# Harmonic synthesis of solutions of elliptic equation with periodic coefficients

Annales de l'institut Fourier (1993)

- Volume: 43, Issue: 3, page 751-768
- ISSN: 0373-0956

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topPalamodov, Victor P.. "Harmonic synthesis of solutions of elliptic equation with periodic coefficients." Annales de l'institut Fourier 43.3 (1993): 751-768. <http://eudml.org/doc/75018>.

@article{Palamodov1993,

abstract = {An elliptic system in $\{\Bbb R\}^ n$, which is invariant under the action of the group $\{\Bbb Z\}^n$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O(\{\rm exp\}( a \vert x \vert ) )$ at infinity can be rewritten in the form of an integral over the family.},

author = {Palamodov, Victor P.},

journal = {Annales de l'institut Fourier},

keywords = {representation of translation group; coherent analytic sheaf; Noether operator for a coherent sheaf; approximation; Floquet solutions; Lasker- Noether decomposition; elliptic system},

language = {eng},

number = {3},

pages = {751-768},

publisher = {Association des Annales de l'Institut Fourier},

title = {Harmonic synthesis of solutions of elliptic equation with periodic coefficients},

url = {http://eudml.org/doc/75018},

volume = {43},

year = {1993},

}

TY - JOUR

AU - Palamodov, Victor P.

TI - Harmonic synthesis of solutions of elliptic equation with periodic coefficients

JO - Annales de l'institut Fourier

PY - 1993

PB - Association des Annales de l'Institut Fourier

VL - 43

IS - 3

SP - 751

EP - 768

AB - An elliptic system in ${\Bbb R}^ n$, which is invariant under the action of the group ${\Bbb Z}^n$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O({\rm exp}( a \vert x \vert ) )$ at infinity can be rewritten in the form of an integral over the family.

LA - eng

KW - representation of translation group; coherent analytic sheaf; Noether operator for a coherent sheaf; approximation; Floquet solutions; Lasker- Noether decomposition; elliptic system

UR - http://eudml.org/doc/75018

ER -

## References

top- [1] V.P. PALAMODOV, Linear differential operators with constant coefficients, Moscow, Nauka, 1967, Springer-Verlag, 1970. Zbl0191.43401
- [2] L. EHRENPREIS, Fourier analysis in several complex variables, N.Y., 1970. Zbl0195.10401MR44 #3066
- [3] I.M. GEL'FAND, Eigenfunction decomposition of equation with periodic coefficients, Doklady ANSSSR, 73, n° 6 (1950), 1117-1120 (Russian).
- [4] P.A. KUCHMENT, Floquet theory for partial differential equations, Russian Math. Surveys, 37, n° 4 (1982), 1-50. Zbl0519.35003MR84b:35018
- [5] L. BUNGART, On analytic fibre bundles I. Holomorphic fiber bundles with infinite dimensional fibres, Topology, 7 (1968), 55-68. Zbl0153.10202MR36 #5390
- [6] V.P. PALAMODOV, The projective limit on the category of linear topological spaces, Mathematics of the USSR Sbornik, 4 (1968), 529-559. Zbl0175.41801
- [7] O. ZARISKI, P. SAMUEL, Commutative algebra, Ch. IV, Van Nostrand, 1958.
- [8] N. BOURBAKI, L'algèbre commutative, Paris, Hermann, 1967.
- [9] V.P. PALAMODOV, Differential operators in coherent analytic sheaves, Mathematics of the USSR Sbornik, 6 (1968), 365-391. Zbl0187.07903MR38 #3471
- [10] Y.T. SIU, Noether-Lasker decomposition of coherent analytic subsheaves, Trans. of A.M.S., 135 (1969), 375-385. Zbl0175.37403MR38 #2340
- [11] R. GUNNING, H. ROSSI, Analytic functions of serveral complex variables, Englewood Cliffs, Prentice Hall, 1965. Zbl0141.08601
- [12] B. MALGRANGE, Existence et approximation des solutions des équations aux dérivées partielles et des équations des convolution, Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 271-355. Zbl0071.09002MR19,280a
- [13] V.P. PALAMODOV, Deformation of complex spaces, Russian Math. Surveys, 31 (1976), 129-197. Zbl0347.32009MR58 #22671

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