Abstract
We present the correlation function of a Gaussian stationary field as the Fourier transform of a spectral measure and construct with its help a (Gaussian) random spectral measure. Then we express a stationary Gaussian field itself as the Fourier transform of this random spectral measure. We also describe the most important properties of spectral and random spectral measures. The proofs heavily depend on a classical result of analysis about the representation of so-called positive definite functions as the Fourier transform of positive measures and on its version about generalized functions. Hence we finish this chapter with a sub-chapter where we discuss these results, called Bochner and Bochner–Schwartz theorems in the literature.
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References
Gelfand, I.M., Vilenkin, N.Ya.: Generalized Functions. IV. Some Applications of Harmonic Analysis. Academic (Harcourt, Brace Jovanovich Publishers), New York (1964)
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Major, P. (2014). Random Spectral Measures. In: Multiple Wiener-Itô Integrals. Lecture Notes in Mathematics, vol 849. Springer, Cham. https://doi.org/10.1007/978-3-319-02642-8_3
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DOI: https://doi.org/10.1007/978-3-319-02642-8_3
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02641-1
Online ISBN: 978-3-319-02642-8
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