Weighted finite Laplace transform operator: spectral analysis and quality of approximation by its eigenfunctions. Issue 9 (2nd September 2018)

• Record Type: Journal Article
• Authors: Bourguiba, NourElHouda; Karoui, Abderrazek
• Is Part Of: Integral transforms and special functions. Volume 29:Issue 9(2018)
• Page Start: 679
• Page End: 698
• Publication Date: 2018-09-02
• Subjects: Weighted finite Laplace transform -- Min–Max theorem -- eigenvalues and eigenfunctions -- special functions -- prolate spheroidal wave functions; 41A30 -- 33E10 -- 34L10 -- 42C10; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2018.1489804
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: For two real numbersc > 0, α > − 1, we study some spectral properties of the weighted finite bilateral Laplace transform operator, defined over the spaceE = L 2 ( I, ω α ), I = [ − 1, 1 ], ω α ( x ) = ( 1 − x 2 ) α, byL c α f ( x ) = ∫ I e c x y f ( y ) ω α ( y ) d y . In particular, we use a technique based on the Min–Max theorem to prove that the sequence of the eigenvalues of this operator has a super-exponential decay rate to zero. Moreover, we give a lower bound with a magnitude of ordere c, for the largest eigenvalue of the operatorL c α . Also, we give some local estimates and bounds of the eigenfunctionsϕ n, c α ofL c α . Moreover, we show that these eigenfunctions are good candidates for the spectral approximation of a function that can be written as a weighted finite Laplace transform of an otherL 2 ( I, ω α ) − function. Finally, we give some numerical examples that illustrate the different results of this work. In particular, we provide an example that illustrate the Laplace-based spectral method, for the inversion of the finite Laplace transform.