On Fourier transforms of functions of the R. Nevanlinna class in the half-plane
HTML articles powered by AMS MathViewer
- by
F. A. Shamoyan
Translated by: A. Plotkin - St. Petersburg Math. J. 20 (2009), 665-680
- DOI: https://doi.org/10.1090/S1061-0022-09-01066-8
- Published electronically: June 2, 2009
- PDF | Request permission
Abstract:
Let $f$ be a function holomorphic in the upper half-plane and belonging to the Nevanlinna class $N(\mathbb {C}_+)$. Assume that \[ \limsup \limits _{y \to +\infty } \frac {\ln |f(iy)|}{y} \le 0 \] and that the boundary values of $f$ on the real axis lie in $L^1(\mathbb {R})$. It is shown that if $\vert \widehat {f}(x)\vert \le \frac {1}{\lambda (|x|)}$, $x\in {\mathbb {R}_-}$, where $\widehat {f}$ is the Fourier transform of $f$ and $\lambda$ is a logarithmically convex positive function on ${\mathbb {R}_+}$, then the condition $\int _{1}^{+\infty }\frac {\ln \lambda (x)}{x^{3/2}} dx=+\infty$ implies that $\widehat {f}(x)=0$ for all $x\in {\mathbb {R}_-}$. Conversely, if one of the conditions listed above fails, then there exists $f\in N(\mathbb {C}_+) \cap L^1(\mathbb {R})$ with $\widehat {f}(x)\ne 0$, $x\in {\mathbb {R}_-}$.References
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- F. A. Shamoyan, On Fourier transform of functions of bounded type, Scientific Conference Dedicated to the Academician I. G. PetrovskiÄ Centenary: Thesis, Bryansk. Gos. Univ., Bryansk, 2001, pp. 27â28. (Russian)
- S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris, 1952 (French). MR 0051893
- Jean-Pierre Kahane and Yitzhak Katznelson, Sur le comportement radial des fonctions analytiques, C. R. Acad. Sci. Paris SĂ©r. A-B 272 (1971), A718âA719 (French). MR 277721
- F. A. Shamoyan, Characterization of the rate of decrease of the Fourier coefficients of functions of bounded type and of a class of analytic functions with infinitely differentiable boundary values, Sibirsk. Mat. Zh. 36 (1995), no. 4, 943â953, v (Russian, with Russian summary); English transl., Siberian Math. J. 36 (1995), no. 4, 816â826. MR 1367262, DOI 10.1007/BF02107340
- Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
- M. M. Dzhrbashyan and A. Ă. Dzhrbashyan, Integral representation for certain classes of analytic functions in the half plane, Dokl. Akad. Nauk SSSR 285 (1985), no. 3, 547â550 (Russian). MR 821337
- N. Bourbaki, ĂlĂ©ments de mathĂ©matique. I: Les structures fondamentales de lâanalyse. Livre IV: Fonctions dâune variable rĂ©elle (thĂ©orie Ă©lĂ©mentaire). Chapitres 1, 2 et 3: DĂ©rievĂ©es. Primitives et intĂ©grales. Fonctions Ă©lĂ©mentaires, ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1074, Hermann, Paris, 1958 (French). DeuxiĂšme Ă©dition. MR 0151354
- Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- M. M. DĆŸrbaĆĄyan, On asymptotic approximation by entire functions in a half plane, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 749â752 (Russian). MR 0086177
- S. N. Mergelyan, Weighted approximations by polynomials, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 5(71), 107â152 (Russian). MR 0083614
- N. I. Ahiezer, Lektsii po teorii approksimatsii, Second, revised and enlarged edition, Izdat. âNaukaâ, Moscow, 1965 (Russian). MR 0188672
- Baltasar R.-Salinas, Functions with null moments, Rev. Acad. Ci. Madrid 49 (1955), 331â368 (Spanish). MR 80174
Bibliographic Information
- F. A. Shamoyan
- Affiliation: Bryansk State University, 241050 Bryansk, Russia
- Email: shamoyan@tu-bryansk.ru
- Received by editor(s): July 5, 2007
- Published electronically: June 2, 2009
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 665-680
- MSC (2000): Primary 30D50
- DOI: https://doi.org/10.1090/S1061-0022-09-01066-8
- MathSciNet review: 2473749