Abstract
Recently renewed interest in the Lobachevsky-type integrals and interesting identities involving the cardinal sine motivate an extension of the classical Parseval formula involving both periodic and non-periodic functions. We develop a version of the Parseval formula that is often more practical in applications and illustrate its use by extending recent results on Lobachevsky-type integrals. Some previously known, interesting identities are re-proved in a more transparent manner and new formulas for integrals involving cardinal sine and Bessel functions are given.
Highlights
The following is known as a Lobachevsky-type integral: ∞ −∞sin πx πx k p(x) dxHere k ∈ N \ {0}, and p : R → C is a periodic, real-valued function with period T > 0 that is assumed to be integrable over a single period
K ∈ N \ {0}, and p : R → C is a periodic, real-valued function with period T > 0 that is assumed to be integrable over a single period
We will base our discussion on the Fourier transform and obtain corresponding identities for all k ∈ N \ {0} and p integrable of arbitrary period T
Summary
The following is known as a Lobachevsky-type integral:. K ∈ N \ {0}, and p : R → C is a periodic, real-valued function with period T > 0 that is assumed to be integrable over a single period. Jolany [2] has published identities for this integral when k is even, with continuous p being of period T = 1, using methods of complex analysis. We will base our discussion on the Fourier transform and obtain corresponding identities for all k ∈ N \ {0} and p integrable of arbitrary period T. The Lobachevsky integral is closely related to the Shannon basis of information theory. Much of our treatment is inspired by identities that are “folklore” in the signal processing community, where the reconstruction formula in the Shannon basis is precisely the cardinal sine expansion [6]
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