On the alternating projections theorem and bivariate stationary stochastic processes

Author:
Habib Salehi

Journal:
Trans. Amer. Math. Soc. **128** (1967), 121-134

MSC:
Primary 60.50; Secondary 47.00

DOI:
https://doi.org/10.1090/S0002-9947-1967-0214135-5

MathSciNet review:
0214135

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we shall first use the theorem of von Neumann on alternating projections to obtain an algorithm for finding the projection of an element *x* in a Hilbert space onto the subspace spanned by -valued orthogonally scattered measures and . We then specialize this algorithm to the case that and are the canonical measures of the components of a bivariate stationary stochastic process (SP), and thereby get an algorithm for finding the best linear predictor in the time domain.

**[1]**A. S. Besicovitch,*A general form of the covering principle and relative differentiation of additive functions*. II, Proc. Cambridge Philos. Soc.**42**(1946), 1-10. MR**0014414 (7:281e)****[2]**H. Cramér,*On the theory of stationary random processes*, Ann. of Math.**41**(1940), 215-230. MR**0000920 (1:150b)****[3]**P. R. Halmos,*Measure theory*, Van Nostrand, Princeton, N. J., 1950. MR**0033869 (11:504d)****[4]**L. H. Koopmans,*On the coefficient of coherence for weakly stationary stochastic processes*, Ann. Math. Statist.**35**(1964), 532-549. MR**0161404 (28:4611)****[5]**P. Masani and J. Robertson,*The time domain analysis of a continuous weakly stationary stochastic process*, Pacific J. Math.**11**(1962), 1361-1378. MR**0149562 (26:7047)****[6]**R. F. Matveev,*On multi-dimensional regular stationary processes*, Theor. Probability Appl.**6**(1961), 149-165.**[7]**J. von Neumann,*Functional operators*, Vol. II, Annals of Mathematics Studies No. 21, Princeton Univ. Press, Princeton, N. J., 1950. MR**0032011 (11:240f)****[8]**F. Riesz and B. Sz.-Nagy,*Functional analysis*, Ungar, New York, 1955. MR**0071727 (17:175i)****[9]**M. Rosenberg,*The square-integrability of matrix-valued functions with respect to a nonnegative hermitian measure*, Duke Math. J.**31**(1964), 291-298. MR**0163346 (29:649)****[10]**H. Salehi,*The Hilbert space of square-integrable matrix-valued functions with respect to a*-*finite nonnegative, hermitian measure and stochastic integrals*, Research memorandum, Michigan State Univ., East Lansing, 1966.**[11]**-,*The prediction theory of multivariate stochastic processes with continuous time*, Doctoral Thesis, Indiana Univ., Bloomington, 1965.**[12]**N. Wiener and P. Masani,*The prediction theory of multivariate stochastic processes*. I, Acta Math.**98**(1957), 111-150; II, Acta Math.**99**(1958), 93-137.

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DOI:
https://doi.org/10.1090/S0002-9947-1967-0214135-5

Article copyright:
© Copyright 1967
American Mathematical Society