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 $x^p \cdot {{df^q} \over {dx^q}}$

What is the Fourier Transform of $x^p \cdot {{df^q} \over {dx^q}}$? This seems like an elementary question, but my CRC book of standard formulae doesn't have it. My attempt is rather trivial, but for ...
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33 views

plz see the problem and give me the solution of proof? [on hold]

At first glance, it looks so easy, but i can't solve this problem easily. (the inequality is estimation ) $$\left \| \sum_{k} \prod_{k} f_{k} \right \|_p \approx < \left \| \sum \left | f_k ...
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2answers
25 views

Fourier transform of this?

I need to prove: $$\frac 1 {\Delta x} \int_0^\infty dk \, e^{-ik(\Delta t-i\epsilon)} \sin k\,\Delta x=\frac 1 {\left(\Delta t-i\epsilon\right)^2 -\Delta x^2} $$ A possible hint could be: ...
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7 views

Discrete time Fourier transform on decimated signals

If I have a signal $x[n]$ and its decimated version, $y[n]=x[2n]$, is there a known expression for the DTFT of $y[n]$, $Y(\theta)$, as a function of $X(\theta)$? Thanks,
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11 views

Proving Inverse DFT

I have trouble understanding the proof I was provided of the IDFT, here is what I have: $$ \nu_n = \frac{n}{\Delta N} \\ x(t) = \int_{-\infty}^{\infty}X(\nu)e^{i2\pi\nu_n t}d\nu \\ $$ the next step I ...
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26 views

A question on Parseval's Theorem (Fourier Transform) [on hold]

How to obtain Fourier Transform using Parseval's relation? Take for example: (sin(at))/at As this example cannot simply be solved using the actual formula for calculating fourier transform. So a ...
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28 views

An exercise regarding fourier inversion formula

I have to solve an exercise which looks like a more general form of Fourier inversion formula. But, I'm having hard time attacking it... A given function f is integrable on the real line and ...
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1answer
32 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
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89 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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1answer
30 views

About convergence in norm of the Fourier Transform

Duoandikoetxea's Fourier Analysis, on page 59 (Corollary 3.7) says that: \begin{equation} \lim_{R \rightarrow \infty}\big\|S_{R}\,f - f\big\|_{p} = 0 \end{equation} for $1<p<\infty$, where ...
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2answers
32 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
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1answer
25 views

About Fourier transform and complex conjugate

why this passage is correct ? \begin{equation*} \mathscr{F}[h(-\tau)] = H^*(f), \end{equation*} when $h(\tau)$ is a real function of real variable $\tau$, and $H^*(f)$ is the complex conjugate of ...
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Is there any mistake? Proof related to the Poisson summation formula.

I need to prove the following statement: Let $\varphi \in C(\Bbb R)$ with compact support. Then, $$ \Big \Vert \sum_{k\in \Bbb Z} \varphi(k) e^{ikx} \Big \Vert_{L_1(0,2\pi)} \leq C \Vert ...
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1answer
28 views

Fourier Analysis Help - $\mathcal L^2$

Let $\{e_k | e_k(x)= e^{ikx}/\sqrt{2\pi\,}\}$ be the orthonormal basis in $\mathcal L^2$ per. I first have to use this basis define two infinite dimensional orthogonal subspaces of $\mathcal L^2$ per. ...
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1answer
20 views

Calculating inverse Fourier Transform without outright integrating

This was one of the later questions in my tutorial which I didnt reach in time. Answers for tutorials aren't posted online however so I tried working through this alone but quickly got stuck ...
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1answer
63 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
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28 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to think ...
3
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1answer
44 views

$|\sum_{k=1}^n \frac{\sin(kx)}{k}| \leq \frac{\pi}{2}+1$

I want to prove that $$\left|\sum_{k=1}^n \frac{\sin(kx)}{k}\right| \leq \frac{\pi}{2}+1.$$ for each $n\in\mathbb N$ and $x\in (0,2\pi)$. I know that the sum is inside is the partial sum of the ...
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24 views

Finite Fourier Transformation [on hold]

I have to solve the boundary value problem described by $\frac{\partial v}{\partial t} =\frac{\partial^2 v}{\partial t^2} $ using finite Fourier Transform where, $0 < x < 6 $, $t >0$, and ...
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31 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
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21 views

Bounds for the solution of heat equation using convolutions [closed]

We know that the solution of the heat equation $$u_t=u_{xx}$$ with initial condition $ u(0,x)=u_0(x)$ is given by $$u(t,x)=(u_o * H_t)(x)$$ where the heat kernel is given by ...
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28 views

Absolute value Sinus Cardinalis integral

The existence of the Fourier Transform integral is conditioned for some functions. An important example is the sinc (sinus cardinalis) function, which although does not satisfy certain conditions, it ...
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1answer
57 views

Example of function not Fourier Invertible in $L^1$

It is well known that, if $1 < p \le 2$, then, for every $f \in L^p$, $$ \int_{[-R,R]^n} e^{-2\pi i x \cdot y} \hat{f}(y) dy \rightarrow f(x) $$ As $R \rightarrow \infty$, in the $L^p$ sense. ...
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25 views

How to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions?

I want to know how to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions. Please help me. Thanks very much.
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1answer
68 views

If $\gamma$ is irrational, then $\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)\to \int_T f(t)\,dt$

I need to show that $$ \lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)=\int_T f(t)\,dt. $$ Here $\gamma$ is any irrational number on the real line and $f(t)$ is any continuous periodic ...
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1answer
38 views

On a property of Fourier coefficients

I need to prove the following: If $(\Phi_n)_{n\ge0}$ is an orthonormal system of integrable functions defined on some interval $[a,b]$, and $(c_n)_n$ is a sequence of reals such that $\sum c_n ...
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Behavior of a Solution to Heat equation Compactly Supported in Time

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
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38 views

An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k \cos(kt) + b_k \sin(kt), \\ a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} x(t) \cos(kt) dt, \\ b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} ...
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41 views

Is the Fourier transform of a continuous and compactly supported function summable?

Let $\varphi$ defined on the real line be continuous and with compact support. What can we say about the summability of $\hat{\varphi}$? I've gone through some theorems such as Parseval's without ...
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1answer
25 views

Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad ...
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1answer
24 views

Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
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1answer
47 views

How to prove for a system with rational stable transfer function, the output is square integrable?

I want to know for a system with rational stable transfer function, i.e. H(jw)=1/[(a1+jw)(a2+jw)...(an+jw)] (a1,a2,..,an>0), why a square integrable (L2 integrable) input must generate a square ...
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Is the following property of a Fourier Transform valid?

We know that $$\mathscr{F}\left\{f*g\right\}=\mathscr{F}\left\{f\right\}\mathscr{F}\{g\}$$ so I was wondering whether the inverse is true: ...
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26 views

Fourier transform of exponential of a function

I am wondering what $\mathcal{F}[\exp(f)]$ is in terms of $\mathcal{F}[f]$. The farthest I have got is using the series expansion of $\exp$, such that I end up with $\mathcal{F}[\exp(f))] = ...
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1answer
33 views

How to solve a particular PDE (which reminds of heat equation)

I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where ...
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1answer
23 views

randomly rough surface by ifft : real output from ifft

I'm trying to generate a randomly rough isotropic surface with predefined roughness amplitudes (standard deviation of heights). Suppose I have the absolute values of fourier components of the surface ...
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20 views

How to get h(t) using direct inverse Fourier transform formula for H(jw)=1/(a+jw) (only when |w|< W)?

I want to get the expression of signal in time domain by using inverse fourier transform. The signal in frequency domain is a little special: H(jw) is 1/(a+jw) only when |w|< W; and when |w|> W ...
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$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
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1answer
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Limit of Cosine and Sine Fourier Transforms

If I define the cosine and sine Fourier transform as (skipping constant prefactors $(2\pi)^{0.5}$): $$\mathcal{F}_C\{f(x)\}=\int_0^{\infty}\,f(x)\,\cos(\omega x)\,dx$$ and ...
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1answer
19 views

Fourier transform of an integrable odd function

I'm trying to prove a proposition about the Fourier transform of an odd function. Let $f\in L^1(\mathbb{R})$ be an odd function. Then there is $M>0$ such that for any $a,A>0$, ...
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1answer
87 views

$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ as $|x| \to \infty$?

Suppose $f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R)\cap L^{\infty}(\mathbb R), (1<p<\infty).$ My Question: Can we expect $\lim_{|x|\to \infty} |f(x)|=0$ ? (In other words, If $f$ and its ...
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17 views

Existence of Fourier Transform for Implicit function

Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by ...
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1answer
42 views

Importance of the phase spectrum

For any given signal using Fourier transform, we can compute it's magnitude and phase spectrum. In that I want to give focus on phase spectrum. But for phase spectrum, I don't have much data ...
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1answer
29 views

How to get $h(t)$ using direct inverse Fourier transform formula for $H(jw)=1/(a+jw)$?

I want to find the inverse Fourier transform of $H(jw)=1/(a+jw)$. We know from the Fourier table that $$ F(e^{-at}) = 1/(a+jw). $$ So that $$ h(t)=e^{-at}. $$ But can we get $h(t)$ directly using ...
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1answer
30 views

question on Fourier Transformation

I have to find the Fourier Sine transform of $f(x)=1$ when $|x|<a$ and $f(x)=0$ when $|x|\ge a$ and hence show that $$\int_0^\infty {\sin(t)\over t} dt =\pi/2$$ and $$\int_0^\infty ...
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1answer
49 views

Fourier coefficients of a symmetric function in $\pi$

I want to show that the Fourier coefficients $\int_{-\pi}^\pi e^{ij \lambda} f'(\lambda) d \lambda$ of the derivative of a continuously differentiable function $f: [ - \pi, \pi] \rightarrow ...
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2answers
20 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
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1answer
29 views

Using fourier transforms to solve odes

In the very last line of the solution that i have given i got the residue to be (e^(iaz))/2i which when multiplied by 2pii gives me pie^(iaz) now from this i don't understand how they got rid of z ...
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33 views

Can a DTFT have a period different of $2\pi$?

I think almost everything is in the title. In an exercise, a DTFT is given : $$X(e^{j\Omega}) = \sin(\Omega) + \cos(\Omega/2)$$ The period of this DTFT is $4\pi$. Is that possible? I mean, the ...
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1answer
17 views

Fourier transform of a product of two rect functions

I am trying to evaluate the following expression $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$ which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, ...