A proof of new summation formulae by using sampling theorems

Zayed, Ahmed I.

doi:10.1090/s0002-9939-1993-1116276-8

Cited by 6 publications

(5 citation statements)

References 16 publications

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“…The results obtained form a stochastic setting counterpart to recent results by Zayed [25,26,24,27], Knockaert [13] and Jankov et al [7]. This paper is organized as follows: in the sequel we give a short account in correlation and spectral theory of stochastic signals, which consists from a necessary introductionary knowledge about different kind stochastic processes appearing in the engineering literature together with associated mathematical models.…”

Section: Introductionmentioning

confidence: 77%

Bessel–Sampling Restoration of Stochastic Signals

2013

APH

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The main aim of this article is to establish sampling series restoration formulae in for a class of stochastic L 2-processes which correlation function possesses integral representation close to a Hankel-type transform which kernel is either Bessel function of the first and second kind J ν ,Y ν respectively. The results obtained belong to the class of irregular sampling formulae and present a stochastic setting counterpart of certain older results by Zayed [25] and of recent results by Knockaert [13] for J-Bessel sampling and of currently established Y-Bessel sampling results by Jankov Maširević et al. [7]. The approach is twofold, we consider sampling series expansion approximation in the mean-square (or L 2) sense and also in the almost sure (or with the probability 1) sense. The main derivation tools are the Piranashvili's extension of the famous Karhunen-Cramér theorem on the integral representation of the correlation functions and the same fashion integral expression for the initial stochastic process itself, a set of integral representation formulae for the Bessel functions of the first and second kind J ν ,Y ν and various properties of Bessel and modified Bessel functions which lead to the so-called Bessel-sampling when the sampling nodes of the initial signal function coincide with a set of zeros of different cylinder functions.

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Section: Introductionmentioning

confidence: 77%

Bessel–Sampling Restoration of Stochastic Signals

2013

APH

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“…Annaby for suggesting the problem and for useful advices during the work. The author also thanks a referee who brought into his attention the references [15][16][17] on the use of the sampling theorems in finding infinite sums.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…One formula of the second work is the expansion ∞ n=−∞ sin[a √ n 2 α 2 + λ 2 ] √ n 2 α 2 + λ 2 = π α J 0 (λa), 0 < α 2π/a, (1.1) where J 0 (x) is the Bessel function of order zero, see also [17,Chapter 7]. In the work of Zayed [15] many summation formulae which contain infinite series with trigonometric and special functions are given. Some of these formulae may be found in the work of Gosper, Ismail and Zhang [8].…”

Section: Introductionmentioning

confidence: 99%

New trigonometric sums by sampling theorem

Hassan

2008

Journal of Mathematical Analysis and Applications

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We use a sampling theorem associated with second-order discrete eigenvalue problems to derive some trigonometric identities extending the results of Byrne and Smith [G.J. Byrne, S.J. Smith, Some integer-valued trigonometric sums, Proc. Edinburg Math. Soc. 40 (1997) 393-401]. We derive both integral and non-integral valued trigonometric sums. We give illustrative examples involving representations of the trigonometric sums n k=0 cot 2m ((2k + 1)π /2(2n + 1)) and n k=0 tan 2m (kπ/(2n + 1)) in an integral-valued polynomial in (2n + 1) of degree 2m, m = 1, 2, . . . .

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“…Bondarenko's recent article [9] and the references therein and Miller's multidimensional expansion [38]. A set of summation formulae of Schlömilch series for Bessel function of the first kind can be found in the literature, such as the Nielsen formula [63, p. 636]; further, we have [58, p. 65], also consult [21,45,50,59,64,65]. Similar summations, for Schlömilch series of Struve function, have been given by Miller [39], consult [59] too.…”

Section: Introductionmentioning

confidence: 99%

Integral representations and summations of the modified Struve function

Baricz

Pogány

2013

Acta Math Hung

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It is known that Struve function Hν and modified Struve function Lν are closely connected to Bessel function of the first kind Jν and to modified Bessel function of the first kind Iν and possess representations through higher transcendental functions like generalized hypergeometric 1 F 2 and Meijer G function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for Lν (x). In this paper firstly, we obtain various another type integral representation formulae for Lν (x) using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schlömilch series built by Iν (x) and Lν (x) which are connected by a Sonin-Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich-Wagner line integral expressions are derived for the Bessel function of the first kind Jν and for an associated generalized Schlömilch series.

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A proof of new summation formulae by using sampling theorems

Cited by 6 publications

References 16 publications

Bessel–Sampling Restoration of Stochastic Signals

Bessel–Sampling Restoration of Stochastic Signals

New trigonometric sums by sampling theorem

Integral representations and summations of the modified Struve function

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