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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Calculation of Fourier transform

How to calculate the Fourier transform of $f(x)=x$. I know using the formula $f(\varepsilon)=\int_xe^{-ix\varepsilon}x \, dx$. But I have problem calculating this complex integral.
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0answers
10 views

Asymptotics of the Fourier transform of a non-analytic function

I have to find the asymptotic behavior of the Fourier transform of a function that is non analytical, but has a cusp. Given $$f(p)=\frac{1}{|p|^\sigma+1}$$ with $\sigma \in \mathbb{R}$, $\sigma >0$,...
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0answers
34 views

FourerIntegrals

Trying to understand a proof about Fourer Intergrals in my book. I can't understand how they got formula (3). As you can see delta W goes to $0$ then $L$ goes to infinty. Doesn't that mean that we get ...
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0answers
32 views

Different Alternate Representations of Functions

Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions? For example, I know of two such ways 1) Taylor Series Expansion 2)...
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27 views

Fourier Transform Properties Established

I have a function, f, in $L_1(R)$. I need to establish the following fourier transforms and I don't know how to do so. Can anyone guide me through one of these that would then help me get the other ...
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4answers
420 views

Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?

If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ ...
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29 views

Describe the Difference as a suitable Orthonormal Family [on hold]

Let $$V_1 = \{f:\mathbb R \to \mathbb C: f (t) = \sum_{k = -\infty}^{\infty}a_k~ \chi_{[k, k+1]} (t)\}$$ with $\displaystyle\sum_{k= -\infty}^{\infty}|a_k|^2 < \infty$ and $$V_2 = \{g:\mathbb R \...
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1answer
104 views

Fourier transform property(uniformly converges) proof

Suppose that f is a 2π-periodic function that satisfies the estimate \begin{equation} |f(x)-f(y)|\leqslant M|x-y|^\alpha \end{equation} for an 0< $\alpha$ <1 Show that $S_n(x)$ converges ...
2
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1answer
74 views

Lacunary Fourier series and Hölder continuity at a point on the circle

Let $(\lambda_{n})$ be lacunary (i.e. $\exists$ constant $q>1$ such that $\lambda_{n+1}>q\lambda_{n}$ for all $n\in\mathbb{N}$); $f\in L^{1}(T)$ with Fourier series $\sum_{n\in\mathbb{N}}a_{n}\...
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0answers
16 views

Connections Between Fourier basis and Discrete Fourier Tansform

I need some help to understand discrete Fourier Transform. Suppose we have a toy daily temperate data. The Fourier basis expansion can be $$ \begin{bmatrix} 1&\cos 0 & \sin 0 \\ 1&\...
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45 views

On the frequency of primes.

Condsider the ("2D") sequence $(mn)_{m,n>1}$ (with $m,n \in \mathbb{N}$). It contains all natural numbers (in various multiplicities) except the primes and the number $1$. Now we construct a ...
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0answers
19 views

Fourier coefficients sum converges

Let $f:[−\pi, \pi] \to \mathbb R$ be continuous with $f(−\pi) = f(\pi)$ extend $f$ to $\mathbb R$ by periodicity conditions. Assume that for some $\alpha > \frac{1}{2}$ we have $$\displaystyle\...
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1answer
49 views

Is going from $V_{\text{L}} = L \frac{di_{\text{L}}}{dt}$ to $\frac{ V_{\text{L}} } {i_L} = L \frac{d}{dt}$ allowed?

The Laplace transform of $\frac{d}{dt} f(t)$ would be sF(s), when f(0)=0, which is something you can find in a Laplace transform table. If there is a rule that prohibits mathematical operations from $...
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1answer
1k views

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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0answers
43 views

Fourier transform of $1/z$

What is Fourier transform of the complex function $f: \mathbb C \to \mathbb C$ defined by $f(z)=1/z$? I want to know how to interpret the integral. Is it equal to $\hat{f} (\xi)= \int_\mathbb C \frac {...
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0answers
30 views

Fourier transform in complex analysis

What does convolution means in complex analysis? In particular, I want to calculate $\varphi \ast 1/z$, where $\varphi$ is the characteristic function of unit ball in $\mathbb{C}$, i.e. $\varphi= \...
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3answers
56 views

find $f\in L^2([0,\pi])$ such that its $L^2$ distance to $\sin(x)$ and $\cos(x)$ are both bounded by specific constants

I want to find all $f\in L^2([0,\pi])$ such that $$ \begin{align} \int_0^\pi\lvert f(x)-\sin(x)\rvert^2\,dx &\le \frac{4\pi}{9}\\ \int_0^\pi\lvert f(x)-\cos(x)\rvert^2\,dx &\le \frac{\pi}{9}\\ ...
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1answer
36 views

Bounds for the Fourier transform of characteristic functions on $\mathbb{Z}/N\mathbb{Z}$ supported on large sets

Suppose $A \subseteq \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$ with $|A| \geq N/2$. Let $$ \hat{A}(h) := \sum_{a \in A} e_N(ha), $$ where $e_N(x) := e^{2\pi i x/N}$. Clearly $|\hat{A}(h)| \leq |A|$ for ...
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1answer
583 views

Spectrum of a convolution operator

Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ that is given by $Tf := f * g$ where $g$ is in $L^2$. How do I now find that the spectrum of $T$ is equal to the essential range ...
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0answers
16 views

Fourier series of two functions composition with small parameter

A function $f(x)$ has fourier series: $$ f(x) = \sum_{n=-\infty}^{\infty}f_n e^{in\omega_x x} $$ where $x \in [x_1, x_2]$, $\omega_x = 2\pi/(x_2 - x_1)$. Now let's say $x = t + \varepsilon g(t)$ ...
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21 views

Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?

Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...
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0answers
41 views

Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
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2answers
3k views

convolution a continuous function?

define $$h(x)=\int_0^{2\pi}f(x-y)g(y)dy=f*g(x)$$ if $f,g \in L^2$ are $2\pi$ periodic, show that h is continuous on $[0,2\pi)$ so let $x_n \to x$, then $$|h(x)-h(x_n)|=|\int f(x-y)g(y)-\int f(x_n-y)...
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1answer
590 views

Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n |\cos(n\...
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2answers
42 views

Complex Frequency Shifting in Fourier Transform

When dealing with Fourier transforms, it is often useful to take advantage of the following property in order to simplify work: $$\mathcal{F}(e^{i\omega_0t}f(t))=G(\omega-\omega_0)$$ where $G(\omega)...
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34answers
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Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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1answer
28 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (...
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1answer
32 views

Convex basis of functions

I'm looking for a set of convex functions which is forms a basis for $C^1(\mathbb{R})$? Most of the basises I know are polynomials or Fourier basis but I was wondering if there was a basis of convex ...
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2answers
405 views

Fourier transform of $e^{-|t|}\sin(t)$

How can i calculate the Fourier transform of $e^{-|t|}\sin(t)$. I guess I need to do something with convolution, but I am not sure. Can somebody show me the way?
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0answers
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Is the Fourier transform a unitary isomorphism between $L^2(\mathbb{T}^n)$ and $\ell^2(\mathbb{T}^n)$

I am reading through Folland's "Real Analysis", and it's clear that if $f\in L^2(\mathbb{T}^n)$, then $\{\hat{f}(\kappa)\}\in\ell^2(\mathbb{T}^n)$, and the norms of those two are equal. However, it's ...
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0answers
32 views

Discrete 2 dimensional Hankel transform

As a side note, I have some experience with discrete/continuous Fourier Transforms in one dimension, but almost none with higher dimensional Fourier/Hankel Transforms. I am attempting to compute the ...
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0answers
33 views

Fourier series for absolute value of sin functiom

If we take the absolute value for sin function, then it becomes even. However, isn't period of this function $\pi$? To find fourier series, 1.Even 2. period $2 \pi$. Can we just treat this ...
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1answer
24 views

Fourier coefficents of harmoinc $L^{1}$ functions in the disk

I just did an exercise in a some lecture notes, my result implied that the Fourier coefficients of an harmonic $L^{1}$ function $u$ was the integer values of the borel measure in the Poisson integral ...
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0answers
21 views

Using 2D Parseval-Plancheler theorem to solve an equation

In the context of a digital communications problem I have to solve the following equation with respect to $\tilde{\tau}$: \begin{eqnarray*} &&Im\Big\{\Big(\int\limits_0^{T_0}r^{*}(t)g(t-\tilde{...
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1answer
341 views

Good Kernel's Properties

I am recently studying properties about a good kernel, and came across a problem. Definition: A kernel $K_\delta$ is 'good' if they are Lebesgue integrable and satisfy the following conditions for $\...
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0answers
14 views

Fourier Transform Spectrometer

Firstly, I was planning on constructing a Fourier Transform Spectrometer for a Physics project at school. Is this feasible? If so, what components could I use to construct it? Secondly, how exactly, ...
2
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1answer
82 views

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

The Plancherel Theorem for the Fourier transform $\hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt$ on $\mathbb{R}$ states that $$ \int_{-\infty}^{\infty}|\hat{f}(s)|^...
3
votes
2answers
142 views

$\lim_{\lambda \to \infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t} $

For a continuous function, $f:[0,b] \to \Bbb{R}$ show that: $$ \lim_{\lambda\to\infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t}\,dt = \frac{\pi}{2}\,f(0) $$ I know it has something to do with the ...
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1answer
44 views

inverse Fourier transform of compactly supported function is in $L^1$

Let $N > d/2$, and $N$ is chosen such that it is an integer. Let $f\in C^N(\mathbb{R^d})$ and $f$ has compact support. I want to show that $f$ is the Fourier transform of a function $g\in L^1(\...
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1answer
511 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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1answer
42 views

definition of Fourier transform questions

I'm completely stumped by the problem below because I haven't attended the lectures which used only Riemann integration and am not sure what the author is getting at. Let $f\in C^1(\mathbb R)\cap ...
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1answer
22 views

Find the spectrum of an operator related to Fourier series

As an exercise, I was told to find the spectrum of the bounded operator $K\in B(L^2[-\pi,\pi])$ defined by $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
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1answer
26 views

Counting solutions by estimating Fourier coefficients

In W. T. Gower's essay The Two Cultures of Mathematics, he mentions the following as an example of a 'general principle' in combinatorics: "If one is counting solutions, inside a given set, to a ...
3
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1answer
554 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
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1answer
34 views

A neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which $\lim_R\int_{-R}^R|f|dx<\infty$

Is there a neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which the limit of Riemann integrals satisfies $\lim_R\int_{-R}^R|f|dx<\infty$ in terms of elements ...
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0answers
13 views

Connecting First Passage Time to Power Spectrum

Let $f$ be a real function. Is there a connection between The first positive abscissa for which its autocorrelation function is equal to zero (which I call the first passage time, fpt) The largest ...
2
votes
1answer
28 views

Complex Fourier series and half-range expansions

I need to find the complex Fourier series for $f(x) = x$, where $0 < x < 2\pi$. I tried to solve this in two different ways, first with even extension, and then with odd, but I did not get the ...
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36 views

Fourier Transform: Musical Instruments cotd.

Upon analysing the Fourier Transform of a musical sound, are there any other applications of the Fourier Transform so obtained? Any ideas would be appreciated. Edit 1: To clarify the situation, I ...
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0answers
29 views

Deducing an equality involving Fourier transform of Schwarz functions

For $\psi,\phi\in\mathscr{S}(\mathbb{R}^n)$ we have $$\psi(0)\int_{\mathbb{R}^n}\mathscr{F}\phi(\xi) \, d\xi = \phi(0) \int_{\mathbb{R}^n} \mathscr{F}\psi(\xi) \, d\xi$$ I want to prove that this ...
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1answer
41 views

Prove the completion of the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ is not separable

Let $G$ be the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ with inner product $$ \left\langle f,g \right\rangle =\lim _{T\rightarrow \infty}\frac 1{2T}\int_{-T}^Tf\bar g .$$ I ...