The Plancherel, Titchmarsh and convolution theorems for the half-Hartley transform. (English) Zbl 1306.44006
This paper is a natural study on the half-Hartley transform, starting with a corresponding Plancherel theorem. The convolution theorem for the half-Hartley transform is constructed using a technique that follows the general convolution method developed for integral transforms of the Mellin convolution-type, and the required mapping properties are studied. An analogue of the Titchmarsh theorem is proved about the absence of divisors of zero in the convolution related to half-Hartley transform. The results are applied to establish solvability conditions and the form of solutions for a homogeneous integral equation of the second kind.
Reviewer’s remark: The results and their proofs are carried out in an elegant manner. Analogous results may be studied for combinations of transforms under initial conditions.
Reviewer’s remark: The results and their proofs are carried out in an elegant manner. Analogous results may be studied for combinations of transforms under initial conditions.
Reviewer: N. R. Mangalambal (Kerala)
MSC:
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 44A35 | Convolution as an integral transform |
| 45E05 | Integral equations with kernels of Cauchy type |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
Hartley transform; convolution method; Plancherel theorem; Titchmarsh theorem; homogeneous integral equation; Mellin convolutionReferences:
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