Geodesic
Special Weber Transform with Nontrivial Kernel
Matematičeskie zametki, Tome 114 (2023) no. 2, pp. 212-228.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the Weber integral transforms $W_{k,k\pm1}$, which have a nontrivial kernel, so that the spectral expansion contains not only the continuous part of the spectrum but also the zero eigenvalue corresponding to the kernel. The inversion formula, the spectral decomposition, and the Plancherel–Parseval equality are derived. These transforms are used in an explicit formula for the solution of the classical nonstationary Stokes problem on the flow past a circular cylinder.
Keywords: Weber transforms, degenerate transform, Stokes problem. 
@article{MZM_2023_114_2_a4,
     author = {A. V. Gorshkov},
     title = {Special {Weber} {Transform} with {Nontrivial} {Kernel}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {212--228},
     publisher = {mathdoc},
     volume = {114},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_114_2_a4/}
}
TY  - JOUR
AU  - A. V. Gorshkov
TI  - Special Weber Transform with Nontrivial Kernel
JO  - Matematičeskie zametki
PY  - 2023
SP  - 212
EP  - 228
VL  - 114
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2023_114_2_a4/
LA  - ru
ID  - MZM_2023_114_2_a4
ER  - 
%0 Journal Article
%A A. V. Gorshkov
%T Special Weber Transform with Nontrivial Kernel
%J Matematičeskie zametki
%D 2023
%P 212-228
%V 114
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2023_114_2_a4/
%G ru
%F MZM_2023_114_2_a4
A. V. Gorshkov. Special Weber Transform with Nontrivial Kernel. Matematičeskie zametki, Tome 114 (2023) no. 2, pp. 212-228. http://geodesic.mathdoc.fr/item/MZM_2023_114_2_a4/

[1] H. Weber, “Ueber eine Darstellung willkürlicher Functionen durch Bessel'sche Functionen”, Math. Ann., 6:2 (1873), 146–161 | DOI | MR

[2] E. C. Titchmarsh, “Weber's integral theorem”, Proc. London Math. Soc. (2), 22 (1924), 15–28 | DOI | MR

[3] J. L. Griffith, “A note on a generalisation of Weber's transform”, J. Proc. Roy. Soc. New South Wales, 90 (1956), 157–162 | MR

[4] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, t. 1, IL, M., 1960 | MR

[5] S. Goldstein, “Some two-dimensional diffusion problems with circular symmetry”, Proc. London Math. Soc. (2), 34:1 (1932), 51–88 | DOI | MR

[6] J. Krajewski, Z. Olesiak, “Associated Weber integral transforms of $W_{\nu-l,j}[\,{;}\,]$ and $W_{\nu-2,j}[\,{;}\,]$ types”, Bull. Acad. Polon. Sci. Ser. Sci. Tech., 30:7–8 (1982), 31–37 | MR

[7] C. Nasim, “Associated Weber integral transforms of arbitrary orders”, Indian J. Pure Appl. Math., 20:11 (1989), 126–1138 | MR

[8] Z. Kabala, G. Cassiani, “Well hydraulics with the Weber–Goldstein transforms”, Transport in Porous Media, 29 (1997), 225–246 | DOI

[9] A. V. Gorshkov, “Associated Weber–Orr transform, Biot–Savart law and explicit form of the solution of 2D Stokes system in exterior of the disc”, J. Math. Fluid Mech., 21:3 (2019), 41 | DOI | MR

[10] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, t. 2, Nauka, M., 1953 | MR

[11] Dzh. N. Vatson, Teoriya besselevykh funktsii, IL, M., 1949 | MR

[12] A. V. Gorshkov, “Boundary stabilization of Stokes system in exterior domains”, J. Math. Fluid Mech., 18:4 (2016), 679–697 | DOI | MR

[13] K. Iosida, Funktsionalnyi analiz, Mir, M., 1967 | MR

[14] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR