A unified construction of product formulas and convolutions for Sturm-Liouville operators

• Type: Article in International Scientific Journal
• Date: 2021

Authors

Sousa, R (Author) Other; Guerra, M (Author) Other; Yakubovich, SB (Author) FCUP

Journal

The Journal is awaiting validation by the Administrative Services.

Title: ANALYSIS AND MATHEMATICAL PHYSICS

Vol. 11

ISSN: 1664-2368

Indexing

ISI Web of Knowledge - 3 Citations

Scopus - 1 Citation

Other information

Authenticus ID: P-00T-RA7

DOI: 10.1007/s13324-021-00520-5

Abstract (EN): We establish a positive product formula for the solutions of the Sturm-Liouville equation l(u) = lambda u, where l belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm-Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm-Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 50