Transformation de Fourier sur les espaces p (L p )
Annales de l'Institut Fourier, Volume 29 (1979) no. 1, pp. 189-206.

We study the Fourier transform on the spaces p (L p ) of functions belonging locally to L p and with p behaviour at infinity. If 1p,p 2, the Fourier transforms of all functions in p (L p ) are in q (L q ). In the other cases, we state some partial results.

We show that the maximal solid subspace of integrable functions on which the Fourier transform can be defined by extension by continuity is the space 2 (L 1 ).

Nous étudions d’abord la transformation de Fourier sur les espaces p (L p ) qui sont formés de fonctions appartenant localement à L p et se comportant à l’infini comme des éléments de p . Si 1p,p 2, les transformées de Fourier des éléments de p (L p ) appartiennent à q (L q ). Dans les autres cas, nous donnons quelques résultats partiels.

Nous montrons ensuite que 2 (L 1 ) est le plus grand espace vectoriel solide de fonctions mesurables sur lequel la transformation de Fourier puisse se définir par prolongement par continuité.

@article{AIF_1979__29_1_189_0,
     author = {Bertrandias, Jean-Paul and Dupuis, Christian},
     title = {Transformation de {Fourier} sur les espaces $\ell ^p(L^p)$},
     journal = {Annales de l'Institut Fourier},
     pages = {189--206},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {1},
     year = {1979},
     doi = {10.5802/aif.734},
     mrnumber = {82a:43006},
     zbl = {0387.42003},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/aif.734/}
}
TY  - JOUR
AU  - Bertrandias, Jean-Paul
AU  - Dupuis, Christian
TI  - Transformation de Fourier sur les espaces $\ell ^p(L^p)$
JO  - Annales de l'Institut Fourier
PY  - 1979
SP  - 189
EP  - 206
VL  - 29
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.734/
DO  - 10.5802/aif.734
LA  - fr
ID  - AIF_1979__29_1_189_0
ER  - 
%0 Journal Article
%A Bertrandias, Jean-Paul
%A Dupuis, Christian
%T Transformation de Fourier sur les espaces $\ell ^p(L^p)$
%J Annales de l'Institut Fourier
%D 1979
%P 189-206
%V 29
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.734/
%R 10.5802/aif.734
%G fr
%F AIF_1979__29_1_189_0
Bertrandias, Jean-Paul; Dupuis, Christian. Transformation de Fourier sur les espaces $\ell ^p(L^p)$. Annales de l'Institut Fourier, Volume 29 (1979) no. 1, pp. 189-206. doi : 10.5802/aif.734. http://www.numdam.org/articles/10.5802/aif.734/

[1] N. Aronszajn, P. Szeptycki, On general integral transformations, Math. Ann., 163 (1966), 127-154. | MR | Zbl

[2] A. Benedek, R. Panzone, The spaces Lp, with mixed norm, Duke Math. Journ., 28 (1961), 301-324. | Zbl

[3] J.P. Bertrandias, Unions et intersections d'espaces Lp sur un espace localement compact, Bull. Sc. Math., 101 (1977) 209-247. | MR | Zbl

[4] J.P. Bertrandias, C. Datry, C. Dupuis, Unions et intersections d'espaces Lp invariantes par translation et convolution, Ann. Inst. Fourier, 28, 2 (1978), 53-84. | Numdam | MR | Zbl

[5] R. Goldberg, On a space of functions of Wiener, Duke Math. Journ., 34 (1967), 683-691. | MR | Zbl

[6] F. Holland, Harmonic analysis on amalgams of Lp and lq, Journ. London Math. Soc., 2, 10 (175), 295-305. | MR | Zbl

[7] F. Holland, On the representation of function as Fourier transforms of unbounded measures, Proc. London Math. Soc., 3, 30 (1975), 347-365. | MR | Zbl

[8] J.P. Kahane, Some random series of function, Heath, 1968. | MR | Zbl

[9] Y. Katznelson, An introduction to harmonic analysis, Wiley, 1968. | MR | Zbl

[10] L. Pigno, Restrictions of Lp transforms, Proc. Ann. Math. Soc., 29 (1971), 511-515. | MR | Zbl

[11] W. Rudin, Fourier analysis on groups, Interscience, 1962. | MR | Zbl

[12] A.B. Simon, Cesàro summability on groups : characterisation and inversion of Fourier transforms, Function algebras (ed. F.T. Birtel) Scott, Foresman, 1966. | Zbl

[13] J, Stewart, Unbounded positive definite functions, Can. Journ. Math., 21 (1969), 1309-1318. | MR | Zbl

[14] P. Szeptycki, Some remarks on the extended domain of Fourier transform, Bull. Ann. Math. Soc., 73 (1967), 398-402. | MR | Zbl

[15] P. Szeptycki, On functions and measures whose Fourier transforms are functions, Math. Ann., 179 (1968), 31-41. | MR | Zbl

[16] N. Wiener, Tauberian theorems, Ann. of Math., 33 (1932), 1-100. | JFM | Zbl

[17] A. Zygmund, Trigonometric series, 2 vol., Cambridge, 1959. | Zbl

Cited by Sources: