On solutions of equations of infinite order in the real domain

V. V. Napalkov

Sbornik. Mathematics, Tome 31 (1977) no. 4, pp. 445-455

Résumé

A homogeneous partial differential equation of infinite order with constant coefficients of the form \begin{equation} L[y]\equiv\sum_{|\alpha|\geqslant0}a_\alpha\frac{\partial^{|\alpha|}}{\partial x^\alpha}\,y(x)=0,\qquad\alpha=(\alpha_1,\dots,\alpha_n), \end{equation} is considered, where $y(x)$ is an infinitely differentiate function that is defined on a convex domain $\Omega\subset R^n$ and satisfies the estimate $$ \max\biggl|\frac{\partial^{|\alpha|}y(x)}{\partial x^\alpha}\biggr|\leqslant Nh^{|\alpha|}M_{|\alpha|},\qquad N=N(K,y),\quad h=h(K,y), $$ on every compact set $K\Subset\Omega$. It is shown under certain conditions on the sequence $M_{|\alpha|}$ that every solution of equation (1) can be approximated by the exponential solutions of this equation. Bibliography: 12 titles.

Bibliographie

[1] M. Neymark, “On the Laplace transform of functionals on classe of infinitely differentiable functions”, Ark. math., 7:6 (1969), 577–594 | DOI | MR | Zbl

[2] Nerasimkhan, Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, izd-vo «Mir», Moskva, 1971

[3] G. Valiron, “Sur les solutions des equations differentielles lineaires d'ordre infini et á coefficients”, Ann. de l'Ecole Norm. Sup., 46:1 (1929), 25–33 | MR

[4] Yu. F. Korobeinik, “O beskonechno differentsiruemykh resheniyakh lineinogo differentsialnogo uravneniya beskonechnogo poryadka”, Sib. matem. zh., 6:3 (1965), 516–527 | MR

[5] A. F. Leontev, “O sposobakh resheniya uravneniya beskonechnogo poryadka v deistvitelnoi oblasti”, Izv. AN SSSR, seriya matem., 34 (1970), 849–880 | MR

[6] V. V. Napalkov, “Ob odnoi teoreme edinstvennosti v teorii funktsii mnogikh kompleksnykh peremennykh i odnorodnykh uravneniyakh tipa svertki v trubchatykh oblastyakh $\mathbf{C}^n$”, Izv. AN SSSR, seriya matem., 40 (1976), 115–132 | MR | Zbl

[7] B. V. Shabat, Vvedenie v kompleksnyi analiz, izd-vo «Nauka», Moskva, 1969 | MR

[8] L. Shvarts, Analiz, t. 1, izd-vo «Mir», Moskva, 1972

[9] B. Malgrange, “Existence et approximation des solution des equations aux derivees partielles et des equations de convolution”, Ann. Inst. Fourier, 6 (1955), 271–354 | MR

[10] A. Martineau, “Equations differentielles d'ordre infini”, Bull. Soc. Math. France, 95 (1967), 109–154 | MR | Zbl

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

[12] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, izd-vo «Nauka», Moskva, 1973 | MR