Weak parallelogram laws on banach spaces and applications to prediction
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This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.
KeywordsParallelogram law Pythagorean theorem Uniform convexity Best predictor Baxter’s inequality Purely nondeterministic
Mathematics Subject Classification46B20 46B25 60G25
The authors are grateful to the Referee for his or her helpful comments and suggestions.
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