×

On multi-dimensional sampling and interpolation. (English) Zbl 1275.42005

The authors prove some interesting theorems and present a proficient survey of the antecendents with 31 terms in the bibliography.
Their abstract summarizes the aim and the results tersely.
“The paper discusses sharp sufficient conditions for interpolation and sampling for functions on \(n\) variables with convex spectrum. When \(n=1\), the classical theorems of A. E. Ingham and A. Beurling state that the critical values in the estimates from above (from below) for the distances between interpolation (sampling) nodes are the same. This is no longer true for \(n>1\). While the critical value for sampling sets remains constant, the one for interpolation grows linearly with the dimension.”

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
41A30 Approximation by other special function classes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baiocchi C., Komornik V., Loreti P.: Théorèmes du type Ingham et application á la thèorie du contrôle. C. R. Acad. Sci. Paris Sér. I Math. 326(4), 453–458 (1998) · Zbl 0924.42021 · doi:10.1016/S0764-4442(97)89791-1
[2] Benedetto, J.J., Wu, H.-Ch.: Non-uniform sampling and spiral MRI reconstruction. In: SPIE-Wavelet Applications in Signal and Image Processing VIII, vol. 4119, pp. 130–141 (2000)
[3] Beurling, A.: Interpolation for an interval in $${\(\backslash\)mathbb{R}\^1}$$ . In: The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhauser, Boston (1989)
[4] Beurling, A.: Balayage of Fourier–Stieltjes transforms. In: The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhauser, Boston (1989) · Zbl 0732.01042
[5] Beurling, A.: Local Harmonic Analysis with some Applications to Differential Operators. In: The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhauser, Boston (1989) · Zbl 0732.01042
[6] Bezuglaya L., Katsnelson V.: The sampling theorem for functions with limited multi-band spectrum. Z. Anal. Anwendungen 12(3), 511–534 (1993) · Zbl 0786.30019
[7] Cassels J.W.S.: An Introduction to the Geometry of Numbers. Springer, Berlin (1971) · Zbl 0209.34401
[8] Clunie J., Rahman Q.I., Walker W.J.: On entire functions of exponential type bounded on the real axis. J. Lond. Math. Soc. (2) 61(1), 163–176 (2000) · Zbl 0946.30018 · doi:10.1112/S0024610799008236
[9] Duffin R.J., Schaeffer A.C.: Some properties of functions of exponential type. Bull. Am. Math. Soc. 44, 236–240 (1938) · Zbl 0018.40901 · doi:10.1090/S0002-9904-1938-06725-0
[10] Higgins J.R.: Sampling Theory in Fourier and Signal Analysis. Foundations. Clarendon Press, Oxford (1996) · Zbl 0872.94010
[11] Hörmander L.: Some inequalities for functions of exponential type. Math. Scand. 3, 21–27 (1955) · Zbl 0065.30302
[12] Ingham A.E.: Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41, 367–379 (1936) · Zbl 0014.21503 · doi:10.1007/BF01180426
[13] Kahane J.-P.: Sur les fonctions moyenne-périodiques bornées. Ann. Inst. Fourier 7, 293–314 (1957) · Zbl 0083.34401 · doi:10.5802/aif.72
[14] Kahane, J.-P.: Fonctions pseudo-périodiques dans $${\(\backslash\)mathbb{R}\^{p}}$$ (French). In: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 274–281. Jerusalem Academic Press/Pergamon, Jerusalem/Oxford (1961)
[15] Kahane J.-P.: Pseudopériodicité et séries de Fourier lacunaires. Ann. Sci. Ecole Norm. Sup. 79, 93–150 (1962) · Zbl 0105.28601
[16] Komornik V., Loreti P.: Fourier series in control theory. Springer Monographs in Mathematics. Springer, Berlin (2005) · Zbl 1094.49002
[17] Landau H.J.: A sparse regular sequence of exponentials closed on large sets. Bull. Am. Math. Soc. 70, 566–569 (1964) · Zbl 0131.06401 · doi:10.1090/S0002-9904-1964-11202-7
[18] Landau H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967) · Zbl 0154.15301 · doi:10.1007/BF02395039
[19] Levin B.Ya.: Lectures on Entire Functions. AMS, Providence (1996) · Zbl 0856.30001
[20] Lyubarskii Yu., Rashkovskii A.: Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons. Ark. Mat. 38(1), 139–170 (2000) · Zbl 1038.42011 · doi:10.1007/BF02384495
[21] Matei B., Meyer Y.: Quasicrystals are sets of stable sampling. C. R. Math. Acad. Sci. Paris 346(23–24), 1235–1238 (2008) · Zbl 1154.42006 · doi:10.1016/j.crma.2008.10.006
[22] Matei B., Meyer Y.: A variant of compressed sensing. Rev. Mat. Iberoam 25(2), 669–692 (2009) · Zbl 1184.42010 · doi:10.4171/RMI/578
[23] Matei B., Meyer Y.: Simple quasicrystals are sets of stable sampling. Complex Var. Elliptic Equ. 55(8–10), 947–964 (2010) · Zbl 1207.94043 · doi:10.1080/17476930903394689
[24] Olevskii A., Ulanovskii A.: Universal sampling of band-limited signals. C. R. Math. Acad. Sci. Paris 342(12), 927–931 (2006) · Zbl 1096.94017 · doi:10.1016/j.crma.2006.04.015
[25] Olevskii A., Ulanovskii A.: Universal sampling and interpolation of band-limited signals. Geom. Funct. Anal. 18(3), 1029–1052 (2008) · Zbl 1169.42014 · doi:10.1007/s00039-008-0674-7
[26] Olevskii A., Ulanovskii A.: On Ingham-type interpolation in $${\(\backslash\)mathbb{R}\^n}$$ . C. R. Math. Acad. Sci. Paris 348(13–14), 807–810 (2010) · Zbl 1202.42024 · doi:10.1016/j.crma.2010.06.007
[27] Rogers C.A.: Packing and Covering. Cambridge University Press, Cambridge (2008)
[28] Seip, K.: Interpolation and sampling in spaces of analytic functions. In: University Lecture Series, vol. 33. American Mathematical Society, Providence (2004) · Zbl 1057.30036
[29] Stein E., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1987) · Zbl 0232.42007
[30] Ulanovskii A.: Sparse systems of functions closed on large sets in $${\(\backslash\)mathbf{R}\^N}$$ . J. Lond. Math. Soc. (2) 63(2), 428–440 (2001) · Zbl 1037.42034 · doi:10.1017/S0024610700001782
[31] Young R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (2001) · Zbl 0981.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.