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Journal of the London Mathematical Society 1988 s2-38(1):133-145; doi:10.1112/jlms/s2-38.1.133
© 1988 by London Mathematical Society
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© Oxford University Press

Best Approximations in Lp (d µ) and Prediction Problems of Szegö, Kolmogorov, Yaglom, and Nakazi

A. G. Miamee and Mohsen Pourahmadi

Department of Mathematics, Hampton University Hampton, Virginia 23668, USA
Department of Mathematical Sciences, Northern Illinois University DeKalb, Illinois 60115, USA

For a non-negative weight function W we prove a general prediction theorem which for a subspace M sub Lp(W), 1 < p < {infty}, and h{varepsilon}L{infty} provides formulae expressing inff{varepsilon}M||h–f||p, w and PM (metric projection) in terms of inff{varepsilon}N||h–f||q, w and PNh, where N (related to M) is a subspace of Lq(W8), 1/p+1/q = 1 and s = – 1/(p– 1). It is shown that this general prediction theorem subsumes and generalizes to p != 2 the prediction theorems of Szegö (1918), Kolmogorov (1941), Yaglom (1963), and Nakazi (1984).


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