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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How to prove that inverse Fourier transform of “1” is delta funstion?

$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.
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81 views

How do you prove that $\lim f(x) = 0$, when $f$ is rapidly decreasing?

Let $f: \Bbb{R} \to \Bbb{R}$ be rapidly decreasing in the sense than $\sup_{x \in \Bbb{R}} |x|^k |f^{(\ell)}(x)| \lt \infty$ for all $k, \ell \geq 0$, where $f^{(\ell)}$ is the $\ell$th derivative. ...
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32 views

Magnitude of Fourier coefficients when $||f||_2 \leq 1$

Let $f \in L_2[- \pi, \pi] $ so that $||f||_2 \leq 1$. Can I say anything about $f$'s Fourier coefficients' magnitude without assuming anything else about $f$? To be more accurate: The squared norm ...
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Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?

It should not be a good argument but rather a short one and one that convinces a physicist ( so no need for mathematical rigor ) that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$ ...
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1answer
216 views

Bound of power series coefficients of a growth-order-one entire function

An entire function $f(z)$ satisfies $$|f(z)| \leq A_\varepsilon e^{2\pi(M+\varepsilon)|z|}$$ for every positive $\varepsilon$. I want to show that $$\limsup_{n \to \infty}\ [f^{(n)}(0)]^{1/n} \leq 2\...
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603 views

How do you prove translation invarianccae of Fourier transform?

Let $f$ be a rapidly decreasing function in the sense that it lies in the Schwartz space $\mathcal{S}(\Bbb{R})$. Then $\widehat{f(x+h)} = \hat{f}(\omega) e^{i 2 \pi h \omega}$, where $\hat{f}(\omega)$...
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207 views

Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?

I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm ...
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1answer
485 views

How do you show that $f(x) = e^{-x^2}$ is in the Schwartz space $\mathcal{S}(\Bbb{R})$?

The Schwartz space $\mathcal{S}(\Bbb{R})$ consists of all indefinitely differentiable functions $f$ such that for all $\ell, k \geq 0$, we have $$ \sup_{x \in \Bbb{R}} |x|^k |f^{(\ell)}(x)| \lt \infty ...
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88 views

Representing real function as integral over trigonometric functions

Since one can clearly express any function $g(x)$ as $$\int_0^{\infty} A(k)\cos(kx)dk+\int_0^{\infty} B(k)\sin(kx)dk,$$ how would $G(k)$ relate to $A(k)$ and $B(k)$? In other words, what would how ...
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54 views

How to perform division in modular arithmetic on complex exponentials: controversy, bugfix required

I have a complex exponent with prime divisor 7: $e^{\frac{2\pi i \cdot 2}{7}}$ and want to take it to the power 1/3: $e^{\frac{2\pi i \cdot 2}{7 \cdot 3}}$ (I'm learning, how to work with division ...
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729 views

Fourier transform of a complex exponential with quadratic argument

I'm a PhD student who is starting to work right now in the well-established field of ultra-fast optics. The thing is that, in most of the papers I have been reading during the past few days, there is ...
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1answer
47 views

Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
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59 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in \ell^2(\...
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1answer
88 views

How to model a stochastic process, continuous in stepsize, which converges against a simple random walk?

I want to compute the probability distribution for a stochastic process with discrete number of steps, where each real value has a nonvanishing probability to be the next stepsize. And I want to ...
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1answer
34 views

Fourier series on $\mathbb{R}^n$

In setting up a Fourier series on $\mathbb{R}^n$, we use that for $l, m \in \mathbb{Z}^n$ $$\int_{[0,2\pi]^n}e^{i\langle x, l-m\rangle }=\begin{cases} 0 &\text{ if }l\neq m \\ (2\pi)^n & \text{...
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49 views

Why include negative $n$ in a Fourier expression?

If we make the Fourier series $$\sum_{n=-\infty}^\infty a_n e^{in\theta},$$ what is the point of explicitly including the negative terms? It seems just using evenness and oddness of cosine and sine ...
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99 views

Is $f$ an absolutely continuous function?

$$f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^{n}\pi x)}{n\cdot2^{n}}, \,\,\,\,\,\, x\in [-1, 1].$$ Is $f$ an absolutely continuous function? If yes how can I show it? If not how about the total ...
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49 views

Finding real part of fourier series

I have encountered the following problem in one of my textbooks but I'm not really getting anywhere: Let $f$ be complex-valued and piecewise continuous on the interval $[-\pi,\pi]$. Find the complex ...
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77 views

How to prove Fourier inverse transform worked?

$$g(t)=\int\limits_{-\infty}^{\infty}g(f)e^{i\omega t}df$$ $g(t)$ is a function of time, $g(f)$ is a function of frequency, $e^{i\omega t}$ represent wave, and $\omega = 2\pi f$, the angular ...
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281 views

compare wavelet and Fourier transform

i would like to compare each other wavelet and Fourier transform on given signal,let us consider following signal ...
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237 views

Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since $$\sin{...
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71 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) \...
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87 views

What are the properties of the fourier transform of a phase-only function?

Given a function of the form: $$ f(x) = e^{i\phi(x)} | \phi(x)\in\Re $$ What are the properties of its Fourier transform? For instance, purely real functions have Fourier transforms with symmetric ...
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99 views

Constructing an L2 function from an entire function bounded on R

I have an entire function $f(z)$ of exponential type $\tau\geq0$ that is bounded on $\mathbb{R}$ and zero at every member of the complex sequence $\{\lambda_n\}$. What I want is an entire function of ...
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190 views

Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \begin{equation} \hat{y}...
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Q: Bases and Frames using Fourier Series

Define $w: \Bbb R \rightarrow \Bbb C$ by \begin{equation} w(t) =\begin{cases} 1/\sqrt{2\pi} & t \in [0, 2\pi)\\ 0 & \text{otherwise}. \end{cases} \end{equation} and for $n \in \Bbb ...
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408 views

About the order of the $L^1$ norm of the Dirichlet kernel.

Reading this text from Wikipedia, I found the following statement about the Dirichlet kernel: $$\| D_n \|_{L^1} \approx \log n, $$ where $\approx$ denotes "is of the order". I think that this mean ...
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1answer
46 views

Function with Fourier coefficients extend onto closed disk

Let $f$ be a continuous function with period $2\pi$. Define $$u(r,\theta)=\sum_{n=-\infty}^\infty r^{|n|}\hat{f}(n)e^{in\theta}$$ for $r\in[0,1)$, where $\hat{f}(n)$ is the $n$th Fourier coefficient ...
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368 views

Convergence in $L^1$ norm of Poisson kernel

Consider the Poisson kernel given by $$P_r(\theta)=\sum_{n=-\infty}^\infty r^{|n|}e^{in\theta}=\frac{1-r^2}{(1-r)^2+2r(1-\cos\theta)}$$ Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$, meaning that $f$ is ...
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51 views

Showing Airy's Integral (Fourier Transform of e^{-ip^3/3}) Converges

Airy's integral (a.k.a. $\widehat{e^{-ip^3/3}}$ times some constant multiple) is given by $\displaystyle\text{Ai}(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ip^3/3}dp$. Although it looks ...
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1answer
61 views

Sum of exponentials with Fourier coefficient

Let $f$ be a continuous function with period $2\pi$. Define $$u(r,\theta)=\sum_{n=-\infty}^\infty r^{|n|}\hat{f}(n)e^{in\theta}$$ for $r\in[0,1)$, where $\hat{f}(n)$ is the $n$th Fourier coefficient ...
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114 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
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1answer
71 views

Confused by a proof in Strichartz' book on Fourier Transforms

Hi I'm confused by a proof on page 53 in Strichartz book on Fourier Transforms. Specifically, in the first equation on page 53, why is it valid to interchange the action of the distribution with the ...
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1answer
65 views

Cesaro mean approaching average of left and right limits

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$, where $\mathbb{R}/2\pi\mathbb{Z}$ means that $f$ is periodic with period $2\pi$. Let $\sigma_N$ denote the Cesaro mean of the Fourier series of $f$. Suppose ...
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Need help with Placherel's Theorem

We know that the law of conservation of energy dictates that the energy carried by a waveform in the time domain must equal the energy contained in its power spectrum in the frequency domain. How ...
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74 views

Compute the Fourier Transform of $\frac1{x^2-2x+2}.$

Compute the Fourier Transform of $$\frac1{x^2-2x+2}.$$ My Answer: The denominator has roots at $1+i, 1-i$. We try to calculate the answer by the residue theorem. If $\xi\leq0$, we can say that $$\...
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1answer
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If $f \in S(\mathbb R)$, can we say $\widehat{|f|} \in L^{1}(\mathbb R)$?

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$, $\hat{f} (y) : = \int ...
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Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so $$2I=\int_{-\...
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(Real Discrete) Fourier Series: Normalisation Factor

If you have the equation: $$f(t) = \sum_{k=0}^N \left( A_k \cos \omega_k t + B \sin \omega_k t\right)$$ To compute the $A_k$ Fourier coefficient you have two cases: $$A_k = \color{\red}{{2 \over T}}...
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120 views

converse of Weyl criterion

Let $f \in L^1([0,1))$,suppose for all equi-distributed sequence $\{a_n\}_{n=1}^{\infty}$ in $[0,1)$,we have $$\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{k=1}^Nf(a_k)=\int_0^1f.$$ Do we have that $f$...
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55 views

Is $f$ is non-prime, Can we say $|f|$ is also non-prime ; in convolution algebra?

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < \...
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Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
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655 views

Condition on Fourier coefficients for real-valued function

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is real-valued if and only if $...
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1answer
882 views

Is it possible to do a half-range sine expansion on the sine function?

Suppose $f(x)=sinx,\quad 0<x<π$. Can you do a half-range sine expansion on f(x)? I tried, but I got $a_0=a_n=b_n=0$. If you requrie me to show my steps (i.e. I should have not gotten $a_0=a_n=...
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1answer
121 views

Confustion with Fourier Transform of harmonic functions

Note: this is homework. Say we assume a time-varying momentum is of the form $p(t) = \Re \{p(w) e^{-iwt} \}$ Now we would like to know the solution of the following equation, in frequency space: $ ...
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1answer
359 views

How to interpret the results of 2D Fourier Transform on an image?

I have a class where we're studying signals processing (mostly filtering of sounds and images) and while I kind of understand the results of a Fourier Transform for sounds I don't really get the ...
2
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1answer
94 views

Relation between the behavior of the Fourier transform of function and its absolute function

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}.$ We define the Fourier transform of $f$ as follows: $$...
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1answer
81 views

Metric on an infinite dimensional space with equivalence relation.

In analyzing a problem I've come across a space defined by the following equivalence relation: $(\cdots, x_{-2}, x_{-1}, x_0, x_1, x_2, \cdots) \sim (\cdots, z^{-2}x_{-2}, z^{-1}x_{-1}, x_0, zx_1, z^...
1
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1answer
172 views

Conditions for $o(|u|^{-1})$ decay of the Fourier transform of a bounded variation function

As the question suggests I am looking for a (not very restrictive) condition on a function of bounded variation so that its Fourier transform is $o(|u|^{-1})$ as $|u| \to \infty$. Let me elaborate on ...
5
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2answers
159 views

Using a FT to solve Heat Eqn

Use an appropriate Fourier transform to solve the inhomogeneous heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \delta^{\prime}(x)$$ on $−\infty<x<\infty$ ...