Weak parallelogram laws on banach spaces and applications to prediction
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Weak parallelogram laws on banach spaces and applications to prediction R. Cheng · W. T. Ross
© Akadémiai Kiadó, Budapest, Hungary 2015
Abstract 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. Keywords Parallelogram law · Pythagorean theorem · Uniform convexity · Best predictor · Baxter’s inequality · Purely nondeterministic Mathematics Subject Classification
46B20 · 46B25 · 60G25
1 Introduction The familiar parallelogram law states that for any vectors x and y in a Hilbert space H , we have x + y2 + x − y2 = 2x2 + 2y2
(1.1)
If this condition is imposed on a normed space, then in fact the polarization identity x, y =
x + y2 − x − y2 i x − y2 − i x + y2 +i 4 4
R. Cheng (B) Department of Mathematics and Statistics, Old Dominion University, Engineering and Computational Sciences Building, Norfolk, VA 23529, USA e-mail: [email protected] W. T. Ross Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA e-mail: [email protected]
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R. Cheng, W. T. Ross
(assuming complex scalars) determines an inner product, with x2 = x, x for all vectors x. Thus, the parallelogram law would have to be altered or weakened in some way in order to apply to a more general normed space. One example of this is furnished by Clarkson’s inequalities, which constitute parallelogram laws of sorts for the spaces L p = L p (, , μ), where 1 < p < ∞ and (, , μ) is any measure space (see, for instance, [7, p. 119]). If 1 < p ≤ 2, then p p p p x + y p + x − y p ≥ 2 p−1 x p + y p (1.2) for all x and y in L p ; and if 2 ≤ p < ∞, then
p p p p x + y p + x − y p ≤ 2 p−1 x p + y p
(1.3)
for all x and y in L p . Another example comes from Bynum and Drew [5] and Bynum [4]. They discovered what they call weak parallelogram laws for L p : If 1 < p ≤ 2, then (1.4) x + y2p + ( p − 1)x − y2p ≤ 2 x2p + y2p for all x and y in L p ; and if 2 ≤ p < ∞, then
x + y2p + ( p − 1)x − y2p ≥ 2 x2p + y2p
(1.5)
for all x and y in L p . That is, they are able to impose a version of condition (1.1) on the space L p , at the cost of introducing a constant factor ( p − 1), and weakening the equation to an inequality. In both examples, the resulting inequalities tell us something about the geometry of the space, such as smoothness and convexity properties. Guided by these two examples, and in the interest of pursuing a parallelogram law for general normed linear spaces, let us adopt the following terminology. Definition 1.1 Let C > 0, and 1 < p < ∞. A Banach Space X satisfies a p-lower weak parallelogram law with c
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