Functional Analysis by Elias M. Stein & Rami Shakarchi

Author:Elias M. Stein & Rami Shakarchi
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2011-04-26T16:00:00+00:00

belongs to Lp([0, 2π]) for every p < ∞.

(b) If then for almost every t [0, 1] the series (10) is not the Fourier series of an integrable function.

The proof is based on Khinchin’s inequality, which like Lemma 1.6 is a further exploitation of the independence of the Rademacher functions.

Suppose {an} are complex numbers with Let with F taken as the L2 limit on L2([0, 1]) of the partial sums.

Lemma 1.8 For each p < ∞ there is a bound Ap so that

for all F Lp([0, 1]) of the form

It clearly suffices to prove the corresponding statement when the an are assumed real and have been normalized so that

Now observe that the defining property (3) shows that whenever {fn} is a sequence of mutually independent (real-valued) functions, so is the sequence {Φn(fn)}, with {Φn} any sequence of continuous functions from to . As a result the functions are mutually independent. Thus if then

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