Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Fourier transform of $L^2$ function

Let us the definition of Fourier transform $$\hat f(\lambda) = \int_{-\infty}^\infty f(t) \exp(- i \lambda t) dt$$ How do I change this expression if $f(t)\in L^2[-\pi,\pi]$? $$\hat f(\lambda) = ...
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2answers
55 views

Calculating $\int_{-\pi}^{+\pi} e^{ixt} e^{-i \omega t} dt$

We know that Fourier Transform of $e^{ixt}$, where $x$ is a real parameter, $t\in \mathbb R$ is $$\int_{-\infty}^{+\infty} e^{ixt} e^{-i \omega t} dt=\int_{-\infty}^{+\infty} e^{ixt-i \omega t} ...
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Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
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Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
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1answer
42 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
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1answer
30 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
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1answer
24 views

How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in ...
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1answer
21 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
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1answer
92 views

Is $\hat S$ a function?

Let $S(x)=\sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x)$. Find the Fourier transformation of $S(x)$. Is $\hat{S}$ a function? $$\hat{S}=\int_R \sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x) ...
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Which of the functions below define a tempered distribution? The functions in a, b, c are defined in R.

Which of the functions below define a tempered distribution? The functions in a, b, c are defined in R. a) $f(x)=x^3+3x$ $$|T(\phi)| \leq \int_R |(x^3+3x)\phi(x)|dx=\int_R |x(x^2+3)\phi(x)|dx \leq ...
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0answers
29 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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76 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
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36 views

Use the Fourier inversion formula to compute h(x) when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$

Use the Fourier inversion formula to compute $h(x)$ when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(x)=\frac{1}{(1+y^2)^2}=\frac{1}{1+y^2} \times \frac{1}{1+y^2}=\widehat{\frac{1}{2}e^{-|x|}} \times ...
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In general, let $f \in L^2(-1,1)$ and let $g:R \to R$, $g(x)=f( \{ x \} )$. How are the Fourier transforms of f and g related?

Recall that $\{ x \}$ is a decimal part of a real number $x$. For example, if $x=3.41$, then $\{ x \}=-0.41$. Part A: Let $f(x)=2x$, with $x \in (-1,1)$ and let $g:R \to R$, $g(x)=2 \{ x \}$. Sketch ...
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Find the Fourier Transform of a Riesz Basis.

Find the Fourier Transform of a Riesz Basis for $L^2[-\pi,\pi]$: $e^{ix_nt}$, where $x_n$ is a sequence of real numbers, $t\in \mathbb R$. I state that I am not very expert in Riesz Basis and Fourier ...
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2answers
50 views

Find the Fourier Transform of $e^{i x t}$.

Find the Fourier Transform of $e^{ixt}$, where $x$ is a real parameter, $t\in \mathbb R$. I started writing: $$\int_{-\infty}^\infty e^{ixt} e^{-i\omega t}dt$$ but I do not know how to go on!
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1answer
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can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
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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 ...
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$\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with ...
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1answer
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Not understanding one step in derivation of Dirichlet kernel

I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the ...
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2answers
96 views

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$. So if I create a number line marking a-b and a+b. If that the integral above has 5 different answers depending on where (-t,t) is located on the number ...
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1answer
32 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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2answers
66 views

Check: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. [closed]

Part A: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. Part B: Generalize the previous problem and deduce a formula for the Fourier transform of a polynomial of degree m. ...
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Check: Find $h(x)$ when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$.

Find h(x) when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(y)=\frac{1}{(1+y^2)^2} =\frac{1}{1+y^2} \times \frac{1}{1+y^2} =\hat{\frac{1}{2}e^{-|x|}} \times \hat{\frac{1}{2}e^{-|x|}}$$ Let ...
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1answer
49 views

In signal processing, every where you see infinity. Why?

Everywhere, in signal processing you see infinity. For example, in Fouriers, correlations. But no body would live to see infinity. Why do we aritificially talk about infinite time signals and then ...
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13 views

impulses in domain region and something in frequency region

I am studying about the fourier series and fourier transform. and I am suffering to deal with the picture that I will show you I am sure that the signal upper in the picture has periodicity and ...
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1answer
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Fractional powers of the operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$, $Bf = f-f^{''}$.

Consider the linear operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$ defined by the following mapping: $Bf = f-f^{\prime\prime} \equiv (I-\Delta)f$, where $\Delta$ is the Laplace operator that ...
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3answers
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Characterization of $\mathscr{S}(\mathbb R^n)$?

Consider the vector space $$\displaystyle\mathscr{S}(\mathbb R^n)=\{f\in C^\infty(\mathbb R^n): \lim_{|x|\to \infty} |x^\alpha \partial^\beta \phi(x)|=0, \forall \alpha, \beta\in\mathbb N_0^n\}.$$ ...
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Showing that Gaussians are eigenfunctions of the Fourier transform

I'm having a bit of trouble on this problem: I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar ...
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the multidimensional Hilbert transforms or partial Hilbert transforms

The one-dimensional Hilbert transform can be defined by the convolution $Hf:=f*\frac{1}{\pi x}$, or can be given by Fourier multiplier $(Hf)\hat{\,}(\xi)=-i\mathrm{sgn}(\xi)\hat{f}(\xi)$. ...
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1answer
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To What Extent is the Fourier Inversion Theorem Due to the Self-Adjointedness of the Laplacian

I've tried looking this up (I looked at various spectral theorems) but couldn't find anything that talks about the connection between Fourier transforms and the eigenfunctions of the Laplacian (we may ...
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1answer
35 views

What is Fourier Space

I know a some basics stuff regarding Fourier Analysis (Fourier series and Fourier transforms), but I've seen the term "Fourier Space" come up and I'm having trouble finding a definition for what this ...
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A continuous function which does not converge to its Fourier series. [duplicate]

Where can I find an example (or the theorem) for a continuous function which does not converge pointwise to its Fourier series, as well as its explanation? I would prefer a web page or a free site or ...
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Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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1answer
35 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
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Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
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Eigenvalues of rotation invariant operators on 2-sphere

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore ...
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Interpreting Fourier transform frequency graph

I've been trying to understand Fourier transform for some time now and I think I've perhaps finally got the idea now. What I would like to do now is to make an example of Fourier transform for ...
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1answer
63 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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Given Tf(x), find the equivalent operator m(k)f^(k) in the Fourier transform sense.

Let $f\in L^2(\mathbb R)$, let $$ g(x) = Tf(x) = \int^{x+1}_{x} f(s)ds $$ Find $m\in L^\infty(\mathbb R)$ such that $\hat g(k)=m(k) \hat f(k)$. Use this to show that $T$ is a bounded linear operator ...
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Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
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1answer
223 views

Sobolev inequality

If $f\in H^2(\mathbb R^2)$, I want to show that $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$ For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb ...
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1answer
76 views

Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
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26 views

fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
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0answers
14 views

Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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1answer
29 views

How to interpret the imaginary part of an inverse fourier transform

The fourier transform of arbitrary real data can (usually will) result in complex data. If the real data represents samples in time, then the complex FT data represent frequencies with the magnitudes ...
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15 views

Nyquist stability criterion with Fourier Transform

I want to ask if I have a system of partial differential delay equations that I have linearized and then found the transfer function in fourier space, is it possible to analyze this using the Nyquist ...
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0answers
24 views

Significance of Mutual Coherence

I am reading about compressive sensing and I am not able to understand the physical significance of mutual coherence. For tow matrices $\Phi$ and $\Psi$, mutual coherence is defined as $\mu(\Phi, ...
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1answer
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Let $f(x)=(1+|x|^2)^{-a}$, with $x\in \Bbb R^n$ and $a>0$.

Let $f(x)=(1+|x|^2)^{-a}$, with $x\in \Bbb R^n$ and $a>0$. Show that $f(x)$ is a constant multiple of $g(x)$. Let $f(x)=(1+|x|^2)^{-a}$, with $x\in \Bbb R^n$ and $a>0$. Show that $f(x)$ is a ...