Geodesic
The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1
Canadian journal of mathematics, Tome 38 (1986) no. 2, pp. 328-359

Voir la notice de l'article provenant de la source Cambridge University Press

The Fourier transform of the surface measure on the unit sphere in R n + 1, as is well-known, equals the Bessel function Its behaviour at infinity is described by an asymptotic expansion The purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in R n + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2]. 
Marshall, Bernard. The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1. Canadian journal of mathematics, Tome 38 (1986) no. 2, pp. 328-359. doi: 10.4153/CJM-1986-016-7
@article{10_4153_CJM_1986_016_7,
     author = {Marshall, Bernard},
     title = {The {Fourier} {Transforms} of {Smooth} {Measures} on {Hypersurfaces} of {Rn} + 1},
     journal = {Canadian journal of mathematics},
     pages = {328--359},
     year = {1986},
     volume = {38},
     number = {2},
     doi = {10.4153/CJM-1986-016-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1986-016-7/}
}
TY  - JOUR
AU  - Marshall, Bernard
TI  - The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1
JO  - Canadian journal of mathematics
PY  - 1986
SP  - 328
EP  - 359
VL  - 38
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1986-016-7/
DO  - 10.4153/CJM-1986-016-7
ID  - 10_4153_CJM_1986_016_7
ER  - 
%0 Journal Article
%A Marshall, Bernard
%T The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1
%J Canadian journal of mathematics
%D 1986
%P 328-359
%V 38
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1986-016-7/
%R 10.4153/CJM-1986-016-7
%F 10_4153_CJM_1986_016_7

[1] 1. Greenleaf, A., Principal curvature and harmonic analysis, Indiana Math. J. 30 (1981), 519–537. Google Scholar

[2] 2. Herz, C., Fourier transforms related to convex sets, Ann. of Math. 75 (1962), 1–36. Google Scholar

[3] 3. Hlawka, E., Über Integrale auf konvexen Körpern I, Monatsh. Math. 54 (1950), 1–36; II, ibid. 54 (1950) 81–99. Google Scholar

[4] 4. Littman, W., Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766–770. Google Scholar

[5] 5. Marshall, B., Estimates for solutions of wave equations with vanishing curvature, Can J. Math. 57 (1985). Google Scholar

[6] 6. Randol, B., On the Fourier transform of the indicator function of a planar set, Trans. Amer. Math. Soc. 139 (1969), 271–278. Google Scholar

[7] 7. Randol, B., On the asymptotic behaviour of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc. 139 (1969), 279–285. Google Scholar

[8] 8. Svensson, I., Estimates for the Fourier transform of the characteristic function of a convex set, Arkiv för Matematik 9 (1971), 11–22. Google Scholar

Cité par Sources :