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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Non zero components in DFT

I wanted to do a simple example of DFT computation using the following python code (numpy + scipy). I am posting here because I am sure my problem is more related to my comprehension of the DFT ...
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Sequence of complex $L_1$ functions with $L_1$-divergent limit of the fourier operator

In a task I should first show that $\mathcal{F}(e^{-\frac{x^2}{2 k}})=\sqrt{k} e^{-\frac{k \xi ^2}{2}}$ if $\text{Re}(k)>0$. Then they say that one may conclude that there is a sequence $(f_k)_{k ...
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311 views

Fourier transform of Kronecker deltas

I have a binary 2D image that consists of 95% black pixels with a few white pixels scattered about, and I want to convolve it with a 2D gaussian kernel. I'm hoping to exploit its sparsity to improve ...
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280 views

Fourier transform of function involving $\log$

I found the following problem which I am unable to solve. Calculate the following integral $$\int_{\mathbb{R}} \frac{d\omega}{2\pi} \log (1 + i a/\omega ) e^{-i \omega t}$$ for $a>0$ and ...
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Proving the locus of a Fourier series is a system of perpendicular lines [closed]

From "Fourier's series and integrals" by H.S. Carslaw, there is the following question: Prove the zero locus of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} \sin(n x) \sin(n y) = 0$ is ...
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692 views

FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions

In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ arising from a charge distribution $\rho(\mathbf{r})$ is $$ \Delta ...
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$f,g: \mathbb{R} \to \mathbb{C}, 2\pi $ periodic. Prove: if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$

Let $f,g: \mathbb{R} \to \mathbb{C}$ $2\pi$ periodic , Riemann integrable in $[0, 2\pi]$. I need to prove that if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$. We ...
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Prove: Fourier series of $e^{\cos x} \sin (\sin x)$ is $\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$

I'd love your help with proving that the following series $$\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$$ is the Fourier series of $e^{\cos x} \sin (\sin x)$. I tried to find $\hat f(n)$ using ...
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507 views

Relation between Fourier transform and Fourier series

Let $f$ be a function on $\mathbb R^n$ whose Fourier transform $\hat f$ exists. Is there any relation between integrability of $\hat f$ and summability of the series $\sum_{n \in \mathbb Z^n} \hat ...
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On rate of decay of integrals (fourier transform)

Given, $f(x)$ is a monotonically decreasing function, and $f(x) \geq 0$ for all $x \in [a,b]$. Suppose that for $f(x)$ the following holds (Riemann-Lebesgue Lemma): $$\lim_{n\to\infty} \int_a^b ...
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How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
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197 views

$f,g$ are two continuous functions with period$=1$, are the Fourier coefficients $f*g=f(n)g(n)$?

Let $f,g$ be two continuous functions with period$=1$. Are the Fourier coefficients of $f*g$ are given by the products $f(n)g(n)$ (of the $n$-th coefficient in each series)? Thanks!
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81 views

question on fourier transform.

I ask myself what $$ {\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)'(\xi) )(s) $$ is. If it was just about $$ {\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)(\xi) )(s) $$ it would ...
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271 views

Conditions for a finite Fourier series

Under what circumstances is the Fourier series of a function guaranteed to have a finite number of coefficients?
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186 views

Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$?

Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $$\tilde f(k) = \frac{\mathcal ...
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165 views

How many terms in a series expansion

General: If $f \in C^1$ is a periodic function defined over some multi-dimensional space, then it should be possible to express $f$ as a FINITE fourier series. is this true of any periodic basis? is ...
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373 views

Approximation of smooth periodic functions by trigonometric polynomials

I know that there is a following version of the Weierstrass approximation theorem for functions on $[a,b]$. For every $f:[a,b] \rightarrow \mathbb{R}$ there is a sequence $(P_n)$ of polynomials such ...
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90 views

A condition for $\hat f$ to be integrable

Let $f \in L^1 (\mathbb R^n)$. Suppose that $f$ is continuous at zero and that the fourier transform $\hat f$ of $f$ is non-negative. Does this imply that $\hat f \in L^1$ (and hence, by the inversion ...
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101 views

Dealing with integrals and Fourier transforms.

I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int ...
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1k views

n-dimensional Fourier integral

let be $ \int_{R^{n}}dVf(r)e^{i(k.r)} $ the n-dimensional Fourier integral. $ dV=dxdydz.... $ the volume and $ (k.r)= \sum_{n} k_{n}.x_{n} $ is the scalar product of the position vector 'r' and the ...
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Pointwise, uniformly and absolute convergence of Fourier series

I'd really love your help with this one: I got this Fourier series for $f(x)=x$ in $[-\pi,\pi]$: $$\sum_{1}^{\infty}\frac{2(-1)^{n+1}\sin(nx)}{n}$$ and I need to check if it's (i) pointwise ...
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Why does the following Fourier series does not converge for $x \in R$, and does for $x \in [0,2\pi]$?

I would really love your help with the following facts that I can't understand. I can't understand why the following Fourier series does not converge: $$\sum_{0}^{\infty}\frac{e^{inx}}{n^2}.$$ 1.If ...
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Solving the heat equation with Fourier Transformations

Can anyone help me with this IVP heat equation problem? I have $$u_t-u_{xx}=g(x,t)$$ where $x \in \mathbb{R}$, $t>0$, $u(x,0)=0$ So i've found by taking a Fourier transformation that ...
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440 views

Fourier series - what is the difference between the Fourier series of $f(x)=x$ in $x \in [0,2\pi]$ and in $x\in [-\pi,\pi]$?

I was asked to compute the fourier series of $f(x)=x$ in two different intervals: $x \in [0,2\pi]$ and $x\in [-\pi,\pi]$. What is the real difference between the two series, cause we know that for ...
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136 views

Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component. Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get ...
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355 views

Fourier transform (logarithm) question

Can we think, at least in the sense of distribution, about the Fourier transform of $\log(s+x^{2})$? Here '$s$' is a real and positive parameter However ...
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Intuition behind decay of Fourier coefficients

Many other posts have discussed the standard result that the smoothness of a function is related to the rate at which its Fourier coefficients decay. For example, there are proofs that show that if ...
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When is the convolution with a tempered distribution again a tempered distribution?

If $f$ is a Schwartz function on $\mathbb R^n$ and $g \in L^1(\mathbb R^n)$, then if $g$ is the Poisson kernel, is $f\ast g$ a Schwartz function? are there any known sufficient conditions on $g$ to ...
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218 views

On covering lemma and Calderón–Zygmund decomposition

I am working on something which needs to understand covering lemmas and Calderón–Zygmund decomposition. These type of lemmas are as in the following link ...
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81 views

Deduce the global differential equation from the pointwisely defined equation in Fourier space

Let $G\in \mathcal{F}(\mathbb{R}^{n+1})'$ be a distribution on the space of spatial Fourier transform'able function, ie an $L^1_{\mathrm{loc}}(\mathbb{R^{n+1}})$ function, $G = G(t,\xi)$. Assume ...
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How can I use the time-frequency uncertainty principle?

I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second. I have been told that, if I want to ...
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202 views

How can the FT be used to find the FS of the periodized signal?

Let's say I know the Fourier transform of a function that is $0$ outside some interval, for example a triangle wave. How can this be used to find the Fourier series of the related periodic function, ...
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979 views

Convergence of Fourier series for $|\sin{x}|$

I was solving this question I saw in a textbook. The question is : Calculate the Fourier series for $ f(x) = |\sin x| $ for $-\pi \leq x \leq \pi$. Which I had $ f(x) = \frac{a_{0}}{2} + \sum ...
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Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?

Assume that the functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are integrable and equal to zero on $(-\infty,0)$, (i.e $f,g \in L^+$). Then by Titchmarsh's theorem: $f*g$ is zero almost everywhere ...
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Tempered Distribution Calculation

I hope you don't mind that rather than typing this question up I took a screenshot and uploaded it: http://www.math.ualberta.ca/~schlitt/stackexchangeproblems/tempered-distributions-calc.png The ...
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91 views

Asymptotic boundary on Fourier coefficients of absolutely continuous function

Let $f$ be absolutely continuous. Prove that $\hat{f}(n)=o\left(\frac{1}{n}\right)$. Any hint will be appreciated, thanks.
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524 views

A generalized version of the Riemann-Lebesgue lemma

Let $f \in L^1 [0,2\pi ] $ and let $ g $ be bounded and $2\pi$-periodic. Prove that $$ \hat{f}(0)\cdot\hat{g}(0)=\lim_{n\to\infty}\frac{1}{2\pi}\intop_0^{2\pi}f(t)g(nt)dt $$ where $\hat{f}(0)$ ...
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247 views

Convergence of Fourier Series

Is there an $f\in L^1(\mathbb{T})$ whose Fourier series converges a.e. on $\mathbb{T}$ but not a.e. to $f$?
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359 views

Problem concerning continuous probability distribution

How do you prove that the real part of the characteristic function of the continuous probability distribution $f(x)$ is a characteristic function, but the imaginary part is not? The second part is ...
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195 views

Type of *the* discrete sine transform

When one refers to "the" discrete cosine transform (DCT), it is generally assumed that one means the type II DCT, where the input array $x_k$ ($0 \leq k < N-1$) is even about $k=-1/2$ and ...
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419 views

Applying Plancherel's theorem to a simple function

Let $f(t) = \frac{1}{25}e^{-(t-11)^2}-\frac{1}{36}e^{-(t-13)^2}$. Using the Wiki definition of the Fourier transform pair, I calculated $\hat{f}$ in Mathematica as $$\hat{f}(\omega) = ...
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Finding Transfer Function with an intermediate variable

How do I find the transfer function (using the bilateral z-transform) of the problem below. A stable LTI system with input x[n] and output y[n] is modeled by the difference equations c[n] + ...
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548 views

Fourier transform in Mathematica

When I calculate the Fourier transform of the function $$f(t) = \mathrm e^{-|t|/\tau} \text{ with } \tau >0$$ in Mathematica once via the function FourierTransform and once by hand, I get different ...
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Slowly varying vectors and coefficients of a sine transform

Let $u_k$ be the vector in $\mathbb{R}^n$ whose $i$'th entry is $\sin(\pi ki/n)$. The vectors $u_1,\ldots, u_n$ are orthogonal and correspondingly every vector in $\mathbb{R}^n$ can be decomposed as a ...
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Sample Sinusoidal Signal, Determine its Frequency

I am tasked with simulating an analog signal, and sampling it isochronously to determine its frequency. By simulate I mean we are using math.c's sin function, so no need (I think) to dive into the ...
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Does the phrase “instantaneous frequency” make sense?

I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a ...
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348 views

Is there an expression for the inverse Fourier transform of a log-normal function?

i.e. is there a simple solution for the following integral? $$\int_{-\infty}^{\infty} \exp(-\log^2(|\omega|/\omega_0)) \; \exp(i \omega t) \; d\omega$$ where $\omega_0 > 0$ Failing that, is ...
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178 views

If the Fourier transform of a signed measure is identically zero, is the same true of the measure?

I am trying to prove the following seemingly obvious fact: Let $\mu$ be a finite signed measure on $\mathbb R$. Suppose that $\hat\mu(u) = \int_\mathbb R e^{iux} d\mu(x) = 0$ for all $u$. Then ...
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Fourier transform of $f(x) = e^{-x^2}$ [duplicate]

Possible Duplicate: Fourier Transform of complicated product: $(1+x)^2 e^{-x^2/2}$ I calculate the Fourier Transform of $f(x)$ by $$\mathbb{F}(t) =\int_{-\infty}^{\infty}e^{-x^2} \cdot ...
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368 views

Fourier-like expansion of a closed curve in 2D

Fourier expansion can be used to represent any periodic function in one variable. Closed surfaces in 3D can be built out of spherical harmonics. Is there a similar expansion to represent a curve of ...