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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multipliers on $H^{1}$

I'm begining to study the hardy space $H^{p}(\mathbb{R}^{n})$. First recall that a $L^{\infty}$ function is called a $H^{1}$ multiplier if the associated operator ...
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126 views

using central limit theorem

I recently got a tute question which I don't know how to proceed with and I believe that the tutor won't provide solution... The question is Pick a real number randomly (according to the uniform ...
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1answer
175 views

Question on Schwartz function

Let T be the operator $(2\pi)^ \frac{-n}{2}\mathcal{F}$. Show that, for $k = 0, 1, 2, 3,$ there exists a polynomial $p_k(x)$ of degree $k$ and a complex number $c_k$ such that ...
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Fourier transform of even/odd function

How can I show that the Fourier transform of an even integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is even real-valued function? And the Fourier transform of an odd integrable function ...
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Poisson's summation formula

It is said that the Fourier transform $\hat{f}(\omega)$ of a function $f(t)$ and the Fourier transform $\hat{b}(\omega)$ of its samples $b(k)=f(t)|_{t=k}$ are related by Poisson's summation formula ...
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fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series)

In order to find $f(x)$ so that $F(u)=\sum\limits_{x=0}^\infty f(x)\sin\dfrac{\pi ux}{L}$ and $F(u)=\sum\limits_{x=0}^\infty f(x)\cos\dfrac{\pi ux}{L}$ , we can borrow the idea from Fourier sine ...
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368 views

What's the Fourier transform of these functions?

The Fourier transform of $|x|^{\alpha}$. This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; ...
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174 views

Windowed Fourier transform of Gaussian distributed random time series

If I have a discrete time series $x(t_i)$, and each of the $x(t_{i})$ are normally distributed, i.e., come from a Gaussian distribution with mean zero and variance one, would a windowed finite Fourier ...
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202 views

Continuous probability measures on the unit circle

Is there a continuous probability measure on the unit circle in the complex plane - $\sigma$ with full support, such that $\hat{\sigma}(n_k)\rightarrow1$ as $k\rightarrow\infty$ for some increasing ...
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1answer
850 views

Is there a time-domain proof of Nyquist sampling theorem?

For a continuous-time signal $x(t)$ that is bandlimited (in the baseband) to $[-W,W]$, the standard proof of Nyquist sampling theorem proceeds in the frequency domain by examining the Fourier ...
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65 views

Help me to understand the Gaussian blurring (2)

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$\begin{align*} p_{i,j} &= \frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy,\\ \Delta_{i,j} &= ...
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97 views

Absolute continuity with respect to a Rajchman measure

A measure $\sigma$ on $\mathbb{T}$ (the unit circle in the complex plain) is called a Rajchman measure if $ \hat{\sigma}(n)\rightarrow0$ as $|n| \rightarrow \infty$. I want to prove that if ...
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120 views

finding the fourier coefficients of $f(x) = \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}$

This is what i know so far: The given series uniformly converges by the M-test and that i can swap the integration and the sum when calculating the coefficients. Apparently i am supposed to use the ...
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2answers
71 views

Product in frequency domain of discrete data.

I have discrete data x[n]=[1 2 2] and h[n]=[3 4 1]. I can find their frequency counterparts using the fourier series x[iw]=[5 -1 -1] & h[iw]=[8 (1/2-3*sqrt(3)*i/2) (1/2+3*sqrt(3)*i/2)]. How can I ...
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1answer
111 views

is there a nice way to find the fourier transform of…

I am looking for a nice way to calculate the FT of the following function $f(x)=\biggl(\sum_{n=1}^{c}~a_n~e^{-\frac{i}{2}~x~b_n}\biggr)^d$, where $d,c>0$, $a_n$ and $b_n$ are real coefficients, ...
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376 views

The boundedness of an integral

Is there a constant $C$ which is independent of real numbers $a,b,N$, such that $$\left| {\int_{-N}^N \dfrac{e^{i(ax^2+bx)}-1}{x}dx} \right| \le C?$$
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1answer
167 views

Bounds on integral

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where ...
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142 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 $G$ ...
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207 views

Determining sparse frequency distribution via discrete Fourier transform

Consider the function $$f(t) = 2 \sin(t)+\sin(2t)+25 \sin(400t)$$ (for example). In this case, how many samples of this function would I have to take, and at what sampling frequency, to determine the ...
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79 views

Bounds on Fourier coefficients of Euclidean distance functions

I am interested in the bounding the Fourier coefficients $a_{m,n}$ of the function $f(x,y)=\sqrt{x^2+y^2}$ defined on the interval $[-1,1]^2$. I am specifically interested in understanding the ...
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483 views

How to find this moment generating function

I am trying to find the moment generating function of a random variable $X$, which has probability density function given by $$f_{X}\left( x\right) =\dfrac {\lambda ^{2}x} {e^{\lambda x}}$$ Where ...
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7k views

Derivative of convolution

Assume that $f(x),g(x)$ are positive and are in $L^1$. Moreover, they are differentiable and their derivative is integrable. Let $h(x)=f(x)*g(x)$, the convolution of $f$ and $g$. Does the derivative ...
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1answer
182 views

Definite integral of a complex function on a real interval

How does $$\int\limits_{k_0-(\Delta k/2)}^{k_0+(\Delta k/2)}e^{ikx}dk$$ equal to $(e^{ik_0x}/x)2\sin (\Delta k \cdot x/2)$?
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$\int_{-\infty}^\infty e^{ikx}dx$ equals what?

What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? I saw this in the Fourier transform, and am ...
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3answers
364 views

Easy Fourier series example: where is my mistake?

I'm doing exercise 15 on page 255 in Kreyszig: To illustrate that a Fourier series of a function $f$ may converge even at a point where $f$ is discontinuous, find the Fourier series of $$ f(x) = ...
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1answer
642 views

Discrete Fourier Transform interpreted in terms of sampling period

I understand that $$X(k) = \sum_{n=0}^{N-1}x_ne^{-i2\pi \frac{k}{N} n}$$ and $$x(n) = \frac{1}{N}\sum_{k=0}^{N-1}X_k e^{i2\pi \frac{k}{N} n}$$ are the discrete Fourier transform and inverse discrete ...
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150 views

Where can I find a clear explanation of the phase vocoder?

There are many explanations of the phase vocoder online, but they are either a bit light on the details "just unwrap the phase!" or using signal processing terminology and electrical diagrams (this ...
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2answers
229 views

Inequality regarding difference of characteristic functions

We want to show if $X, Y$ are random variables defined on a common probability space, with characteristic functions $f, g$ respectively, then the following inequality is valid: $$\sup |f(x)-g(x)| \le ...
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472 views

About completeness of the Fourier series.

The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta + \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n ...
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72 views

About the orthonormal decomposition of $L^2 (-\pi , \pi) $

For any $f \in L^2 (-\pi, \pi)$, prove that there exists unique orthonormal decomposition with even functions and odd functions : $$ L^2 ( -\pi , \pi) = L^2 _{odd} (-\pi , \pi ) \oplus L^2_{even} ...
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How can I prove $\lim_{n \to \infty} \int_{0}^{\pi/2} f(x) \sin ((2n+1) x) dx =0 $?

For continuous $f$, $f \in L^2$, prove that $$\lim_{n \to \infty} \int_{0}^{\pi/2} f(x) \sin ((2n+1) x) dx =0 $$
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What is the unit of the FFT output?

Consider a signal, f(t), with impulse samples taken N times, i.e f[0],f[1],f[2],...f[N-1] Let us perform FFT on it. Now, we have the amplitude on the y-axis and the frequency on the x-axis. I want to ...
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1answer
100 views

Continuity of Translation of the Trigonometric Polynomials

Had a question from Katznelson recorded in my journal which is still bugging me; I believe I have solved the following exercise subject to a minor point: Let $B$ be a Banach spach on $\mathbb{T}$ with ...
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1answer
42 views

equivalence in Fourier space

I have a comprehension problem regarding Fourier transforms. So far I know, the Fourier transform can be defined on the whole Schwartz space $\mathcal{S}(\mathbb{R})$ and is bijective on it. So I have ...
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1answer
110 views

Ill posed PDE problem using Fourier transform.

I have got this question in my exam and i was not able to solve it . The hint that i had gotten was to use Fourier transform and solve it . But i couldn't . . Can anyone help me . Thank you .
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293 views

Reduced frequency range FFT

Generally when one takes the FFT of a signal it "works" over the whole bandwidth dividing up the spectrum into chunks given by the resolution. If the bandwidth of the signal is 10khz and your ...
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2answers
164 views

Fourier series on $\mathbb T$ and $S^1$

From my lecture notes: "The notation $\mathbb T$ will be used for the additive circle and $S^1$ for the multiplicative circle." What I understand: As a topological group, $S^1$ has the subspace ...
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1answer
377 views

Fourier and half range series for $\sin x$

Expand in Fourier series of $f(x) = \sin x$ for $0<x<l$. Deduce the result \[ \frac1{1 \cdot 3} - \frac{1}{3\cdot5} +\frac{1}{5\cdot 7} - \cdots = \pi-\frac{2}{4}. \] Obtain half range ...
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472 views

What exactly is a Haar measure

I've come across at least 3 definitions, for example: Taken from here where $\Gamma$ is a topological group. Apparently, this definition doesn't require the Haar measure to be finite on compact ...
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1answer
178 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 ...
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2answers
139 views

Continuous Random Variable with constant moments?

I would like to know if there exists a measure $\rho$ on the positive real line such that its moments $\int_0^{\infty} x^j d\rho(x)$ are equal to a constant (for example equal to one) for all ...
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1answer
169 views

What is the upper bound for the rate of decay of fourier coefficients in the 2-dimensional case?

I am studying and trying to do something with fourier analysis, though I have little background and do not know much about the subject. I believe, and please correct me if I am wrong that if your ...
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1answer
55 views

Uniform convergence and convergence in $S'(\mathbb{R}^n)$

Let $$\hat{f_\epsilon}: \xi \mapsto \exp(-\epsilon |\xi|) \frac{\sin(|\xi|t)}{|\xi| t}$$ denote to the Fourier transform of $f$. How do I see $\hat{f_\epsilon}$ converges uniformly on ...
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2answers
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Fourier transform of function composition

Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually? I know you can do this for the sum, the ...
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1answer
828 views

Two Fourier Transform Questions

I am reading a book by Zoran Gajic titled Linear Dynamic Systems and Signals. There were two problems in chapter 3 that I was curious about. The first question asks for the Fourier Transform of a ...
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1answer
449 views

Fourier Transform of a derivative + Bochner's theorem about positive definite functions

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
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216 views

Are the amplitudes of these frequency spikes equal to 1 when the real part of the complex number “s” is equal to one half?

Over at stack overflow I asked a question about how to plot the Riemann zeta zero spectrum from the von Mangoldt function. Then I asked a question about calculating the Riemann zeta function at the ...
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1answer
509 views

Fourier transform of a product of sinusoidal functions

Given the function: $$x(t)=A_1\cos(\omega_1 t+\phi)$$ its Fourier transform is: $$X(\omega)=\sqrt{\frac{\pi}{2}}A_1 e^{-i\phi}\delta(\omega_1-\omega)+\sqrt{\frac{\pi}{2}}A_1 ...
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3answers
172 views

basis functions do not lie in the space they form

For example, any continuous function in $\mathbb{L}^2(-\infty,\infty)$ space can be expanded by delta functions $\delta(x-a)$ or Fourier basis $e^{ikx}$. However, the basis functions, both ...
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2answers
725 views

Fourier transform eigenvalues

Since I've studied the Fourier transform extension to the Hilbert space $L^2$, I wondered if there is a complete study relative to its eigenvalues. I know that its adjoint operator is the inverse ...