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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Proving $f(t) = \sum\limits_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}$ is Lipschitz

I have a homework problem which consists of two parts, the first of which I have been staring at for several days with very little (constructive) progress. I need to show that the function $$f(t) = ...
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Can a DTFT have a period different of $2\pi$?

I think almost everything is in the title. In an exercise, a DTFT is given : $$X(e^{j\Omega}) = \sin(\Omega) + \cos(\Omega/2)$$ The period of this DTFT is $4\pi$. Is that possible? I mean, the ...
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Fourier transform of a product of two rect functions

I am trying to evaluate the following expression $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$ which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, ...
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43 views

Fourier transform in three dimensions getting out of hand

I have the following integral I wish to compute, it transforms a quantum position wave function into momentum space: $$\phi(\mathbf p)=\int\frac{\mathrm d^3r}{(2\pi\hbar)^{3/2}}e^{-i\mathbf{p\cdot ...
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Is there an explicit formula for the Fourier transform of $(z-|\xi|^{\alpha})^{-1}$ on $\mathbb{R}$?

Let $0<\alpha< 2$, and $z=\lambda+i\mu$, where $\mu\ne 0$. Consider the following Fourier transform on $\mathbb{R}$ $$g(z,x)=\int_{\mathbb{R}}\frac{e^{ix\cdot\xi}}{(z-|\xi|^{\alpha})}d\xi$$ ...
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450 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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Multiple convolutions

Let $\phi(x)=1$ on [0,1] and 0 anywhere else. Is there a was to say what the support of the n-times convolution of phi with itself, that I wann to denote by $B_n$, is? In especially is it possible to ...
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36 views

Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
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29 views

Fourier series sketching

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ...
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59 views

Solving the Wave Equation using Fourier Transforms

The problem is: \begin{equation} u_{tt} -c^2u_{xx} - a^2 u = 0 \end{equation} with $\hspace{2mm}-\infty < x < \infty $, $ \hspace{2mm} u(x,t) \hspace{2mm}$ bounded as $ x \rightarrow \pm ...
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Fourier transform and $L^1,$ $L^2$ convergence

Let $\phi \in L^2(\mathbb{R})$ and $\hat{\phi}$ be the Fourier transform of $\phi.$ Does this mean that $\sum_{m \in \mathbb{Z}} |\hat{\phi}(x + 2 \pi m)|^2$ converges in the $L^1$ sense on each ...
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50 views

Calculate $\int_{-T}^T {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau$.

Let $\lambda$ and $\nu$ be real numbers. Then, it has \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= ...
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292 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
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What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
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48 views

How i can find the fourier transform of $\frac{\sinh(ax)}{\sinh(\pi x)}$ where,$ |a| < \pi$

Using a rectangular contour in the complex plane, bypassing the poles at $z=0$ and $z=i$, i got $$\int_{-\infty}^{+\infty}\frac{\sinh(ax)e^{ikx}}{\sinh(\pi x)}dx - ...
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154 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
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360 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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382 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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21 views

Please explain the $*$-operator in $x^*[n]$

I have to calculate the $IDFT$ for a signal $y_2[n]$: \begin{align*} y_2[n] = DFT^{-1} \Big\{ \Im m \{ \tilde{X}[k] \} \Big\} \end{align*} and I am allowed to use some formulas from a collection ...
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33 views

Fourier transformations and the inversion formula

I am working through the above question in preparation for an upcoming exam. I have completed part (a) and quoted the inversion formula for part (b), but I cannot see how to find a form to evaluate ...
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34 views

Books covering the basics of Fourier Transform for image processing

I am studying computer science and I would like to improve myself on the subject of image processing. There is just one obstacle, Fourier transformations. Is there any material which covers basics of ...
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133 views

Is there a way to relate prime numbers and the fourier transform

According to what I know about Fourier transforms, any continuous periodic signal can be represented as a combination of sine and cosine functions. To me, this looks analogous to the "Fundamental ...
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Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
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80 views

What are Basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But what are basis images actually?
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Fourier inversion Lemma (Lars Hörmander)

I always like to have more than one proof for the same theorem. The other day I was browsing through my copy of Lars Hörmander's book on PDE (volume 1). When proving the fourier inversion formula (on ...
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31 views

Fourier Transform method to solve a parabolic PDE in $\mathbb{R^n}$

Let $b\in \mathbb{R^n}$ and $c>0$. Assume $g \in C(R^n)$ has compact support and $f = f(x,t)$, $f \in C_1^2(R^n \times [0,\infty))$ has compact support. I'm trying to solve the following IBP via ...
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Fourier transform and recursion

Starting with the first derivative of a continuous general function $u(x)$, say $\frac {du}{dx}$ and I take the Fourier Transform of it, I know the solution is $i\cdot k\cdot U$, where $ U$ is the ...
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19 views

Relationship between Inverse Fourier and Inverse Laplace Transform?

Suppose we are given a fourier transform $$ F(\omega) = \frac{1}{\omega^2+4} $$ Can we use inverse laplace tranform by taking $i\omega = p$ to find the inverse fourier transform? I did this and got ...
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71 views

Have some queries about Fourier Transform

I have some queries about the Fourier transform In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational ...
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How to choose $f\in C_{c}^{\infty}(\mathbb R)$ so that $ \hat{g}\in \ell^{1}(\mathbb Z)$, where $g(x)=f(x+2\pi)$?

Suppose $K$ is compact proper subset of $[0, 2\pi]$ with the property $K\subset V \subset [0, 2\pi]$ where $V$ is open . My Question: Is it possible to choose $f\in C_{c}^{\infty}(\mathbb R)$ such ...
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Why are these functions called “kernels”?

In the last years while studying numerical analysis I came across different "kernels", like the Dirichlet Kernel $$D_n(x) = \sum_{k=-n}^n e^{ikx}$$ the Fejer-Kernel $$F_n(x) = \frac{1}{n} ...
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How do I find the Fourier transform of $\mathcal{F}[\log(a^2+s^2)](s)$

For $a>0$ i have managed to show that this is the Fourier transform of the function. $$ \mathcal{F}[e^{-a|x|}](s) = \frac {2a}{\sqrt{2{\pi}}(a^2+s^2)}. $$ How do I now use this to find the ...
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1answer
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Show that an analytic function in a strip has a complex Fourier expansion

I am a self-studier, and this is a homework problem from a course in Complex Analysis. First, let me give a plug for the course as it is outstanding. Taught by Jerry Shurman at Reed. Great lecture ...
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Mode of convergence for partial Fourier series in $B( L_p[-\pi; \pi ])$, $p \in [1; \infty]$

Which mode of convergence takes place, strong, weak, or in norm? If we have sequence of continuous linear operators in $L_p[-\pi; \pi]$: $(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) ...
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280 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
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26 views

Convergence of $S_{n}(f;t)$

Let $f \in L^P(T)$ for some $p>1$. If $n$th Fourier partial sum $S_n(f;t)$ converges almost everywhere as $n \rightarrow \infty$, does the limit have to be $f(t)$ almost everywhere? I am trying to ...
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Decay of Fourier Transform

I encountered the following statement, and I cannot see why it is true(if it is). Suppose $f$ is a nonnegative, bounded, compactly supported and measurable function with the following properties: ...
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Fourier Transform - Conditions

I've been looking at the topic in the Farlow book on PDEs, and it says: "The major drawback of the Fourier Transform is that not all the functions can be transformed; for example even simple ...
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Fourier Analysis question on convolution [closed]

Let $f(x) = \operatorname{sinc}(x)^2$. Find $(f*f)(x)$? This is what I tried $f(x)=\mathrm{sinc}(x)^2$ $$ \begin{align} f\ast f(x) &=\int f(u)\cdot f(x-u)\,\mathrm{d}u\\ ...
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Frequency domain analysis with limited time serie

I have a 1Hz measurement signal for a length of about 60 seconds. I strongly suspect/expect there to be 6 DOF +1 frequencies to be disrupting this signal (due to the physical nature of my ...
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Maximal Magnitude of Fourier Transform

Assume you are given a length-$n$ vector $x\in\mathbb{C}$ with elements $x_0$ through $x_{n-1}$. Define the Fourier transform of $x$ as $$ X(e^{j\theta}) = \sum_{k=0}^{n-1} x_k e^{-jk\theta}. $$ I'm ...
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spherical wave expansion

In the paper, Sheen, David M., Douglas L. McMakin, and Thomas E. Hall. "Three-dimensional millimeter-wave imaging for concealed weapon detection." Microwave Theory and Techniques, IEEE Transactions ...
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What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different ...
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Deriving existence of classical Fourier transform via the space of temperate distributions

If for some measurable function $f:{\bf R}^n\rightarrow{\bf R}$ the functional $h\mapsto\int fh$ is in ${\scr S}'$ (space of temperate distributions) and there exists some measurable $g$ such that the ...
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Does a Plancherel Style Theorem for the Hardy Space $\mathbb{H}^2$ on the Unit Circle Exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathbb{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
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1answer
45 views

Evaluate two dimensional frequency domain for single point

I need to compute one specific value in the original domain from the 2D frequency domain data I have. I can't just use IFFT for a whole set for performance reasons. I know how to do this in 1D by ...
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Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
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1answer
71 views

How could I continue to show the inequality?

Let $g: [0, \pi]\rightarrow \mathbb{R}$ a $C^{\infty}$ function for which the following stands: $$g(0)=0 \ \ , \ \ g(\pi)=0$$ I have to show that $$\int_0^{\pi}g^2(x)dx \leq ...
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34 views

Differentiate between Fourier analysis and Fourier decomposition

I am a beginner. I am confused between two terms i.e. Fourier analysis and Fourier decomposition.I don't understand when to use Fourier analysis term and when to use Fourier decomposition term. It ...
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1answer
31 views

How can we find the sums ?

We have the function $$g: [0, 2\pi] \rightarrow \mathbb{R} \\ g(x)=\frac{(x-\pi)^2}{4}, x \in [0, 2\pi]$$ I found that the Fourier series of $g$ is the following: $$g \sim ...