The Hilbert space L²([0,2pi], R) and Fourier series.

Thread starter quasar987
Start date Jan 24, 2008
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Fourier Fourier series Hilbert Hilbert space Series Space

Jan 24, 2008

quasar987

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Homework Statement

Is there a way to prove that E={sin(nx), cos(nx): n in N u {0}} is a maximal orthonormal basis for the Hilbert space L²([0,2pi], R) of square integrable functions (actually the equivalence classes "modulo equal almost everywhere" of the square integrable functions)?

I mean, I am asking if we can show directly, using the definition or some other characterization, that E is a Hilbert space basis for L², so that we can conclude that L² functions are equal to their Fourier series. In other words, we can't use the fact that L² functions converge to their Fourier series to show that E is maximal.

Homework Equations

Relevant characterizations of "E is a hilbert space basis" that I am aware of:

(1) E is a maximal orthonormal set
(2) the orthogonal complement of E is trivial
(3) the span of E is dense

The Attempt at a Solution

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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