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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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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Principal value methods for fourier laplace etc.

I recently saw this here: $$\int_{-\infty}^{\infty} \frac{1}{\omega^2} e^{j\omega t} d\omega$$ and I was unable to understand how such an integral could be computed. I want to learn about this ...
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74 views

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the ...
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14 views

Convergence of a subsequence in $(C(\mathbb{T}), \|\cdot\|_2)$

Problem: Define, $ \mathbb{T} := \mathbb{R}/{2\pi\mathbb{Z}} $. Consider a sequence of functions $(g_n)_{n\in \mathbb{N}} \in C^4(\mathbb{T})$ such that, $ \sup_{n \in \mathbb{N}}(\| g_n \|_2 + \| ...
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A problem similar to $L^2$ Fourier transform, but in the setting of Borel measure.

Problem: Let $\mu$ be a finite Borel measure on the real axis, supported on a countable set $\mathbb{Q} \subset \mathbb{R}$ (I'm not sure whether here $\mathbb{Q}$ is all rational numbers ). And let ...
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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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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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2answers
36 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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10 views

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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26 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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61 views

What is the 3D fourier transform of a spherical shell?

I am trying to build up intuition about what the fourier transform of a spherical shell will look like but I can't say I'm making much progress. I've also tried to dumb down the problem in 2D and ...
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31 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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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 ...
3
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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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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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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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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
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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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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2answers
116 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

We know that a Fourier series for signal $x(t)$ is given as $$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$ So my question is what ...
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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
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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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 ...
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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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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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80 views

Features of phase and magnitude spectrum?

I have read in many books that whether the signal is 1D or multidimensional , The magnitude spectrum tells you how strong are the harmonics in the signal and The phase spectrum tells where this ...
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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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24 views

Finite Fourier Transformation [closed]

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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1answer
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What will happen if we try to reconstruct signal using phase only or magnitude only?

I am studying Fourier Transform and it's inverse. We get phase and magnitude from Fourier transform and reconstruct it back from both together My question is that What will happen if we try ...
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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 (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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1answer
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$|\sum_{k=1}^n \frac{\sin(kx)}{k}| \leq \frac{\pi}{2}+1$

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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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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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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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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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
30 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
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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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
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Judging whether a function is not in the range of Fourier transformation

(1) First, I have to show that if f is an odd function that is integrable on the rea line, then there exists a positive number M such that for any a,A (where A is bigger) the following holds. (2) ...
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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
48 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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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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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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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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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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Why fourier transformation use complex number?

I know that the Fourier transform is as follows:$$\hat{f}(\xi)= \int_{-\infty}^{\infty}\exp(-\mathrm ix\xi)f(x)\mathrm{d}x$$ but I couldent understand why should use complex number $i$ in the ...
2
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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))] = ...
2
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2answers
102 views

Fourier Decompositon problem

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
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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 ...