Geodesic
An equation of convolution type on convex domains in $\mathbf R^2$
Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 169-184 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $D$ be a convex domain in $\mathbf R^2$, and let $C^{(k)}(D)$, $k=(k_1,\,k_2)$, be the space of functions $f(x)$, continuous in $D$ together with their partial derivatives $$ \frac{\partial^{n_1+n_2}}{\partial x_1^{n_1}\partial x_2^{n_2}}f, $$ $n_1\leqslant k_1$, $n_2\leqslant k_2$. This space is provided with the natural topology of uniform convergence of functions and corresponding derivatives on compact subsets of $D$. Consider in $C^{(k)}(D)$ the homogeneous convolution equation $\mu*f=0$, where $\mu$ is a continuous linear functional on $C^{(k)}(D)$. It is proved that every solution of this equation from the space $C^{(k)}(D)$ can be approximated in the topology of $C^{(k)}(D)$ by a linear combination of exponential polynomials satisfying this equation. Bibliography: 15 titles. 
@article{SM_1974_23_2_a1,
     author = {V. V. Napalkov},
     title = {An equation of convolution type on convex domains in~$\mathbf R^2$},
     journal = {Sbornik. Mathematics},
     pages = {169--184},
     year = {1974},
     volume = {23},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_23_2_a1/}
}
TY  - JOUR
AU  - V. V. Napalkov
TI  - An equation of convolution type on convex domains in $\mathbf R^2$
JO  - Sbornik. Mathematics
PY  - 1974
SP  - 169
EP  - 184
VL  - 23
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1974_23_2_a1/
LA  - en
ID  - SM_1974_23_2_a1
ER  - 
%0 Journal Article
%A V. V. Napalkov
%T An equation of convolution type on convex domains in $\mathbf R^2$
%J Sbornik. Mathematics
%D 1974
%P 169-184
%V 23
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1974_23_2_a1/
%G en
%F SM_1974_23_2_a1
V. V. Napalkov. An equation of convolution type on convex domains in $\mathbf R^2$. Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 169-184. http://geodesic.mathdoc.fr/item/SM_1974_23_2_a1/

[1] J. Delsarte, “Les fonctions “Moyenne-periodiques””, J. Math. pures et appl., 14 (1935), 403–453 | Zbl

[2] L. Schwartz, “Approximation d'une function quelconque par des sommes d'exponentielles imagin”, Ann. sci. Univ. Toulouse, ser. 4, 6 (1943), 111–174 | MR

[3] L. Schwartz, Etude des sommes d'exponentielles, deuxieme edition, Act. scient. et jndustr., 1959

[4] J. P. Kahane, “Travaux de Beurling et P. Malliavin”, Sem. Bourbaki, 14-e annee, 1961/62, expose 225 | Zbl

[5] A. F. Leontev, “O svoistvakh posledovatelnostei polinomov Dirikhle, skhodyaschikhsya na intervale mnimoi osi”, Izv. AN SSSR, seriya matem., 29 (1965), 269–328 | MR

[6] B. Malgrange, “Existence et approximation des solution des equations aux dériveés partielles et des equations de convolution”, Ann. Inst. Fourier, 6 (1956), 271–355 | MR | Zbl

[7] L. Ehrenpreis, “Solutions of some problems of division, II”, Amer. J. Math., 77:2 (1955), 286–292 | DOI | MR | Zbl

[8] L. Ehrenpreis, “Solutions of some problems of division, IV”, Amer. J. Math., 82:3 (1960), 522–588 | DOI | MR | Zbl

[9] L. Hörmander, “Convolution equations in convex domains”, Invent. Math., 4 (1968), 306–317 | DOI | MR | Zbl

[10] L. Khërmander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, izd-vo «Mir», Moskva, 1968 | MR

[11] A. Zigmund, Trigonometricheskie ryady, t. 2, izd-vo «Mir», Moskva, 1965 | MR

[12] V. V. Napalkov, “Uravnenie tipa svertki v trubchatykh oblastyakh $\mathbf C^2$”, Izv. AN SSSR, seriya matem., 38:2 (1974), 446–456 | MR | Zbl

[13] V. S. Vladimirov, Metody teorii funktsii miogikh kompleksnykh peremennykh, izd-vo «Nauka», Moskva, 1964 | MR

[14] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, izd-vo «Nauka», Moskva, 1971 | MR

[15] P. Shapiro, Teoriya giperfunktsii, izd-vo «Mir», Moskva, 1972 | MR | Zbl