St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Some functional-difference equations solvable in finitary functions


Author: E. A. Gorin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 18 (2006), nomer 5.
Journal: St. Petersburg Math. J. 18 (2007), 779-796
MSC (2000): Primary 34K99, 32A15
Published electronically: August 9, 2007
MathSciNet review: 2301043
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following equation is considered: $ q(-i\partial/\partial x)u(x)=(f*u)(Ax)$, where $ q$ is a polynomial with complex coefficients, $ f$ is a compactly supported distribution, and $ A:\mathbb{R}^n\to\mathbb{R}^n$ is a linear operator whose complexification has no spectrum in the closed unit disk. It turns out that this equation has a (smooth) solution $ u(x)$ with compact support. In the one-dimensional case, this problem was treated earlier in detail by V. A. Rvachev and V. L. Rvachev and their numerous students.


References [Enhancements On Off] (What's this?)

  • 1. I. M. Vinogradov, V. I. Bitjuckov, and Ju. V. Prohorov (eds.), Matematicheskaya entsiklopediya. Tom 1: A–G, “Sovet. Èntsiklopediya”, Moscow, 1977 (Russian). MR 470361
  • 2. V. L. Rvachev and V. A. Rvachev, Nonclassical methods of approximation theory in boundary value problems, ``Naukova Dumka'', Kiev, 1979. (Russian)
  • 3. Yu. G. Stoyan, V. S. Protsenko, G. P. Man′ko, I. V. Goncharyuk, L. V. Kurpa, V. A. Rvachëv, N. S. Sinekop, I. B. Sirodzha, A. N. Shevchenko, and T. I. Sheĭko, Teoriya R-funktsii i aktualnye problemy prikladnoi matematiki, “Naukova Dumka”, Kiev, 1986 (Russian). MR 896485
  • 4. V. A. Rvachëv, Compactly-supported solutions of functional-differential equations and their applications, Uspekhi Mat. Nauk 45 (1990), no. 1(271), 77–103, 222 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 1, 87–120. MR 1050928, 10.1070/RM1990v045n01ABEH002324
  • 5. V. M. Kolodyazhniĭ and V. O. Rvachov, Some properties of atomic functions of several variables, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 1 (2005), 12–20 (Ukrainian, with English summary). MR 2150996
  • 6. E. A. Gorin, On finite solutions for some functional-differential equations, Uspekhi Mat. Nauk 36 (1981), no. 4, 211-212. (Russian)
  • 7. M. Hervé, Several complex variables. Local theory, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1963. MR 0151632
  • 8. Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
  • 9. B. V. Shabat, Vvedenie v kompleksnyi analiz. Chast II, 3rd ed., “Nauka”, Moscow, 1985 (Russian). Funktsii neskolkikh peremennykh. [Functions of several variables]. MR 831938
    B. V. Shabat, Introduction to complex analysis. Part II, Translations of Mathematical Monographs, vol. 110, American Mathematical Society, Providence, RI, 1992. Functions of several variables; Translated from the third (1985) Russian edition by J. S. Joel. MR 1192135
  • 10. E. M. Chirka, Kompleksnye analiticheskie mnozhestva, “Nauka”, Moscow, 1985 (Russian). MR 784797
    E. M. Chirka, Complex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by R. A. M. Hoksbergen. MR 1111477
  • 11. G. R. Belitskiĭ and Yu. I. Lyubich, Normy matrits i ikh prilozheniya, “Naukova Dumka”, Kiev, 1984 (Russian). MR 799924
  • 12. B. Ya. Levin, Distribution of zeros of entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0087740
    B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
  • 13. E. A. Gorin, Estimates for the involution of decomposable elements of a complex Banach algebra, Funktsional. Anal. i Prilozhen. 39 (2005), no. 4, 14–31, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 39 (2005), no. 4, 256–270. MR 2197511, 10.1007/s10688-005-0047-z
  • 14. E. A. Gorin, Bernstein inequalities from the perspective of operator theory, Vestnik Khar′kov. Gos. Univ. 205 (1980), 77–105, 140 (Russian). MR 643356
  • 15. G. E. Šilov, Matematicheskii analiz: Vtoroi spetsialnyikurs, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0219869
  • 16. Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278
  • 17. Z. I. \cyr{B}orevich and I. R. \cyr{S}hafarevich, Teoriya chisel, 2nd ed., Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0352038
    Z. I. Borevitch and I. R. Chafarevitch, Théorie des nombres, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1993 (French). Translated from the Russian by Myriam Verley and Jean-Luc Verley; Reprint of the 1967 French translation. MR 1355542
  • 18. L. I. Ronkin, Entire functions, Current problems in mathematics. Fundamental directions, Vol. 9 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 5–36, 292 (Russian). MR 860608
  • 19. John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
  • 20. N. Ja. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0209523
    N. Ja. Vilenkin, Special functions and the theory of group representations, Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. MR 0229863
  • 21. Ingrid Daubechies and Jeffrey C. Lagarias, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), no. 4, 1031–1079. MR 1166574, 10.1137/0523059
  • 22. Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, 10.1090/memo/0453
  • 23. G. A. Derfel, A probabilistic method for studying a class of functional-differential equations, Ukrain. Mat. Zh. 41 (1989), no. 10, 1322–1327, 1436 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 10, 1137–1141 (1990). MR 1034672, 10.1007/BF01057249
  • 24. Yuval Peres and Boris Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. 3 (1996), no. 2, 231–239. MR 1386842, 10.4310/MRL.1996.v3.n2.a8
  • 25. V. Yu. Protasov, On the problem of the asymptotics of the partition function, Mat. Zametki 76 (2004), no. 1, 151–156 (Russian); English transl., Math. Notes 76 (2004), no. 1-2, 144–149. MR 2099853, 10.1023/B:MATN.0000036752.47140.98

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 34K99, 32A15

Retrieve articles in all journals with MSC (2000): 34K99, 32A15


Additional Information

E. A. Gorin
Affiliation: Department of Mathematics, Moscow State Pedagogical University, Ulitsa Malaya Pirogovskaya 1, Moscow 119882, Russia
Email: evgeny.gorin@mtu-net.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-07-00973-9
Keywords: Finitary solution, entire function of exponential type, refinement equation
Received by editor(s): April 22, 2006
Published electronically: August 9, 2007
Dedicated: Dedicated to the 100th anniversary of B. Ya. Levin’s birth
Article copyright: © Copyright 2007 American Mathematical Society