1
vote
1answer
24 views

Hilbert space (nonseparable): ONB

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
2
votes
1answer
34 views

Series convergence in Hilbert space and dual.

I'd like to prove that: $$ \|u_\varepsilon-f\|_*\rightarrow0 \quad\text{in }V^* $$ with $V$ Hilbert and $V^*$ its dual. In particular $u_\varepsilon\in V$. From the precedent points of the proof I ...
3
votes
1answer
83 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
1
vote
1answer
33 views

If $f$ is identically zero then the coefficients are all zero

I am looking at the space: $$A:=\left\{f(x)=\sum_{k\in\mathbb{Z}}{a_ne^{inx}}:(a_n)_{n\in\mathbb{Z}}\in l^1(\mathbb{Z})\right\}$$ I want to say the following: if $f\equiv0$, then $a_n=0$ for all ...
1
vote
1answer
140 views

What is this Hilbert space?

The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means $\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$ where $\hat f$ is the fourier transform of $f$: $\hat ...
5
votes
2answers
769 views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
0
votes
1answer
64 views

Recovering an Operator on $L^2$ Given its Action as a Composition on the Spectrum of $f$

Let $\ell^2 = \ell^2(\mathbb{Z})$ denote the Hilbert space of square summable complex sequences on $\mathbb{Z}$ and suppose that $\sigma:\mathbb{Z} \to \mathbb{Z}$ is a function such that the linear ...
2
votes
2answers
285 views

Approximating $|x|$ by a linear combination of $1, \cos x, \sin x, \cos 2x, \sin 2x$

Let $\phi(x) = |x|$ for $x \in (-\pi, \pi)$. Suppose we approximate $\phi(x)$ by a linear combination of the functions $\{1, \cos x, \sin x, \cos 2x, \sin 2x\}$. What linear combination of the form: ...
7
votes
3answers
314 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...