Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Integral representation of functions and distributions positive definite relative to the orthogonal group
HTML articles powered by AMS MathViewer

by A. E. Nussbaum PDF
Trans. Amer. Math. Soc. 175 (1973), 355-387 Request permission

Abstract:

A continuous function f on an open ball B in ${R^N}$ is called positive definite relative to the orthogonal group $O(N)$ if f is radial and $\smallint \smallint f(x - y)\phi (x)\overline {\phi (y)} \;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. It is shown that f is positive definite in B relative to $O(N)$ if and only if f has an integral representation $f(x) = \smallint {e^{ix \cdot t}}d{\mu _1}(t) + \smallint {e^{x \cdot t}}d{\mu _2}(t)$, where ${\mu _1}$ and ${\mu _2}$ are bounded, positive, rotation invariant Radon measures on ${R^N}$ and ${\mu _2}$ may be taken to be zero if, in addition to f being positive definite relative to $O(N),\smallint \smallint f(x - y)( - \Delta \phi )(x)\phi (y)\;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. Both conditions are satisfied if f is a radial positive definite function in B. Thus the theorem yields as a special case Rudin’s theorem on the extension of radial positive definite functions. The result is extended further to distributions.
References
  • Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
  • Salomon Bochner, Lectures on Fourier integrals. With an author’s supplement on monotonic functions, Stieltjes integrals, and harmonic analysis, Annals of Mathematics Studies, No. 42, Princeton University Press, Princeton, N.J., 1959. Translated by Morris Tenenbaum and Harry Pollard. MR 0107124
  • A. P. Calderón and R. Pepinsky, On the phase of Fourier coefficients for positive real periodic functions, Computing Methods and the Phase Problem in X-ray Crystal Analysis, Department of Physics, Pennsylvania State College, State College, Pa., 1952, pp. 339-348.
  • Allen Devinatz, On the extensions of positive definite functions, Acta Math. 102 (1959), 109–134. MR 109992, DOI 10.1007/BF02559570
  • G. I. Èskin, A sufficient condition for the solvability of a multi-dimensional problem of moments, Soviet Math. Dokl. 1 (1960), 895–898. MR 0121660
  • Lars Gårding, Applications of the theory of direct integrals of Hilbert spaces to some integral and differential operators, The Institute for Fluid Dynamics and Applied Mathematics, Lecture series, no. 11, University of Maryland, College Park, Md., 1954. MR 0071627
  • I. M. Gel′fand and A. G. Kostyučenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 349–352 (Russian). MR 0073136
  • I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
  • Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
  • L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221.
  • M. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940), 17–22 (French). MR 0004333
  • M. Krein, On a general method of decomposing Hermite-positive nuclei into elementary products, C. R. (Doklady) Acad. Sci. URSS (N.S.) 53 (1946), 3–6. MR 0018342
  • Béla Sz.-Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 39, Springer-Verlag, Berlin-New York, 1967 (German). Berichtigter Nachdruck. MR 0213890
  • John von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401–485. MR 29101, DOI 10.2307/1969463
  • A. E. Nussbaum, Radial exponentially convex functions, J. Analyse Math. 25 (1972), 277–288. MR 302835, DOI 10.1007/BF02790041
  • Albrecht Pietsch, Nukleare lokalkonvexe Räume, Schriftenreihe Inst. Math. Deutsch. Akad. Wiss. Berlin, Reihe A, Reine Mathematik, Heft 1, Akademie-Verlag, Berlin, 1965 (German). MR 0181888
  • Walter Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963), 532–539. MR 151796
  • Walter Rudin, An extension theorem for positive-definite functions, Duke Math. J. 37 (1970), 49–53. MR 254514
  • I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. of Math. (2) 39 (1938), no. 4, 811–841. MR 1503439, DOI 10.2307/1968466
  • François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A35
  • Retrieve articles in all journals with MSC: 43A35
Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 175 (1973), 355-387
  • MSC: Primary 43A35
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0333600-6
  • MathSciNet review: 0333600