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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DFT Vs DCT - spectrum differences

When you are transforming a signal in to a frequency domain using both DFT and DCT, say for a function sin(x), how the spectrum will be? will both frequency values (of DCT and DFT) same? Can anyone ...
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81 views

DFT example in textbook

There is an example of the use of the DFT formula in my textbook which I don't quite follow. The text goes as follows: Let us define the $N$-periodic and anti-Hermitian series $g_n$ where $g_n = f_n ...
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47 views

Monotonically decreasing Fourier transform

What would be the conditions on $f(x)$ such that it's Fourier transform $F(k)$ would be monotonically decreasing from $k=0$ to half range ($F(0)$ would be the maximum, and it would "fall" on both ...
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40 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
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82 views

Discrete Fourier Transform on a shifted frequency grid

I use the discrete Fourier transform in 3D to solve my model partially in real space and partially in Fourier space. The DFT pair is defined as \begin{equation} ...
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189 views

Parseval's theorem, DFT to FT

So I am working with DFTs for the first time and I have some problems. I have a discrete signal to this signal i want to apply a DFT, then I want to use the output in an integral. So $S(t)$ is my ...
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41 views

Checking if a function is in the Schwartz space of rapidly decreasing functions.

Is there any neat bi-implication other than the definition that I can use to check this? This question was motivated by a question that asked if $ f(x) = e^{-|x|^3}$ was in S. It isn't infinitely ...
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130 views

Fourier Transform over function depend on time and frequency

In my task I need to perform Inverse Fourier Transform from spectrum that depend on time and frequency arguments simultaneously. E.g., I have a discrete spectrum of some function $S(t, f)$ with $2N$ ...
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50 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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75 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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83 views

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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37 views

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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55 views

(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 ...
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117 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 ...
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183 views

Exponential decay only from one side : what translation in Fourier transform?

In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) : if $T$ is a tempered ...
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71 views

How many terms are there in a truncated Fourier series of order $N$ for a function $f: \mathbb R^n \to R$

Let $S_N(f)$ be the truncated Fourier series of order $N$ for $$ f: \mathbb R ^n \to \mathbb R. $$ How many Fourier coefficient does $S_N(f)$ contains? I'm not clear on how multidimensional Fourier ...
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39 views

Approximation of a given function by rational functions

Given a function $1/\sqrt{x^2−k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the $L_p (p=2 \ or\ ∞)$ norm of ...
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62 views

Inverse Fourier transform of the projection

$\renewcommand{div}{\mathop{\mathrm{div}}}$ Let $a(x) \colon \mathbb R^n \to \mathbb R^n$ be a smooth vector field with compact support. Let $\hat a(\xi)$ be it's Fourier transform: $$ \hat a(\xi) ...
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49 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
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26 views

Is it possible to calculate a single frequency bin in $O(\log N)$ time (considering the $N\log N$ performance of FFT algorithms)?

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...
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52 views

Question on Fourier Transform

Fourier transform on $f$: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ $\xi\in\mathbb{R}^d$. How to show that $$\hat{f}(\xi)=\frac{1}{2}\int_{\mathbb{R}^d}[f(x)-f(x-\xi')]e^{-2\pi ...
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34 views

Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
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100 views

Definition of $L^p(\mathbb T)$ with $\mathbb T$ the unit circle

I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation ...
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128 views

Accuracy of Real FFTs

I'm not sure if this is the best place to ask, please advice if not. I'm performing the following test, to check the output of real-FFTs, and I'm getting surprisingly high errors. So a real-FFT only ...
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51 views

Moments of Fourier transform

Fix a smooth $\mathcal{C}^{\infty}$ compactly supported function $f$ with the support of $f$ being the unit interval $(-1,1)$ and with $\hat{f} \geq 0$. Is it true that as $k$ goes to infinity $$ ...
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79 views

Function supported on [-1,1] with arbitrary prescribed sub-exponential Fourier decay?

Given $g:[1,\infty) \rightarrow (0,\infty)$ with $g(t) = o(t)$, does there exist $f:\mathbb{R} \rightarrow [0,\infty)$ with support contained in $[-1,1]$ such that $$ \widehat{f}(y) = ...
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159 views

Closed formula Fourier transform of complex exponential

Is there a closed formula for the Fourier transform of the function \begin{equation} f(t) = e^{2\pi i \sqrt{1-t^2}}, \end{equation} where the square root for $|t|>1$ is $i \sqrt{t^2-1}$. This ...
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145 views

Inverse Fourier-cosine transfrom

Suppose we have a function $F(x)$ given by the integral: $$F(x)=\int_{0}^{\infty}f(t)\frac{\cos(t\log x)}{t}dt\;\;\;\;\;(x>1)$$ This looks tantalizingly like a Fourier-cosine transform of ...
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154 views

Fourier transform of projection of spherical cap

I am currently trying to derive an analytical expression for the Fourier transform of the projection of a spherical cap of the unit sphere onto the xy-plane. Setting up the integration in cylindrical ...
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40 views

Fourier transform of a sequence by a matrix

Let $n$ be a positive integer and $H$ the Hilbert space $\ell^2(\mathbb Z^n,\mathbb C^n)$. For $u\in H$, denote by $\mathcal{F}(u)$ the Fourier transform of $u$, defined by $\displaystyle ...
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31 views

An alternative proof without multiplier theorem

Given $f\in \mathcal{S}'$ and $\phi\in C_0^\infty$, with $\operatorname{Supp}\phi = \{1 \leq \xi \leq 2\}$. Define $\phi_j = \phi(2^{-j}\cdot)$. We have if $\phi_k*f\in L^p$, then for all ...
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82 views

The Envelope (wave front) of a span of plane waves.

Imagine we have for every real direction $\mathbf n \in \mathbb R^3$ a plane wave, be it either quantum, eletric, magnetic, acoustic or in my case some sheets of paper I am holding up. Each of these ...
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63 views

What sequence has this Discrete Fourier Transform?

Suppose $$ x[n]= \begin{cases} x_i &, i \in P\\ 0 &, i \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_i \geq 0$. Suppose these equalities hold : $$ ...
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94 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
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49 views

A Fourier Coefficients (Series) problem

Let $\alpha$ be a number such that $\alpha/\pi$ is not rational. Prove that (1) $$\lim_{N\to\infty}\sum_{n=1}^{N} e^{ik(x+n\alpha)}=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{ikt}\,dt,$$ (2) for any ...
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65 views

The Sinc Function and the Partition of Unity Property

For this question purposes a function, $ f \left( t \right) $ is said to hold the Partition of Unity if: $$ \forall t, \ \sum_{n \in \mathbb{Z}} f \left ( t - n \right ) = 1 $$ Using the Poisson ...
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28 views

Interpolation of linear operators

If $T$ is a bounded linear operator from $L^{p_1}$ into a homogeneous Lipschitz space of order, say $\lambda.$ Further if $T$ is also bounded from $L^{p_2}$ into $L^{q}$ for some $p_1,p_2,$ and ...
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208 views

Convolution property in terms of fft (matlab)

I am working on some signal processing and I have the following data: ...
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56 views

Fourier Analysis and Groups

Consider the Fourier Transform and Inverse Fourier Transform: $$\mathcal{F} f(-s) = \int_{-\infty}^{\infty} e^{-2 \pi i (-s)t} f(t) \ dt = \int_{-\infty}^{\infty} e^{2 \pi ist} f(t) \ dt = ...
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152 views

Is this implication correct? (Proof of Bessel's inequality)

I'm just curious if this implication is correct: $$0 \le\int_a^b \left( f(x)-S_n(x) \right)^2dx \Rightarrow 0 \le \lim_{n\rightarrow \infty}\int_a^b \left( f(x)-S_n(x) \right)^2dx.$$ What needs to ...
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364 views

Help with Hermitian Symmetry and its inverse Fourier transform in MATLAB.

I have tried to impose Hermitian symmetry on the complex number $z$ which is varies with $x$. I need to take its inverse Fourier transform. A hermitian symmetry should give a real valued inverse FT. ...
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212 views

Convert to displacement time signal (from accelerometer) using velocity time signal or displacement frequency signal

Firstly, I try to use ifft to convert displacement frequency into displacement time but the ifft fails to give the original sign.The code as below: ...
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191 views

Fourier transform of convolution in a finite range

Can anyone help me evaluate the Fourier transform of of the following function, $t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $f(t) = \int_{t_0}^t ...
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56 views

Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
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57 views

Fourier Transform of $(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$

I would like to calculate the FT of the following function: $$(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$$ Any hint is highly appreciated!
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86 views

2D Fourier transform of exponentials and cosines

I would like to know the 2D FT of the following functions: 1.$$\exp\left(-\frac{(x-y+a)^2}{b^2}\right)$$ ...
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72 views

Inverse fourier transform involving exponentials

I would like to calculate the inverse fourier transform of: $$\hat{f}(k)=\exp(-a k^2+ikv)\cdot \frac{\sinh(m\sqrt{(b+ck^2+ikf)^2-d})}{\sqrt{(b+ck^2+ikf)^2-d}}$$ Any clue? Thanks!
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122 views

Discrete fractional fourier transform

I have written a code for producing matrix of fractional fourier transform with the help of eigen vectors of fourier transfom matrix. Does anyone know the elements of this matrix ( for example a 4 by ...
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93 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
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37 views

About this question,I have done something,but I can't go on.

About this question,I have done something,but I can't go on.Firstly,you can examine the validity of my work.If it is right,please go on to compute.If it is a wrong way,please give me your idea.