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: where is my mistake?

I am trying to do a fourier-transform of the function $$\psi(x,t=0) = \frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}e^{-\frac{x^2}{4\sigma^2}}e^{ik_0x}$$ My calculation is $$\int_{-\infty}^\infty ...
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51 views

Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
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97 views

Reconstruction formula for a function of moderate decrease

I want to show that, for a function $f$ of moderate decrease and $\hat{f}(\xi)$ supported in $I=[-\frac{1}{2},\frac{1}{2}]$, $$f(x) = \sum^{\infty}_{n=-\infty} f(n)K(x-n)$$ where $K(y) ...
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135 views

Finding the natural frequency of a system via a Fourier Transform

This is a mechanical engineering question with its roots in Mathematics and so I felt this was the best stack exchange site to post my question. In short, I have used a device to measure the ...
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24 views

A Question on Fourier analysis

The following is an exercise in Knapp's book, representation theory of Semisimple Groups. For $f\in C_c^\infty(\mathbb R^2-\{0\})$ with norm $$\|f\|^2=\int_{-\infty}^\infty\int_{-\infty}^\infty ...
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95 views

Is $\int_0^\infty \sin(Kx)f_K(x)\mathrm dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\mathrm dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...
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68 views

On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$ Can we use residue formula?
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24 views

Decay of Fourier coefficients of $\frac{1}{f}$

Let $\alpha > 0$ and define \begin{equation*} \mathbb{H}^{\alpha}\left[-\pi,\pi\right] = \left\{f:\left[-\pi,\pi\right]\mapsto\mathbb{R} \;s.t.\; \sum\limits_{n\in\mathbb{Z}} \left\lvert ...
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40 views

Uniform Boundedness in N of $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$

Show that $\int_0^\infty \frac{\sin(x)}{x}\,\mathrm{d}x = \frac{\pi}{2}$, and using that show that $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$ is uniformly bounded in N and ...
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104 views

Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
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52 views

Rate of convergence of Fourier series

I am having a bit of a confusion regarding convergence results. Suppose $f$ is Lipschitz, or $f \in C^\infty$ and let $S_{N}f$ be its truncated Fourier series. In the wikipedia page ...
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32 views

Fourier Transform of $sin(5t - \frac{\pi}{4})U(t+8)$

I have this function $$ sin(5t - \frac{\pi}{4})U(t+8) $$ I know the Fourier Transform of $sin(5t - \frac{\pi}{4})$, which is $$ \frac{e^{-\frac{\pi^2}{2}fj}}{2j}\left [\delta (f-\frac{5}{2 \pi}) ...
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38 views

Rearrangement of Fourier Series Sum to attain convergence

Let $f$ be a continuous function with diverging partial Fourier sums $S_N(f)(0)$ : $$ f(\theta) = \sum \limits_{k=1}^\infty \alpha_k P_{N_k}(\theta)$$ Let $f(x) \sim \sum \limits_{n=-\infty}^\infty ...
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42 views

How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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88 views

Spatio-temporal triple correlation

I would like to simplify if possible the spatio-temporal triple correlation of the following function: $$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$ where $\delta$ is the Dirac ...
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32 views

Why is this an inverse fourier cosine transform?

I would like to understand the principle of Fourier transform spectroscopy. This is explained in Wikipedia. I did all the modeling of the system and I got the same formula: $$ I(p,\tilde{\nu}) = ...
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71 views

Inverse Fourier transform of two variable function $F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}$

I am trying to find the inverse Fourier transform of: $$ F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}, $$ where $k^2 = k_x^2 +k_y^2 +k_z^2 = k_\rho ^2 +k_z^2 $ is a constant. I am getting confused as to ...
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71 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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42 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral ...
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46 views

Fourier transform evaluation

Let $x\in \mathbb{R}^n$ and $h(x)=\left\{\begin{array}{cc} 1 & ,|x|<1\\ 0 & ,|x|\geq 1 \end{array}\right.$ I want to find the Fourier transform , $\hat{h}(\xi).$ Here is how I proceed ...
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52 views

A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
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73 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
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70 views

Is the definition of DTFT using $\omega$ wrong?

I'll briefly explain this problem I faced. Let's take this simple signal: $$x(n)=\cos(\pi n)$$ The signal is not absolutely summable, however we can define its DTFT in terms of distributions. That ...
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96 views

Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
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18 views

Fourier Operator and roots of Identity operator

I have seen that if Fourier operator is defined by $$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 ...
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69 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
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92 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
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53 views

Number of roots of sine-and-cosine expression

Is there an easy proof of the following fact? Let $a_0, \ldots, a_n, b_1, \ldots, b_n$ be real numbers, not all zero. Then, the function $$a_0 + a_1 \cos x + b_1 \sin x + a_2\cos 2x+b_2\sin ...
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18 views

Fast Fourier Transform for non-trigoniometric bases

The fast fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other basis, e.g. orthogonal polynomial bases ...
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42 views

Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
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74 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

I asked yesterday on math.stackexchange a question and received no answer. Since I'm very interested in an answer, I'm reposting it here: "Let $k$ be a fixed integer, and $\mathcal{F}$ the set of ...
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76 views

Fourier- Lebesgue space and Fourier transform

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
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51 views

What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
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43 views

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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82 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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48 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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41 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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90 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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199 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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137 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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56 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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78 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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63 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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187 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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74 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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40 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 ...