Tagged Questions
1
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1answer
117 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
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2answers
90 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
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1answer
112 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
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1answer
133 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
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1answer
101 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
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1answer
70 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?
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2answers
276 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
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1answer
109 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 ...
6
votes
3answers
291 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
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2answers
358 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
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3answers
157 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 ...
