an:00495718
Zbl 0831.35038
Oskolkov, K. I.
Vinogradov series in the Cauchy problem for equations of SchrÃ¶dinger type
EN
Proc. Steklov Inst. Math. 200, 291-315 (1993); translation from Tr. Mat. Inst. Steklova 200, 265-288 (1991).
1991
a
35J10 42A16 11L15
Vinogradov extensions; equations with constant coefficients
Let \(f(x)\) be a periodic complex valued function with period 1 of the real variable \(x\). Then the Vinogradov extensions of \(f\) is the trigonometric series
\[
V(f; x_r, \ldots, x_1) = \sum^\infty_{n = - \infty} \widehat f(n) e(n^r x_r + \cdots + nx_1),
\]
where the \(\widehat f(n)\) are the Fourier coefficients of \(f\). Such series and their applications were studied in a previous paper by the author [Tr. Mat. Inst. Steklova 190, 186-221 (1989; Zbl 0707.11059)]. After a survey of results on the relation between a function and its Vinogradov extension under various assumptions, the author turns to the main topic of this paper: partial differential equations of the SchrÃ¶dinger type in the space variable \(x\) and in the time variable \(t\), the initial condition being given by the function \(f\). In the paper mentioned above, the case of equations with constant coefficients was considered. Now the coefficients are allowed to depend on \(x\) and \(t\). The (generalized) solution can be expressed in terms of the Vinogradov extension of \(f\) even in this general case. The author discusses first the case \(f \in L^2\), and then the more restricted case where \(f\) is of bounded variation.
For the entire collection see [Zbl 0774.00012].
M.Jutila (Turku)
0707.11059