Abstract
This paper presents an elementary proof of the following theorem: Given {r j } m j =1 with m=d+1, fix\(fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } \) and let Q=[−R, R]d. Then any f∈ L2(Q) is completely determined by its averages over cubes of side rj that are completely contained in Q and have edges parallel to the coordinate axes if and only if rj/rk is irrational for j≠k. Whend=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.
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References
Berenstein, C. andTaylor, B. A., The “three squares” theorem for continuous functions,Arch. Rational Mech. Anal. 63 (1977), 253–259.
Berenstein, C., Gay, R. andYger, A., The three squares theorem, a local version, inAnalysis and Partial Differential Equations (Sadosky, C. ed.), pp. 35–50, Marcel Dekker, New York, 1990.
Berenstein, C. andYger, A., Analytic Bezout identities,Adv. in Appl. Math. 10 (1989), 51–74.
Beurling, A., Mittag-Leffler lectures on harmonic analysis, inCollected Works of Arne Beurling, vol.2, pp. 341–365, Birkhäuser, Boston, Mass., 1989.
Beurling, A. andMalliavin, P., On the closure of characters and the zeros of entire functions,Acta. Math. 118 (1967), 79–93.
Casey, S. andWalnut, D., Systems of convolution equations, deconvolution, Shannon sampling and the wavelet and Gabor transforms,SIAM Rev. 36 (1994), 537–577.
Feichtinger, H. andStrohmer, T., A Kaczmarz-based approach to nonperiodic sampling on unions of rectangular lattices, inSamp TA — Sampling Theory and Applications (Bilinskis. I. and Marvasti, F., eds.), pp. 32–37, Institute of Electronics-and Computer Science, Riga, 1995.
Hörmander, L., Generators for some rings of analytic functions,Bull. Amer. Math. Soc. 73 (1967), 943–949.
Jaffard, S., A density criterion for frames of complex exponentials,Michigan Math. J. 38 (1991), 339–348.
Kuipers, L. andNiederreiter, H.,Uniform Distributions of Sequences, Wiley, New York, 1974.
Landau, H., Necessary density conditions for sampling and interpolation of certain entire functions,Acta Math. 117 (1967), 37–52.
Petersen, E. andMeisters, G., Non-Liouville numbers and a theorem of Hörmander,J. Funct. Anal. 29 (1978), 142–150.
Seip, K., On the connection between exponential bases and certain related sequences inL 2 (−π, π),J. Funct. Anal. 130 (1995), 131–160.
Seip, K., On Gröchenig, Heil, and Walnut's proof of the local three squares theorem,Ark. Mat. 38 (2000), 93–96.
Walnut, D., Sampling and the multisensor deconvolution problem, inFourier Analysis (Bray, W. O., Milojević, P. S. and Stanojević, Č. V., eds.), Marcel Dekker, New York, 1994.
Walnut, D., Nonperiodic sampling of bandlimited functions on unions of rectangular lattices,J. Fourier Anal. Appl. 2 (1996), 435–452.
Walnut, D., Solutions to deconvolution equations using nonperiodic sampling,J. Fourier Anal. Appl. 4 (1998) 669–709.
Young, R., On complete biorthogonal systems,Proc. Amer. Math. Soc. 83 (1981), 537–540.
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The third author was supported by NSF Grant DMS9500909 and gratefully acknowledges the support of Bill Moran and the Mathematics Department of Flinders University of South Australia where certain portions of this work were completed. The authors thank the referee for valuable comments and suggestions.
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Gröchenig, K., Heil, C. & Walnut, D. Nonperiodic sampling and the local three squares theorem. Ark. Mat. 38, 77–92 (2000). https://doi.org/10.1007/BF02384491
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DOI: https://doi.org/10.1007/BF02384491