Parseval-Goldstein type theorems for integral transforms in a general setting

Abstract

This research paper explores Parseval-Goldstein type relations concerning general integral operators. It investigates the continuity properties of these operators and their adjoints over Lebesgue spaces. Through rigorous analysis, the study elucidates the intricate connections between these operators and sheds light on their behaviour within functional spaces. By exploring the convergence and stability of these relations, the paper contributes to a deeper understanding of integral operators behaviour and their implications in various mathematical contexts. The paper also examines specific cases of the main index transforms, including the KontorovichLebedev transform, the Mehler-Fock transform of general order, the index 2𝐹1-transform, the Lebedev-Skalskaya transforms and the index Whittaker transform, as well as operators with complex Gaussian kernels, contributing valuable insights into their behaviour and applications.

Keywords

• Integral operators
• weighted Lebesgue spaces
• Parseval-Goldstein relations
• index transforms
• Gaussian kernels

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