Divided differences and restriction operator on Paley–Wiener spaces $${PW_{\tau}^{p}}$$ for N–Carleson sequences
Tóm tắt
For a sequence of complex numbers
$${\Lambda}$$
we consider the restriction operator
$${R_{\Lambda}}$$
defined on Paley–Wiener spaces
$${PW_{\tau}^{p}}$$
(1 < p < ∞). Lyubarskii and Seip gave necessary and sufficient conditions on
$${\Lambda}$$
for
$${R_{\Lambda}}$$
to be an isomorphism between
$${PW_{\tau}^{p}}$$
and a certain weighted l
p
space. The Carleson condition appears to be necessary. We extend their result to N–Carleson sequences (finite unions of N disjoint Carleson sequences). More precisely, we give necessary and sufficient conditions for
$${R_{\Lambda}}$$
to be an isomorphism between
$${PW_{\tau}^{p}}$$
and an appropriate sequence space involving divided differences.
Tài liệu tham khảo
Aleksandrov A.B.: A simple proof of a theorem of Volberg and Treil on the embedding of coinvariant subspaces of the shift operator. J. Math. Sci 85(2), 1773–1778 (1997)
Avdonin S.A., Ivanov S.A.: Exponential Riesz bases of subspaces and divided differences. St. Petersbourg. Math. J 93(3), 339–351 (2001)
Bruna J., Nicolau A., Oyma K.: A note on interpolation in the Hardy spaces of the unit disc. Proc. Am. Math. Soc 124(4), 1197–1204 (1996)
Carleson L.: An interpolation problem for bounded analytic functions. Am. J. Math 80, 921–930 (1958)
Garnett, J.B.: Bounded analytic functions (Revised first edition), Graduate Texts Math. 236 (2007). Springer-Verlag, Berlin. First edition in Pure and applied Mathematics, 86 (1981). Academic Press, New York
Gaunard, F.: Problèmes d’interpolation dans les espaces de Paley-Wiener et applications en théorie du contrôle, thèse de l’Université Bordeaux 1, (2011)
Hartmann, A.: Interpolation libre et caractérisation des traces des fonctions holomorphes sur les réunions finies de suites de Carleson. Thèse de l’Université Bordeaux 1 (1996)
Hartmann A.: Une approche de l’interpolation libre généralisée par la théorie des opérateurs et caractérisations des traces \({H^{p}|\Lambda}\) . J. Oper. Theory 35(2), 281–316 (1996)
Hunt R., Muckenhoupt B., Wheeden R.: Weighted norm inequalities for the conjugate Hilbert transform. Proc. Am. Math. Soc 176, 227–251 (1973)
Hruscev S.V., Nikolskii N.K., Pavlov B.S.: Unconditional bases of exponentials and of reproducing kernels in Complex analysis and spectral theory. Lectures Notes Math. 864, 214–335 (1981)
Levin, B.Y.: Lectures on entire functions. Math. Monographs 150. American Mathematical Society (1996)
Lyubarskii Y.L., Seip K.: Complete interpolating sequences for Paley–Wiener spaces and Muckenhoupt’s (A p ) condition. Rev. Mat. Iber 13(2), 361–376 (1997)
Minkin A.M.: The reflection of indices and unconditionnal bases of exponentials. St. Petersburg Math. J 3(5), 1043–1064 (1992)
Nikolskii, N.K.: A treatise on the shift operator. Grundlehren der mathematischen Wissenschaften, vol. 273. Springer, Berlin (1986)
Nikolskii, N.K.: Operators, functions and systems: an easy reading. volume 1.Mathematical Surveys and Monographs, vol. 92. American Mathematical Society, Prividence
Nikolskii, N.K.: Operators, functions and systems: an easy reading, volume 2. Mathematical Surveys and Monographs vol. 93. American Mathematical Society, Providence (2002)
Seip K.: Developments from nonharmonic Fourier series. Proc. Int. Congr. Math. Doc. Math. Extra II, 713–722 (1998)
Seip, K.: Interpolation and sampling in spaces of analytic functions. Univ. Lect. Series 33. American Mathemetical Society, Providence (2004)
Shapiro H.S., Shields A.L.: On some interpolation problems for analytic functions. Am. J. Math 83, 513–532 (1961)
Treil, S.R., Volberg, A.L.: Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator. Algebra i Analiz 7–6 (1995), 205–226; translation in St. Petersburg Math. J. 7–6 (1996), 1017–1032
Vasyunin V.I.: Traces of bounded analytic functions on finite unions of Carleson sets. J. Soviet Math 27(1), 2448–2450 (1984)