Positive definite functions and generalizations, an historical survey

2 fi COS Xi + i X 6 S i n i = °i = l ' ' i = l ' Likewise it is easily verified directly that e is p.d. for real \ , but it is not so straightforward to see that such functions as e~H e~*, and (1 4x)" ? e p.d. These and other examples are discussed in § 3. Positive definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics since the beginning of this century. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, embedding problems, information theory, and other areas. Their history constitutes a good illustration of the words of Hobson [51, p. 290] : "Not only are special results, obtained independently of one another, frequently seen to be really included in

[1]  C. Carathéodory.  Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen , 1907 .

[2]  J. Mercer.  Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations , 1909 .

[3]  J. Schur.  Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. , 1911 .

[4]  O. Toeplitz,et al.  Über die fourier’sche entwickelung positiver funktionen , 1911 .

[5]  F. Riesz,et al.  Sur certains systèmes singuliers d'équations intégrales , 1911 .

[6]  E. Fischer,et al.  Über das carathéodory’sche problem, potenzreihen mit positivem reellen teil betreffend , 1911 .

[7]  M. Mathias.  Über positive Fourier-Integrale , 1923 .

[8]  M. Stone.  On One-Parameter Unitary Groups in Hilbert Space , 1932 .

[9]  A. Khintchine.  Korrelationstheorie der stationären stochastischen Prozesse , 1934 .

[10]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[11]  S. Bochner.  Hilbert Distances and Positive Definite Functions , 1941 .

[12]  I. J. Schoenberg,et al.  Fourier integrals and metric geometry , 1941 .

[13]  W. Ambrose.  Spectral resolution of groups of unitary operators , 1944 .

[14]  H. Cartan,et al.  Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts , 1947 .

[15]  Reduction of Bochner's Theorem to Stone's Theorem , 1948 .

[16]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[17]  A remark on theorems of Stone and Bochner , 1951 .

[18]  S. Bochner,et al.  Vorlesungen über Fouriersche Integrale , 1952 .

[19]  L. H. Loomis.  An Introduction to Abstract Harmonic Analysis , 1953 .

[20]  M. M. Crum.  On Positive‐Definite Functions , 1956 .

[21]  A. Devinatz.  On the extensions of positive definite functions , 1959 .

[22]  On Extrapolating a Positive Definite Function from a Finite Interval. , 1959 .

[23]  Some Aspects of Analysis and Probability , 1959 .

[24]  A. Devestatz.  On Measurable Positive Definite Operator Functions , 1960 .

[25]  Positive Definite Functions of a Real Variable , 1960 .

[26]  Quelques problèmes non résolus de la théorie des fonctions caractéristiques , 1961 .

[27]  M. G. Kreĭn,et al.  Some questions in the theory of moments , 1962 .

[28]  W. Rudin.  The extension problem for positive-definite functions , 1963 .

[29]  R. Bucy,et al.  Extreme positive definite functions and Choquet's Representation Theorem , 1965 .

[30]  I. Glicksberg,et al.  The decomposition of certain group representations , 1965 .

[31]  Naĭ mark's moment theorem. , 1966 .

[32]  A Probabilistic Proof of Bochner's Theorem on Positive Definite Functions , 1967 .

[33]  K. Urbanik.  Positive definite sequences , 1967 .

[34]  Positiv definite Funktionen auf einem unendlich dimensionalen Vektorraum , 1968 .

[35]  Functions Positive Definite in the Space C , 1968 .

[36]  Integrally positive-definite functions on groups , 1968 .

[37]  On tempered distributions and Bochner-Schwartz theorem on arbitrary locally compact Abelian groups , 1968 .

[38]  A note on positive definite functions , 1970 .

[39]  Algebraic models for probability measures associated with stochastic processes , 1971 .

[40]  Multipliers, positive functionals, positive-definite functions, and Fourier-Stieltjes transforms , 1971 .

[41]  Counterexample to a Conjecture on Positive Definite Functions , 1972, Canadian Journal of Mathematics.

[42]  Bochner's theorem in infinite dimensions. , 1972 .

[43]  Integral representation of functions and distributions positive definite relative to the orthogonal group , 1973 .

[44]  J. G. D. Lamadrid,et al.  Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups , 1974 .