Section 4.2

Bessel's Inequality, Parseval's Theorem, and the Riesz-Fischer Theorem

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Problem 4.2.1
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(a)

Prove that in any inner product space (V,⟨⋅,⋅⟩),f(V,\langle \cdot , \cdot \rangle ), f and gg orthogonal implies

∥f∥2+∥g∥2=∥f+g∥2. \left\| f\right\| ^{2}+\left\| g\right\| ^{2}=\left\| f+g\right\| ^{2}.

Here, as usual, ∥⋅∥=⟨⋅,⋅⟩\left\| \cdot \right\| =\sqrt{\langle \cdot , \cdot \rangle }.

(b)

Prove Parseval's theorem.

Problem 4.2.2

Assume that {fn}n=1∞\left\{ f_{n}\right\}_{n=1}^{\infty } is a sequence in L2L^{2} and that fn→ff_{n} \rightarrow f in mean. Prove that {∥fn∥2}n=1∞\left\{ \left\| f_{n}\right\|_{2}\right\}_{n=1}^{\infty } is a bounded sequence of real numbers.

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Problem 4.2.3

Assume that f1,f2,…,fnf_{1}, f_{2}, \ldots , f_{n} is an orthonormal family in an inner product space. Prove that f1,f2,…,fnf_{1}, f_{2}, \ldots , f_{n} are linearly independent.

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Problem 4.2.4

For ff and gg in an inner product space, g≠0g \neq 0, the projection of ff on gg is the vector

⟨f,g⟩∥g∥2g \frac{\langle f, g\rangle }{\left\| g\right\| ^{2}} g

Show that the two vectors

⟨f,g⟩∥g∥2g and f−⟨f,g⟩∥g∥2g \frac{\langle f, g\rangle }{\left\| g\right\| ^{2}} g \quad \text{ and } \quad f-\frac{\langle f, g\rangle }{\left\| g\right\| ^{2}} g

are orthogonal.

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Problem 4.2.5
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(a)

Show that the classical Fourier series of f(x)=xf(x)=x is

2∑n=1∞(−1)n+1nsin⁡(nx) 2 \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n} \sin (n x)
(b)

Use your work in (a), together with Parseval's identity, to obtain Euler's remarkable identity

∑n=1∞1n2=π26 \sum _{n=1}^{\infty } \frac{1}{n^{2}}=\frac{\pi ^{2}}{6}

(Note: The same procedure can be applied to the Fourier series of x2x^{2} to obtain

∑n=1∞1n4=π490 \sum _{n=1}^{\infty } \frac{1}{n^{4}}=\frac{\pi ^{4}}{90}

and so on!)