Weighted Paley-Wiener spaces

Lyubarskii, Yurii¹ ; Seip, Kristian¹

¹ Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway

Journal of the American Mathematical Society, Tome 15 (2002) no. 4, pp. 979-1006

Résumé

We study problems of sampling and interpolation in a wide class of weighted spaces of entire functions. These weights are characterized by the property that their natural regularization as the envelop of the unit ball of the corresponding space is equivalent to the original weight. We give an independent description of such weights and also show that, in a sense, this is the widest class of weights and associated spaces for which results on sets of uniqueness, sampling, and interpolation related to the classical Paley-Wiener spaces can be extended in a direct and natural way, keeping the basic features of the theory intact. One of the basic tools for our study is the De Brange theory of spaces of entire functions.

Détail

DOI : 10.1090/S0894-0347-02-00397-1

Bibliographie

[1] Arf, Cahit Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper J. Reine Angew. Math. 1939 1 44

[2] Beurling, Arne The collected works of Arne Beurling. Vol. 1 1989

[3] Beurling, A., Malliavin, P. On Fourier transforms of measures with compact support Acta Math. 1962 291 309

[4] De Branges, Louis Hilbert spaces of entire functions 1968

[5] Flornes, Kristin M. Sampling and interpolation in the Paley-Wiener spaces 𝐿_{𝜋}^{𝑝},0<𝑝≤1 Publ. Mat. 1998 103 118

[6] Hunt, Richard, Muckenhoupt, Benjamin, Wheeden, Richard Weighted norm inequalities for the conjugate function and Hilbert transform Trans. Amer. Math. Soc. 1973 227 251

[7] Hruå¡ÄÃ«V, S. V., Nikol′Skiä­, N. K., Pavlov, B. S. Unconditional bases of exponentials and of reproducing kernels 1981 214 335

[8] Krasichkov-Ternovskiä­, I. F. Interpretation of the Beurling-Malliavin theorem on the radius of completeness Mat. Sb. 1989 397 423

[9] Levin, B. Ya. Lectures on entire functions 1996

[10] Lin, Peng, Rochberg, Richard Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights Pacific J. Math. 1996 127 146

[11] Lyubarskii, Yurii I., Seip, Kristian Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt’s (𝐴_{𝑝}) condition Rev. Mat. Iberoamericana 1997 361 376

[12] Mergelyan, S. N. Weighted approximations by polynomials 1958 59 106

[13] Ortega-Cerdã , Joaquim, Seip, Kristian Multipliers for entire functions and an interpolation problem of Beurling J. Funct. Anal. 1999 400 415

[14] Dunford, Nelson A mean ergodic theorem Duke Math. J. 1939 635 646

[15] Vol′Berg, A. L. Thin and thick families of rational fractions 1981 440 480