A simple range characterization for spherical mean transform in even dimensions
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918005480357888 |
|---|---|
| author | Agrawal, Divyansh Ambartsoumian, Gaik Krishnan, Venkateswaran P. Singhal, Nisha |
| author_facet | Agrawal, Divyansh Ambartsoumian, Gaik Krishnan, Venkateswaran P. Singhal, Nisha |
| contents | The paper presents a new and simple range characterization for the spherical mean transform of functions supported in the unit ball in even dimensions. It complements the previous work of the same authors, where they solved an analogous problem in odd dimensions. The range description in even dimensions consists of symmetry relations, using a special kind of elliptic integrals involving the coefficients of the spherical harmonics expansion of the function in the range of the transform. The article also introduces a pair of original identities involving normalized Bessel functions of the first and the second kind. The first result is an integral cross-product identity for Bessel functions of integer order, complementing a similar relation for Bessel functions of half-integer order obtained in the aforementioned work of the same authors. The second result is a new Nicholson-type identity. Both of these relations can be considered as important standalone results in the theory of special functions. Finally, as part of the proof of one of the theorems, the authors derive an interesting equality involving elliptic integrals, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21824 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A simple range characterization for spherical mean transform in even dimensions Agrawal, Divyansh Ambartsoumian, Gaik Krishnan, Venkateswaran P. Singhal, Nisha Classical Analysis and ODEs Mathematical Physics Functional Analysis 44A12, 44A15, 44A20, 45Q05, 33C10, 33C55, 33E05 The paper presents a new and simple range characterization for the spherical mean transform of functions supported in the unit ball in even dimensions. It complements the previous work of the same authors, where they solved an analogous problem in odd dimensions. The range description in even dimensions consists of symmetry relations, using a special kind of elliptic integrals involving the coefficients of the spherical harmonics expansion of the function in the range of the transform. The article also introduces a pair of original identities involving normalized Bessel functions of the first and the second kind. The first result is an integral cross-product identity for Bessel functions of integer order, complementing a similar relation for Bessel functions of half-integer order obtained in the aforementioned work of the same authors. The second result is a new Nicholson-type identity. Both of these relations can be considered as important standalone results in the theory of special functions. Finally, as part of the proof of one of the theorems, the authors derive an interesting equality involving elliptic integrals, which may be of independent interest. |
| title | A simple range characterization for spherical mean transform in even dimensions |
| topic | Classical Analysis and ODEs Mathematical Physics Functional Analysis 44A12, 44A15, 44A20, 45Q05, 33C10, 33C55, 33E05 |
| url | https://arxiv.org/abs/2504.21824 |