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On the identity of minimal and maximal realizations related to Fourier series operators. (English) Zbl 0754.42009
The paper studies the Fourier series operator \[ (L(x,D)\varphi)(x):=(2\pi)^{-n}\sum_{l\in \mathbb{Z}^ n} L(x,l)\varphi_ l e^{i(l,x)} \] on a certain subspace \(B^ \pi_{p,k}\) of the periodic distributions defined on \(\mathbb{R}^ n\). Here \(\varphi_ l\) denotes the Fourier coefficient of the distribution \(\varphi\). Under certain conditions on the symbol \(L(x,l)\), the author studies the identity of the maximal and minimal realizations of the operator \(L(x,D)\).
MSC:
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
47G30 Pseudodifferential operators
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References:
[1] BEALS R.: Advanced Mathematical Analysis. Graduate texts in mathematics. Springer-Verlag, New York-Heidelberg-Berlin, 1973. · Zbl 0294.46029
[2] BERGH J., LÖFSTRÖM J.: Interpolation Spaces. Die Grundlehren der mathematischen Wissenschaften 223. Springer-Verlag, Berlin-Heidelberg-New York, 1976. · Zbl 0344.46071
[3] BERS L., SCHECHTER M.: Elliptic equations. Partial differential equations, by L. Bers, F.John, and M. Schechter. Lectures in applied mathematics (summer seminar, Boulder (Colorado), 1957 III. Interscience Publishers, a division of John Willey and Sons, Inc, New York-London-Sydney, 1964, pp. 131-299. · Zbl 0126.00207
[4] EELLS J.: Elliptic operators on manifolds. Complex analysis and its applications I. International Centre for Theoretical Physics, Trieste. lnternatinal Atomic Energy Agency, Viena, 1976. · Zbl 0349.58008
[5] HÖRMANDER L.: Linear Partial Differential Operators. Die Grundlehren der mathematischen Wissenschaften 116. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. · Zbl 0108.09301
[6] KULESHOV A. A.: Linear equations in space of periodic generalized functions. Differ. Uravn. 20 (1984), 308-315. · Zbl 0544.47043
[7] POLIŠČUK V. N., PTAŠNIK B. I.: Periodic solutions of the system of partial differential equations with constant coefficients. Ukr. Math. J. 32 (1980), 239-243.
[8] TERVO J.: Zur Theorie der koerzitiven linearen partiellen Differentialoperatoren. Ann. Acad. Sci. Fenn., Ser. A. I, Diss. 45 (1983). · Zbl 0534.35016
[9] TERVO J.: On compactness of Fourier series operators. Ber. Univ. Jyväskylä Math. Inst. 38 (1987). · Zbl 0661.47039
[10] YOSIDA K.: Functional Analysis. Die Grundlehren der mathematishen Wissenschaften 123. Springer-Verlag, Berlin-Heidelberg-New York, 1974. · Zbl 0286.46002
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