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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Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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18 views

Wiener's Tauberian theorem for positive-value functions

The elementary Wiener's Tauberian theorem goes as follows: Theorem (Rudin, Functional Analysis Th. 9.5): Suppose $K \in L^1(\mathbb{R}^n)$ and $Y$ is the smallest closed translation-invariant ...
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39 views

Completeness of a set of functions

We know Fourier basis $e^{i k \xi \cdot x}$ forms a $L^2$ complete basis, $|\xi| = 1$ and $k\in \mathbb{N}$. I would like to know if $k$ belongs to a open interval, do we still have the completeness. ...
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36 views

Bound on the following integral using integration by parts

Let $W: \mathbb{R}^s \rightarrow [0,1]$ be a smooth function supported on $[0,1]^s$ that satisfies $$ \left| \frac{\partial^k }{\partial x_{i_1} \cdots \partial x_{i_k}} W(\mathbf{x}) \right| \leq ...
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59 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
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1answer
36 views

Fast Fourier Transform as Matrix Factorization

I'm given a vector of length 4 and three matrices that correspond to a Fast Fourier Transform, I'm not exactly sure which one, but I guess it's supposed to be the Cooley-Tukey algorithm. Here is the ...
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17 views

How to compute a complex-valued function specifying its amplitude and the ampltiude of its Fourier transform

I want to set the amplitude of a complex function (independent of its phase), and also the amplitude of its Fourier transform (again independent of its phase). Given these two functions (amplitude of ...
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25 views

Equivalence of $H^{1/2}(S^{1})$ norm with integral

I've been attempting to show that the Sobolev space norm $$\|f\|^{2}_{H^{1/2}} := |\hat{f}(0)|^{2} + \sum_{n \neq 0} n |\hat{f}(n)|^{2}, $$ for $f$ on the circle, is equivalent to the integral $$I(f) ...
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22 views

Fourier sine series

Compute the Fourier sine series of $f(t)=t$ over the interval $[1,3]$. The question I have is that over $[-L,L]$, the cosine series is $0$ but does this still apply over the interval $[1,3]$? So ...
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45 views

Solving a simple Schrodinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occuring in the simplest Schrodinger equation possible: $\partial_t \psi ...
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1answer
43 views

Some questions Regarding Fourier Transform

I have some questions regarding Fourier Transform. I have studied a lot of books for Fourier Transform and currently, I am confused weather I should use Riemann integral or Lebesgue integral for the ...
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12 views

Find appropriate sine and cosine terms and coefficient formulas to approximate a function on [1,3]

This is probably a ridiculous/simple question but I am still having trouble! Find appropriate sine and cosine terms and coefficient formulas to approximate a function on [1,3]. I understand how to ...
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13 views

How to create obtain aliased version of $f(t)$ by upsampling whenever $f(t)$ at every $t$ is available

Suppose there is original complex-valued $f(t)$ with $t$ ranging from $-\infty$ to $\infty$. It is possible obtain samples from original $f(t)$ at every $t$ with some negligible error. If one ...
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110 views

Evaluating the integral $\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$

I want to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$$ for all $t \in \mathbb{R}$. I would preferably do it using the tools of complex analysis, but since I ...
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14 views

Proving Div-Curl Lemma through Paraproducts

I want to to prove the classical div-curl lemma of Coifman-Lions-Meyer-Semmes in the following form: Div-Curl Lemma. Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a Schwartz function such ...
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25 views

Fourier Transform of 1/x

I am trying to calculate the Fourier Transform of $h(x)=1/x$ So I was thinking: Since the FT of $g(x)=sign(x)$ is $G(f)=1/(i\pi f)$ , then using property of duality gives that $G(x)=1/(i\pi x)$ ...
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15 views

Writing matrix representation of multiplication operator

For a given $m(x)\in L^2(0,1)$, let's write the multiplication operator $M\colon L^2(0,1)\longrightarrow L^2(0,1)$ as $Mf(x)=m(x)f(x)$. To write the matrix representation of this operator we need a ...
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20 views

Fourier shift theorem

I have a problem that can be roughly simplified as follows. Let's have a function of one variable in the Fourier space that has this form $F(k)=G(k)/k$, where $k$ is the wavevector. I sample the ...
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1answer
43 views

Fourier series of: $[\log(\sin x)]^2$

What is the Fourier expansion of: $${ \left[ \log\left( \sin x \right) \right] }^{ 2 }$$ This is a well known Fourier series: $$-\log(\sin x ...
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1answer
28 views

How out of fourier line calculate below sums? [duplicate]

I put it down there (fourier line) since www page does not work. http://imgur.com/gallery/53n9qIp/new ..... 3 sums to calculate:
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26 views

Let $A^T = A$, find $\mathcal{F}^\pm e^{-\langle Ax, x\rangle}$.

Let $A^T = A\in \mathbb{R}^{n\times n}$ and $\langle Ax,x\rangle \geqslant \alpha \|x\|^2$ for a certain fixed $\alpha>0$. Prove: 1) $e^{-\langle Ax,x\rangle} \in L^1(\mathbb{R}^n)$ ...
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16 views

Let $\phi: x\mapsto Ax+b$, calculate $\mathcal{F}^\pm (f\circ \phi)(y).$

Let $\phi: \mathbb{R}^n\to \mathbb{R}^n; x\mapsto Ax+b$, where $A$ is non-singular and $b\in \mathbb{R}^n$. Show: $$\mathcal{F}^\pm(f\circ \phi)(y) = \frac{1}{\det{A}}e^{\mp i \langle ...
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1answer
41 views

evaluate a Fraunhofer diffraction integral

I need to evaluate the following integral: $$\int_{-\infty}^\infty \text{rect}(\frac{x'}{2w}) \exp\left (\frac{im}{2}\sin \left(k_Gx' \right) \right )\exp\left (-\frac{ik}{z}xx' \right )dx'$$ where ...
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1answer
29 views

Fourier transform this convolution

So we have that $$ g(t) = \frac{1}{T}\int_{t-T}^{t}f(\tau) d\tau $$ for $T>0$ and I'm asked to show that $\left| \hat{g}(w) \right|≤\left| \hat{f}(w) \right|$. The hint I get from the question is ...
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1answer
30 views

Solve ODE by using Fourier Transform

$$- u''(x)+u(x)=f(x)$$ for every $x \in \mathbb{R}$ with $\lim_{|x| \to \infty}u(x)=0, \lim_{|x| \to \infty}u'(x)=0$ and $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$. I need to solve this by using ...
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1answer
16 views

best approximation of $f$ in $L^2([0,2\pi])$ as a linear combination of $\sin(kx)$ (with $k∈\{1,2,3,\ldots,10\}$

Let $f$ be a $2\pi$ periodic function. Assume that $f$ is quadratic integrable in the interval $[0,2\pi]$. Consider $f$ as a vector in the Hilbert space $L^2([0,2\pi])$. Give, based on the Fourier ...
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70 views

Interchange of integral and infinite sum

I'm reading Fourier analysis an introduction by Stein, and I have a problem from section 5.4 about the Poisson kernel. For the following equations \begin{align} ...
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2answers
25 views

Properties of $L_{2}$ Fourier transform

I have a question regarding the $L_{2}$ Fourier transform. I know the fourier operator can be extended to functions in $L_{2}$, and I know Plancherel's formula works as well for functions in $L_{2}$. ...
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36 views

$1 / (2 \pi)$ factor in Fourier transform

I have been unable to see why the $1 / (2 \pi)$ appears in Fourier transform. Would you please justify it to me? Problem: Let $$ f(x) = \int_{-\infty}^{\infty} \mathrm{d} k \, e^{ikx} \tilde{f}(k) ...
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14 views

The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq ...
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1answer
19 views

Why does this Fourier inner product equal this sum?

This is part of a derivation in a text that I am struggling to follow. It says that if we write $e_k(t) = e^{2 \pi i k t}$ then $$\langle \sum_{n=- \infty}^{\infty} \langle f, e_n \rangle ...
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1answer
29 views

Fourier transform of a distribution null on [-1,1]

Here is an interesting problem : Let $f \in \mathcal{C}^0 ( \mathbb{R})$ bounded, and $T_f \in \mathcal{S}'(\mathbb{R})$ defined by $\displaystyle \langle T_f, \phi \rangle = \int_{\mathbb{R}} ...
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1answer
37 views

Calculate $\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}$ using Fourier transformations

Calculate $\left(\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}\right)(y)$ using Fourier transformations. I have found a solution, but my method was very long. How could I shorten the solution? ...
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2answers
35 views

Calculating the convolution of a piecewise function

Let $$f(x) = \begin{cases} \frac{1}{2}, & \text{if $\rvert x\lvert \le 1$ } \\ 0, & \text{otherwise} \end{cases}$$ I want to calculate the convolution of $f$ with itself. I am ...
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13 views

Quesion about Parsevals formula for Fourier-Legendre Series

Question: A function $f(x)$ defined on $(-1,1)$ can be expanded as $$ f(x) \backsim \sum_{n=0}^{\infty} c_nP_n(x) $$ What do Parsevals formula look like for this expansion? My solution: Ok so I ...
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2answers
47 views

Calculating the Inverse Fourier Transform of $\frac{1}{\sqrt{2\pi}k}\sin k$

This used to be part of a longer question that I posted earlier but since that question seemed to long I decided to split it up. Given the function $$f(x) = \begin{cases} \frac{1}{2}, ...
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3answers
58 views

Show that the sum equals $0$ according to Cesàro

I'm stuck at some problems in my Fourier Analys course, maybe you got a clue. If $x \neq n \cdot 2 \pi$, then $$ \frac{1}{2}+ \sum_{k=1}^{\infty}\cos(kx) = 0 \tag{C,1} $$ Solution: So I know where ...
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28 views

Integral of $\sin (e^{x^2})e^{-x^2+ix\lambda}$

Trying to solve this problem : Is $T$ invariant under Fourier transform ? Where : $T= \{f\in \mathcal{C}^{\infty} (\mathbb{R}), \forall n \in \mathbb{N}, |x|^nf(x) \to 0 \; \text{when}\; |x| \to ...
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29 views

Deriving the coefficients during fourier analysis

I'm self-studying Fourier transforms, but I'm stuck on a basic point about integration during the derivation of an expression for the coefficients of the Fourier transform. For a function of period ...
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24 views

Complex coefficient in Fourier series

Why can $$\sum_{n=1}^N a_n \frac{e^{2 \pi i n t} + e^{-2 \pi i n t}}{2} + b_n \frac{e^{2 \pi i n t} - e^{-2 \pi i n t}}{2i} $$ be written as $$ \sum_{n=-N}^N c_n e^{2 \pi i n t}$$ for some setting of ...
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1answer
24 views

Ill-posed integral equation problem using Fourier Transfom

By using the Fourier Transform, show that the following equation $$\int_{-\infty}^{+\infty} K(x-y) g(y) dy = f(x), \qquad -\infty < x < \infty$$ is ill-posed. For overcoming ill-posedness of ...
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101 views

What is the Fourier transform of $\exp(2 \pi i / x)$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
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1answer
26 views

How to find the Fourier's coefficient $a_n$ of the Fourier's series of $\sin(x)$ on $(0,\pi]$, $0$on $(-\pi,0]$

Considering $g(x)$, periodical with a period of $2\pi$ defined by \begin{equation*} g(x)= \begin{cases} 0 & \text{for $x \in (-\pi;0]$} \\ \sin(x) & \text{for $x \in ...
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1answer
40 views

Strange equation after derivative

Assume Fourier integral of this function $ f(x)= \left\{\begin{matrix} 1 , \left | x \right |< 1 & \\ 0 , \left | x \right |> 1 & \end{matrix}\right. $ is this: $$ f(x)= \frac{2}{\pi ...
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1answer
27 views

How to show that the Fourier's series of $f(x)=x$ uniformly converges?

How to show that the Fourier's series of $f(x)=x$ uniformly converges? After finding its coefficient, I got: $$\sum\limits_{n=1}^{+\infty}\frac{2(-1)^{n+1}}{n}\sin(nx)$$ I showed the pointwise ...
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33 views

Pointwise version of Fejer's theorem (convergence of Cesaro means)

Prove a pointwise version of Fejer's theorem: If $f\in \mathscr{R}$ and $f(x+),f(x-)$ exist for some $x$, then $$\lim \limits_{N\to \infty}\sigma_N(f;x)=\frac{f(x+)+f(x-)}{2},$$ where ...
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1answer
22 views

Why are the Fourier's coefficient on $0,2\pi$ and $-\pi,\pi$ the same?

I was given the Following Fourier's coefficient and I was happy with it: $$\left\{ \begin{array}{ll} a_n(f)=\frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos\left(nx\right)\,\mathrm{d}x\\ b_n(f) = ...
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15 views

Why is the square of an image not equvalent to taking the autoconvolution of an image in fourier space?

Al right I know that in order to multiply in the normal domain I have to take the convolution in Fourier domain but when I do so in matlab and invert the result then I come up with nothing but a lot ...
2
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2answers
95 views

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform?

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform? I had an idea in my mind. To use the $\text{sinc}$ function and take its inverse Fourier Transform or something like that. ...
2
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1answer
25 views

Existence of $L^{1}(\mathbb{R}^{n})$ Function Defined via Functional Equation

Perhaps, I'm reading the problem statement wrong, and it's not asking for existence, only uniqueness; but in any case... Problem. Let $g\in L^{1}(\mathbb{R}^{n})$, $\|g\|_{L^{1}}<1$. Prove that ...