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Vinogradov series in the Cauchy problem for equations of Schrödinger type. (English. Russian original) Zbl 0831.35038
Proc. Steklov Inst. Math. 200, 291-315 (1993); translation from Tr. Mat. Inst. Steklova 200, 265-288 (1991).
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 Schrödinger 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].
Reviewer: M.Jutila (Turku)

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 11L15 Weyl sums