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A normed system of functions for operators with mixed derivatives. (Russian) Zbl 0634.35011

An infinite system of functions \(\{f_ n\}\), \(n=0,1,2,..\). is called a normed system of functions with respect to an operator L in a certain domain \(\Omega\), if \(Lf_ n=f_{n-1}\), \(n\neq 0\), \(Lf_ 0=0\) in \(\Omega\). The authors construct a normed system of functions with respect to an operator with mixed derivatives \(L_{m+1}=D_{12...m}+D_{m+1}L_ m\), \(m=1,2,...\), where \(L_ 1=1\), \(D_{12...m}=\partial^ m/\partial x_ 1\partial x_ 2...\partial x_ m\), \(D_{m+1}=\partial /\partial x_{m+1}\) and obtain solutions of the iterated equation (*) \(L^ p_{m+1}u=0\) in closed form. The polynomial and quasipolynomial solutions of equation (*) are investigated.

MSC:

35C05 Solutions to PDEs in closed form
35G05 Linear higher-order PDEs
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