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Extensions of Positive Definite Functions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2160))

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Abstract

Positive-definiteness arises naturally in the theory of the Fourier transform. There are two directions in transform theory. In the present setting, one is straightforward, and the other (Bochner) is deep. First, it is easy to see directly that the Fourier transform of a positive finite measure is a positive definite function; and that it is continuous. The converse result is Bochner’s theorem. It states that any continuous positive definite function on the real line is the Fourier transform of a unique positive and finite measure. However, if some given positive definite function is only partially defined, for example in an interval, or in the planar case, in a disk or a square, then Bochner’s theorem does not apply. One is faced with first seeking a positive definite extension; hence the theme of our monograph.

Mathematics is an experimental science, and definitions do not come first, but later on.

—Oliver Heaviside

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Notes

  1. 1.

    M.H. Stone, Ann. Math. 33 (1932) 643–648M.A. Naimark, Izv. Akad. Nauk SSSR. Ser. Mat. 7 (1943) 237–244 W. Ambrose, Duke Math. J. 11 (1944) 589–595 R. Godement, C.R. Acad. Sci. Paris 218 (1944) 901–903.

  2. 2.

    We refer to Sect. 2.1 for details. A Hermitian operator, also called formally selfadjoint, may well be non-selfadjoint.

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Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, Iowa, USA

    Palle Jorgensen

  2. Department of Mathematics, Wright State University, Dayton, Ohio, USA

    Steen Pedersen

  3. Mathematics Department, Hampton University, Hampton, Virginia, USA

    Feng Tian

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Jorgensen, P., Pedersen, S., Tian, F. (2016). Introduction. In: Extensions of Positive Definite Functions. Lecture Notes in Mathematics, vol 2160. Springer, Cham. https://doi.org/10.1007/978-3-319-39780-1_1

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