Random Spectral Measures

  • Péter Major
Part of the Lecture Notes in Mathematics book series (LNM, volume 849)


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.


Correlation Function Random Field Spectral Measure Gaussian Random Field Positive Definite Function 
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  1. 16.
    Gelfand, I.M., Vilenkin, N.Ya.: Generalized Functions. IV. Some Applications of Harmonic Analysis. Academic (Harcourt, Brace Jovanovich Publishers), New York (1964)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Péter Major
    • 1
  1. 1.Alfréd Rényi Mathematical Institute Hungarian Academy of SciencesBudapestHungary

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