Abstract
This is a survey article on uniqueness, sampling and interpolation problems in complex analysis. Most of these problems are motivated by applications of great practical importance in signal analysis and data transmission, but they also admit other mathematical formulations relating them to fundamental questions about existence of good bases in function spaces. This circle of ideas in complex analysis has experienced in recent years a notorious revitalization, mostly because of its connections with analogous problems in time-frequency and wavelet analysis, some of which will be discussed as well.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Berndtsson, Zeros of analytic functions in several variables, Ark. Mat. 16 (1978), 251–262.
B. Berndtsson and J. Ortega-Cerdà, On interpolation and sampling in Hilbert spaces of analytic functions, J. Reine Angew Math. 464 (1995), 109–128.
B. Berndtsson, An inequality for Fourier-Laplace transforms of entire functions, and the existence of exponential frames in Fock space, J. Funct. Anal. 149 (1997), 83–101.
D. Bernier and K. F. Taylor, Wavelets from square-integrable representations, Siam J. Math. Anal. 27(2) (1996), 594–608.
L. Carleson, P. Malliavin, J. Neuberger and J. Wermer, editors, “The Collected Works of Arne Beurling”, vol. 2, Birkhäuser, 1989, 341–365.
L. de Branges, “Hilbert spaces of entire functions”, Prentice Hall Inc. Englewood Cliffs, N.J., 1968.
I. Daubechies, “Ten Lectures on Wavelets”, SIAM, Philadelhia, Pennsylvania, 1992.
R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.
B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101–121.
H. Führ, Discrete and semidiscrete wavelet transforms in higher dimensions, preprint 1999.
S. V. Khrushchev, N. K. Nikol’skii and B. S. Pavlov, Unconditional bases of exponentials and reproducing kernels, in: “Complex Analysis and spectral theory”, Lecture Notes in Math. vol. 864, Springer-Verlag, Berlin-Heidelberg, 1981, 214–335.
P. Koosis, Leçons sur le théorème de Beurling et Malliavin, Les Publications CRM, Montreal, 1996.
I. Laba, Fuglede’s conjecture for a union of two intervals, preprint Princeton Univ.
H. J. Landau, A sparse regular sequence of exponentials closed on large sets, Bulletin A.M.S. 70 (1964), 566–569.
H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37–52.
N. Lindholm, Sampling in weighted L P spaces of entire functions in ℂn and estimates of the Bergman kernel,preprint Göteborg University.
Y. Lyubarskii and K. Seip, Sampling and interpolating sequences for multibandlimited functions and exponential bases on disconnected sets, J. ofFourier Analysis and Applications 3 (5) (1997), 599–615.
Y. Lyubarskii and K. Seip, Weighted Paley-Wiener spaces, preprint 1999.
Y. Lyubarskii and A. Rashdovskii, Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons, to appear in Arkiv f. Math.
P. McMullen, Convex bodies which tile space by translation, Mathematika 27 (1980), 113–121.
A. M. Minkin, Reflection of exponents, and unconditional bases of exponentials, St. Petersburg Math. J. 3 (1992), 1043–1068.
N. K. Nikol’skii, Bases of exponentials and the values of reproducing kernels, Dokl. Akad. Nauk SSSR 252 (1980), 1316–1320; english transi. in Sov. Math. Dokl. 21 (1980).
J. Ortega-Cerdà and K. Seip, Beurling-type density theorems for weighted L P spaces of entire functions, J. Analyse Mathematique 75 (1998), 247–266.
J. Ortega-Cerdà and K. Seip, Multipliers for entire functions and an interpolation problem of Beurling, J. Funct. Anal. 162 (1999), 400–415.
J. Ortega-Cerdà and K. Seip, On Fourier Frames, preprint 2000.
B. S. Pavlov, Basicity of an exponential system and Muckenhoupt’s condition, Dokl. Akad. Nauk SSSR 247 (1979), 37–40; english transi. in Sov. Math. Dokl. 20 (1979).
J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Applied and Computational Harmonic Analysis 2 (1995), 148–153.
L. I. Ronkin, On discrete uniqueness sets for entire functions of exponential type in several variables, Sib. Mat. Zh. 19(1) (1971), 142–152; english transi. in Sib. Math. J. 19 (1978), 101–108.
K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew Math. 429 (1992), 91–106.
K. Seip and R. Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew Math. 429 (1992), 107–113.
K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), 21–39.
K. Seip, On the connection between exponential bases and certain related sequences in L 2 (-π,π) J. Funct. Anal130 (1995), 131–160.
A. Ulanovskii, Sparse systems of functions closed on large sets in ℝn, to appear in Proc. London Math. Soc.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Bruna, J. (2001). Sampling in Complex and Harmonic Analysis. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8268-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8268-2_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9497-5
Online ISBN: 978-3-0348-8268-2
eBook Packages: Springer Book Archive