window function

Autori
Baricz, Árpád ; Poganj, Tibor

Vrsta, podvrsta
Radovi u časopisima, znanstveni

Izvornik
Constructive approximation (2023)

Status rada
Prihvaćen

Ključne riječi
Generalized Kaiser-Bessel window function ; Modi fied Bessel function ; Moments ; sub-Gaussian distribution ; Cumulative distribution function ; Characteristic function ; Moment generating function ; Differential entropy ; Rényi entropy ; Ricatti dfferential equation ; Wigner's semicircle distribution ; Power semicircle distribution ; Turán type inequalities ; log-concavity ; Generating random variables ; Parameter estimation ; Method of moments

Sažetak
The generalized Kaiser-Bessel window function is defi ned via the modifi ed Bessel function of the fi rst kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser- Bessel distribution, which we de ne via the symmetric form of the generalized Kaiser-Bessel window function. The Kaiser-Bessel distribution resembles to the Bessel distribution of McKay of the fi rst type, it is a platykurtic or sub- Gaussian distribution, it is not infi nitely divisible in the classical sense and it is an extension of the Wigner's semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two di erent ways. In addition, we nd its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modifi ed Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser-Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modifi ed Bessel functions of the fi rst kind. Finally, we also present a modifi ed method of moments to estimate the parameters of the Kaiser- Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser-Bessel distribution. The paper is closed with conclusions and proposals for future works.

Izvorni jezik
Engleski

Znanstvena područja
Matematika, Interdisciplinarne prirodne znanosti, Temeljne tehničke znanosti

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