The convolution of functions and distributions

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Fisher B., TAŞ K.

Journal of Mathematical Analysis and Applications, vol.306, no.1, pp.364-374, 2005 (SCI-Expanded)

• Publication Type: Article / Article
• Volume: 306 Issue: 1
• Publication Date: 2005
• Doi Number: 10.1016/j.jmaa.2005.01.004
• Journal Name: Journal of Mathematical Analysis and Applications
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
• Page Numbers: pp.364-374
• Keywords: Convolution, Dirac delta function, Distribution
• Uşak University Affiliated: No

Abstract

The non-commutative convolution f * g of two distributions f and g in D′ is defined to be the limit of the sequence {(fτn) * g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that x λ * (sgnx x μ) = 2 sin(λπ/2)cos(μπ/2)/sin[(λ + μ)π/2] B(λ + 1, μ + 1) sgn x x λ+μ+1, for - 1 < λ + μ < 0 and λ, μ≠ - 1, - 2,..., where B denotes the Beta function. © 2005 Elsevier Inc. All rights reserved.