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.