Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Convolution roots of radial positive definite functions with compact support


Authors: Werner Ehm, Tilmann Gneiting and Donald Richards
Journal: Trans. Amer. Math. Soc. 356 (2004), 4655-4685
MSC (2000): Primary 42A38, 42A82, 60E10
DOI: https://doi.org/10.1090/S0002-9947-04-03502-0
Published electronically: May 10, 2004
MathSciNet review: 2067138
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A classical theorem of Boas, Kac, and Krein states that a characteristic function $\varphi$ with $\varphi(x) = 0$ for $\vert x\vert \geq \tau$ admits a representation of the form

\begin{displaymath}\varphi(x) = \int u(y) \hspace{0.2mm} \overline{u(y+x)} \, \mathrm{d}y, \qquad x \in \mathbb{R}, \end{displaymath}

where the convolution root $u \in L^2(\mathbb{R})$ is complex-valued with $u(x) = 0$ for $\vert x\vert \geq \tau/2$. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If $\varphi$ is real-valued and even, can the convolution root $u$ be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of $\varphi$ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on $\mathbb{R}^d$ is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if $f$ is a probability density on $\mathbb{R}^d$ whose characteristic function $\varphi$ vanishes outside the unit ball, then

\begin{displaymath}\int \vert x\vert^2 f(x) \, \mathrm{d}x = - \Delta \varphi(0) \geq 4 \, j_{(d-2)/2}^2 \end{displaymath}

where $j_\nu$ denotes the first positive zero of the Bessel function $J_\nu$, and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in $\mathbb{R}^2$ does not exist.


References [Enhancements On Off] (What's this?)

  • 1. N. I. Achieser, Theory of approximation, Frederick Ungar, New York, 1956. MR 20:1872
  • 2. E. J. Akutowicz, On the determination of the phase of a Fourier integral, II, Proc. Amer. Math. Soc. 8 (1957), 234-238. MR 18:895c
  • 3. O. D. Anderson, Bounding sums for the autocorrelations of moving average processes, Biometrika 62 (1975), 706-707. MR 52:12265
  • 4. V. V. Arestov and E. E. Berdysheva, Turán's problem for positive definite functions with support in a hexagon, Proc. Steklov Math. Inst., Suppl., 2001, pp. S20-S29.
  • 5. -, The Turán problem for a class of polytopes, East J. Approx. 8 (2002), 381-388. MR 2003i:42010
  • 6. V. V. Arestov, E. E. Berdysheva, and H. Berens, On pointwise Turán's problem for positive definite functions, East J. Approx. 9 (2003), 31-42. MR 2004b:42014
  • 7. R. Barakat and G. Newsam, Upper and lower bounds on radially symmetric optical transfer functions, Optica Acta 29 (1982), 1191-1204.
  • 8. R. P. Boas, Entire functions, Academic Press, New York, 1954. MR 16:914f
  • 9. R. P. Boas and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189-206, Errata 15 (1948), 107-109. MR 6:265h
  • 10. S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Math. Ann. 108 (1933), 378-410.
  • 11. H. Bohman, Approximate Fourier analysis of distribution functions, Ark. Mat. 4 (1960), 99-157. MR 23:A3963
  • 12. H. Carnal and M. Dozzi, On a decomposition problem for multivariate probability measures, J. Multivariate Anal. 31 (1989), 165-177. MR 91c:60016
  • 13. K. C. Chanda, On bounds of serial correlations, Ann. Math. Statist. 33 (1962), 1457-1460. MR 26:872
  • 14. J.-P. Chilès and P. Delfiner, Geostatistics, Wiley, New York, 1999. MR 2000f:86010
  • 15. H. Cohn, New upper bounds of sphere packings. II, Geom. Topol. 6 (2002), 329-353. MR 2004b:52032
  • 16. H. Cohn and N. Elkies, New upper bounds of sphere packings I, Ann. of Math. 157 (2003), 689-714. MR 2004b:11096
  • 17. R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience Publishers, New York, 1953. MR 16:426a
  • 18. N. Cressie and M. Pavlicová, Calibrated spatial moving average simulations, Stat. Model. 2 (2002), 1-13.
  • 19. N. Davies, M. B. Pate, and M. G. Frost, Maximum autocorrelations for moving average processes, Biometrika 61 (1974), 199-201. MR 51:11889
  • 20. H. Dym and H. P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Academic Press, New York, 1976. MR 56:6829
  • 21. M. L. Eaton, On the projections of isotropic distributions, Ann. Statist. 9 (1981), 391-400. MR 82c:60026
  • 22. W. Feller, An introduction to probability theory and its applications, vol. II, second ed., Wiley, New York, 1971. MR 42:5292
  • 23. B. R. Frieden, Maximum attainable MTF for rotationally symmetric lenses, J. Opt. Soc. Am. 59 (1969), 402-406.
  • 24. A. Garsia, E. Rodemich, and H. Rumsey, On some extremal positive definite functions, J. Math. Mech. 18 (1969), 805-834. MR 40:4682
  • 25. O. Glatter, The interpretation of real-space information from small-angle scattering experiments, J. Appl. Cryst. 12 (1979), 166-175.
  • 26. -, Convolution square-root of band-limited symmetrical functions and its application to small-angle scattering data, J. Appl. Cryst. 14 (1981), 101-108.
  • 27. T. Gneiting, On $\alpha$-symmetric multivariate characteristic functions, J. Multivariate Anal. 64 (1998), 131-147. MR 99h:60025
  • 28. -, Radial positive definite functions generated by Euclid's hat, J. Multivariate Anal. 69 (1999), 88-119. MR 2000g:60022
  • 29. -, Criteria of Pólya type for radial positive definite functions, Proc. Amer. Math. Soc. 129 (2001), 2309-2318. MR 2002b:42018
  • 30. -, Compactly supported correlation functions, J. Multivariate Anal. 83 (2002), 493-508. MR 2003i:60057
  • 31. T. Gneiting, K. Konis, and D. Richards, Experimental approaches to Kuttner's problem, Experiment. Math. 10 (2001), 117-124.
  • 32. B. I. Golubov, On Abel-Poisson type and Riesz means, Anal. Math. 7 (1981), 161-184. MR 83b:42015
  • 33. D. V. Gorbachev, Extremum problem for periodic functions supported in a ball, Math. Notes 69 (2001), 313-319. MR 2002e:42006
  • 34. E. A. Gorin, Extremal rays in cones of entire functions, Abstracts, XIII Soviet Workshop on Operator Theory in Function Spaces, Siberian Academy of Sciences, 1988, pp. 57-58.
  • 35. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, fifth ed., Academic Press, Boston, 1994. MR 94g:00008
  • 36. E. M. Hofstetter, Construction of time-limited functions with specified autocorrelation functions, IEEE Trans. Info. The. IT-10 (1964), 119-126.
  • 37. M. E. H. Ismail and M. E. Muldoon, On the variation with respect to a parameter of zeros of Bessel and q-Bessel functions, J. Math. Anal. Appl. 135 (1988), 187-207. MR 89i:33011
  • 38. A. J. E. M. Janssen, Frequency-domain bounds for non-negative band-limited functions, Philips J. Res. 45 (1990), 325-366.
  • 39. -, Bounds for optical transfer functions: analytical results, Philips J. Res. 45 (1991), 367-411.
  • 40. M. Kanter, Unimodal spectral windows, Statist. Prob. Lett. 34 (1997), 403-411. MR 98e:60025
  • 41. M. N. Kolountzakis and S. G. Révész, On a problem of Turán about positive definite functions, Proc. Amer. Math. Soc. 131 (2003), 3423-3430.
  • 42. -, On pointwise estimates of positive definite functions with given support, preprint (2003); available from http://fourier.math.uoc.gr/mk/publ/.
  • 43. M. Krein, Sur le probléme du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N. S.) 26 (1940), 17-22. MR 2:361h
  • 44. B. Kuttner, On the Riesz means of a Fourier series (II), J. London Math. Soc. 19 (1944), 77-84. MR 7:59d
  • 45. W. Lawton, Uniqueness results for the phase-retrieval problem for radial functions, J. Opt. Soc. Am. 71 (1981), 1519-1522. MR 82m:94010
  • 46. W. Lukosz, Properties of linear low-pass filters for nonnegative signals, J. Opt. Soc. Am. 52 (1962), 827-829.
  • 47. -, Übertragung nichtnegativer Signale durch lineare Filter, Optica Acta 9 (1962), 335-364.
  • 48. A. Mantoglou and J. L. Wilson, The turning bands method for simulation of random fields using line generation by a spectral method, Water Resour. Res. 18 (1982), 1379-1394.
  • 49. R. P. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Am. A7 (1990), 394-411.
  • 50. F. Natterer, The mathematics of computerized tomography, Teubner, Stuttgart, 1986. MR 88m:44008
  • 51. D. S. Oliver, Moving averages for Gaussian simulation in two and three dimensions, Math. Geol. 27 (1995), 939-960. MR 96k:86013
  • 52. R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, American Mathematical Society, New York, 1934. MR 98a:01023
  • 53. A. Papoulis, Apodization of optimum imaging of smooth objects, J. Opt. Soc. Am. 62 (1972), 1423-1429.
  • 54. -, Minimum-bias windows for high-resolution spectral estimates, IEEE Trans. Info. The. IT-19 (1973), 9-12.
  • 55. M. Plancherel and G. Pólya, Fonctions entières et intégrales de Fourier multiples, Comment. Math. Helv. 9 (1937), 224-248.
  • 56. J. Rosenblatt, Phase retrieval, Commun. Math. Phys. 95 (1984), 317-343. MR 86k:82075
  • 57. W. Rudin, An extension theorem for positive-definite functions, Duke Math. J. 95 (1970), 49-53. MR 40:7722
  • 58. I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. of Math. 39 (1938), 811-841.
  • 59. H. Stark and B. Dimitriadis, Minimum-bias spectral estimation with a coherent optical spectrum analyzer, J. Opt. Soc. Am. 65 (1975), 425-431, Errata 65 (1975), 973.
  • 60. A. Walther, The question of phase retrieval in optics, Optica Acta 10 (1963), 41-49. MR 29:890
  • 61. C. S. Williams and O. A. Becklund, Introduction to the optical transfer function, Wiley, New York, 1989.
  • 62. V. P. Zastavnyi, On positive definiteness of some functions, J. Multivariate Anal. 73 (2000), 53-81. MR 2002b:42017
  • 63. -, Positive definite radial functions and splines, Dokl. Math. 66 (2002), 446-449.
  • 64. V. P. Zastavnyi and R. M. Trigub, Positive definite splines of special form, Mat. Sb. 193 (2002), 41-68.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42A38, 42A82, 60E10

Retrieve articles in all journals with MSC (2000): 42A38, 42A82, 60E10


Additional Information

Werner Ehm
Affiliation: Institut für Grenzgebiete der Psychologie und Psychohygiene, Wilhelmstrasse 3a, 79098 Freiburg, Germany
Email: ehm@igpp.de

Tilmann Gneiting
Affiliation: Department of Statistics, University of Washington, Box 354322, Seattle, Washington 98195-4322
Email: tilmann@stat.washington.edu

Donald Richards
Affiliation: Department of Statistics, Pennsylvania State University, 326 Thomas Building, University Park, Pennsylvania 16802-2111
Email: richards@stat.psu.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03502-0
Received by editor(s): April 10, 2003
Received by editor(s) in revised form: September 2, 2003
Published electronically: May 10, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society