1
vote
1answer
68 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
2
votes
1answer
62 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
4
votes
0answers
45 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
0
votes
1answer
54 views

Square Summable functions

Can somebody please help me understand the notion of square summable functions intuitively?? I have been self studying Hilbert Spaces and Fourier Transform for DSP. Any help is appreciated. Thanks.
0
votes
1answer
43 views

Parseval's theorem to $\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$.

Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's ...
2
votes
0answers
37 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
6
votes
1answer
48 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
1
vote
1answer
146 views

l2 norms, rapidly decreasing functions and fourier transforms

Let $f\colon \mathbb{R} \to \mathbb{C}$ be a rapidly decreasing (rd) function. Let $\mathcal{F}(f)$ be the Fourier transform of $f$. It is known that 1) $\| \mathcal{F}(f) \|_2 = \| f \|_2 $ ...
15
votes
4answers
636 views

How to interpret the adjoint?

Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$. I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I ...
5
votes
0answers
103 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
1
vote
1answer
57 views

Show that the subspace A is the whole Hilbert space H

"Let $A$ be a subset in a Hilbert space $H$, such that $x\in H$ and $x \perp A$ imply $x = 0$. (1) Show that the closed subspace that is generated by $A$ is $H$. (2) Let $f(x)$ be a square summable ...
1
vote
1answer
143 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 ...
3
votes
2answers
230 views

Span of functions dense in $L^2$

This is an exercise from Werner's Funktionalanalysis. I have to show that the linear span of the functions $f_n(x)=x^ne^{-x^2/2}, n\geq0$ is dense in $L^2(\mathbb{R})$. The book gives the hint to ...
1
vote
1answer
141 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
3
votes
1answer
174 views

A problem on Fourier transforms and orthogonality

Let $f$ be a square integrable function, strictly positive almost everywhere. Consider the family of functions $f_a=f(x+a)$, where $a$ is any real number. I want to prove that if a function is ...
3
votes
1answer
110 views

What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)

What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle) I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
2
votes
1answer
81 views

Finding an ON basis of $L_2$

The set $\{f_n : n \in \mathbb{Z}\}$ with $f_n(x) = e^{2πinx}$ forms an orthonormal basis of the complex space $L_2([0,1])$. I understand why its ON but not why its a basis?
2
votes
2answers
645 views

Why is Parseval's Equality and Bessel's Inequality Different?

Bessel's Inequality: $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$ Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$
2
votes
1answer
150 views

completeness of orthonormal set

I am currently working through some lecture notes on the Geometry of Hilbert spaces, and I am stuck with the following comment: If we are given the inner product space $C([0,1])$ of continuous ...
7
votes
3answers
319 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 ...
1
vote
2answers
494 views

Does the vector space spanned by a set of orthogonal basis contains the basis vectors themselves always?

I used to think that in any Vector space the space spanned by a set of orthogonal basis vectors contains the basis vectors themselves. But when I consider the vector space $\mathcal{L}^2(\mathbb{R})$ ...
1
vote
3answers
179 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...