Almost Periodic Functions and Weakly Stationary Stochastic Processes

  • Toru Maruyama
Part of the Monographs in Mathematical Economics book series (MOME, volume 2)


It is a basic idea for the classical theory of Fourier series to express periodic functions as compositions of harmonic waves. This idea can be successfully extended to nonperiodic functions by means of Fourier transforms. However, we will be confronted with a lot of obstacles when we consider \(\mathfrak {L}^p\)-function spaces in the case p > 2.


  1. 1.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)zbMATHGoogle Scholar
  2. 2.
    Bochner, S.: Beiträge zur Theorie der fastperiodischen Funktionen, I, II. Math. Ann. 96, 119–147, 383–409 (1927) Google Scholar
  3. 3.
    Bochner, S., von Neumann, J.: Almost periodic functions in a group, II. Trans. Amer. Math. Soc. 37, 21–50 (1935) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bohr, H.: Zur Theorie der fastperiodischen Funktionen. I–III. Acta Math. 45, 29–127 (1925); 46, 101–214 (1925); 47, 237–281 (1926)Google Scholar
  5. 5.
    Bohr, H.: Fastperiodische Funktionen. Springer, Berlin (1932)CrossRefGoogle Scholar
  6. 6.
    Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brooks, Pacific Grove (1988) Google Scholar
  7. 7.
    Dunford, N., Schwartz, J.T.: Linear Operators, Part 1, Interscience, New York (1958) Google Scholar
  8. 8.
    Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge University Press, Cambridge (2004) Google Scholar
  9. 9.
    Kawata, T.: Ohyo Sugaku Gairon (Elements of Applied Mathematics). I, II. Iwanami Shoten, Tokyo (1950, 1952) (Originally published in Japanese)Google Scholar
  10. 10.
    Kawata, T.: On the Fourier series of a stationary stochastic process, I, II. Z. Wahrsch. Verw. Gebiete, 6, 224–245 (1966); 13, 25–38 (1969) Google Scholar
  11. 11.
    Kawata, T.: Teijo Kakuritsu Katei (Stationary Stochastic Processes). Kyoritsu Shuppan, Tokyo (1985) (Originally published in Japanese)Google Scholar
  12. 12.
    Loomis, L.: The spectral characterization of a class of almost periodic functions. Ann. Math. 72, 362–368 (1960) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Malliavin, P.: Integration and Probability. Springer, New York (1995) CrossRefGoogle Scholar
  14. 14.
    Maruyama, T.: Sekibun to Kansu-kaiseki (Integration and Functional Analysis). Springer, Tokyo (2006) (Originally published in Japanese) Google Scholar
  15. 15.
    Maruyama, T.: Fourier analysis of periodic weakly stationary processes. A note on Slutsky’s observation. Adv. Math. Econ. 20, 151–180 (2016)CrossRefGoogle Scholar
  16. 16.
    Rudin, W.: Weak almost periodic functions and Fourier–Stieltjes transforms. Duke Math. J. 26, 215–220 (1959)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rudin, W.: Fourier Analysis on Groups. Interscience, New York (1962) Google Scholar
  18. 18.
    Stromberg, K.R.: An Introduction to Classical Real Analysis. American Mathematical Society, Providence (1981) Google Scholar
  19. 19.
    von Neumann, J.: Almost periodic functions in a group, I. Trans. Amer. Math. Soc. 36, 445–492 (1934) Google Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Toru Maruyama
    • 1
  1. 1.Professor EmeritusKeio UniversityTokyoJapan

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