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 on fractional Sobolev spaces

We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of ...
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Approximation of one function by other using a smooth multiplier function

This problem is from the Book, Harmonic Analysis by Katznelson (Problem 2, Page 160). Suppose $f$, $g\in L^2(\mathbb{R})$ such that $f(x) = 0$ implies $g(x)=0$ for almost all $x\in\mathbb{R}$. Then ...
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Prove of the Parseval's theorem for Discrete Fourier Transform (DFT)

If $x[k]$ and $X[r] $ are the pair of discrete time Fourier sequences, where $x[k]$ is the discrete time sequence and $X[r]$ is its corresponding DFT. Prove that the energy of the aperiodic sequence ...
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How this result in archived in Fourier series

I was reading some notes about functions of symmetry in Fourier series and came across the following result for a function with symmetry of an odd quarter wave $$\begin{align} ...
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Substitution in complex-valued Fourier integral

In Knapp (Representation theory of semisimple groups, 86'), on page 34 it is shown by means of Euclidean Fourier transform that the principal series representation of $SL(2, \mathbb C)$ is irreducible ...
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Intuition Behind the Riesz Transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. ...
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Fourier Transform of $\delta(t-nt)$

Given the discrete signal $x(n)=\begin{bmatrix} \alpha ^n, n\geq 0 \\0, n<0 \end{bmatrix}$ where $\alpha \in (-1,1)$ and some natural number $N$, we know that the discrete signal $y(n)$ (where $0 ...
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Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
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Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
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Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
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Sampling in the time domain vs. sampling in the frequency domain

If I have a sample rate of 2 seconds on a 128 second time window (64 samples total) and then I do a Fourier transform, what is my sample "rate" in the frequency domain? Will I end up with 64 ...
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37 views

Fast Fourier Transform Splitting Algorithm

I'm trying to figure out how the FFT splitting algorithm works. I've pretty much understood the general idea, but when I try to compute it, I get something completely different than what I expect $ ...
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properties of orthonormal systems and hilbert spaces [closed]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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Fourier transforms intuitive explanation

I have read on wikipedia that: The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in ...
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Fourier transform of function composition

Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually? I know you can do this for the sum, the ...
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Linear span of poisson kernels dense in $L^1(\mathbb{T})$

A paper I am reading ("Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in $L^2(\mathbb{T})$" by Sergei Khrushchev...really a great paper) repeatedly mentions ...
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Fourier series phasor form and sin/cos form

can anyone give me a link on how to convert the forms (from phasor to sine/cos and vice versa)? I am new to this and I can't find the convertion table with a valid explaination.
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Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
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Estimate $|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi \lambda_k x)}{\lambda_k}\right|$.

We known that there exists a constant $C\geq 0$ such that for all $n\geq 1$ : $$|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi kx)}{k}\right| \leq C -\log |\sin (\pi x)|, \quad \forall x\in (0,1]. $$ ...
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A Fourier Analysis Question I am stuck at

If $f,g\in C[-\pi,\pi]$,and $f,g$ are $2\pi$ periodic, prove that $$\lim_{n\to\infty}\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)g(nt)\mathbb dt=\big(\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)\mathbb ...
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Parseval's identity does not hold for constucted basis

As part of an exercise, I was asked to show that given an orthonoraml basis $(\varphi_1,\varphi_2,\varphi_3,...)$ in $L_2[-\pi,\pi]$, we can construct an orthonormal basis $(\psi_1,\psi_2,\psi_3,...)$ ...
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how manipulating the coefficients vector effects on the result of DFT?

given: calculate: note that the given DFT is from n order and we want to compute DFT's from 2n order. edit: this is my try of B. i don't see where the given DFT is used and how to proceed: ...
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How to find fourier transform of $e^{-x^2}$?

I want to find the fourier transform of $e^{-x^2} = \int_{-\infty}^{\infty}e^{ikx-x^2}\,dx$ using contour integration. I consider the rectangular contour $C$ with verticies $\pm R, \pm R + ik$ Then ...
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Create periodic function from combining non-periodic functions

I'm studying recurrent neural networks which often use tanh as an activator function which is not periodic. However in research and papers it's shown that these recurrent neural nets can exhibit ...
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Complex exponential argument to a function

In many texts on signal processing, the following notation is used to describe the Fourier transform of a discrete time signal $x$: $$ \hat{X}\left(e^{j\omega}\right) = ...
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Show $\int_{-\pi}^{\pi}[f'(t)]^2 dt \geq \int_{-\pi}^{\pi}[f(t)]^2 dt$

Given that $f$ is $C^1$ on $\mathbb{R}$ with period $2\pi$ and $\int_{-\pi}^{\pi}f(t)dt =0$, show $$\int_{-\pi}^{\pi}[f'(t)]^2 dt \geq \int_{-\pi}^{\pi}[f(t)]^2 dt$$ I've tried using Bessel's ...
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Fourier Transform for Boundary Value Problems

I am trying to understand the problem defined by $$\phi_{xx} + \phi_{yy} = 0, in -\infty \lt x \lt \infty, y \gt 0$$ $$\phi = f(x) \space as \space y \to 0, \phi = 0 \space as \space y \to \infty$$ ...
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Calculating Fourier Transform of $\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$

This question deals with finding the Nyquist Frequency of a given signal. Suppose you have the signal $x(t)=\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$ in the time domain where $T>0$ is some ...
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Fourier transform of $e^{-4\pi ^2 x^2}$

How do you prove $$\int_{-\infty}^{\infty}e^{-(2\pi x + i\xi/2)^2}dx=\int_{-\infty}^{\infty}e^{-(2\pi x)^2}dx$$ for $\xi \in \mathbb{R}$. The Question arises from calculating the Fourier Transform ...
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Harmonic analogue of the Weierstrass approximation theorem

The Weierstrass approximation theorem says that, given any continuous function $f(x)$ on a closed interval, there is a polynomial which approximates it arbitrarily closely. I'm looking for a theorem ...
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What's amiss in this Fourier convergence analysis?

I worked out this solution to this basic PDE Fourier series convergence problem, but I suspect the result is "too easy to be correct," because all the answers point to no restriction on either $m$ and ...
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Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...
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Coefficient calculation on Fourier series !? [closed]

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
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A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
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Pontryagin Dual of the Unit Circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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How to calculate full width half max of this curve

I'm trying to calculate the full width half max of this function and I keep receiving non-sense answers. The function is $F(ω)=A_0 (\frac{\frac{1}{τ}}{(\frac{1}{τ^2})+(ω_0 - ω)^2})$ and I then have ...
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What is a good source for learning fourier transformations as an application

I'm an undergraduate physics major and for my research I need to start learning and understanding fourier transformations for my research. Does anyone know good resources for doing this. I don't need ...
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Fourier transform of the error function, erf (x)

I define $\text{erf}(x):=\frac{2}{\sqrt{\pi}}\int_0^xe^{-\xi^2}d\xi$. What is its Fourier transform (unitary, ordinary frequency)? That is, simplify ...
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Periodic convolution of functions

Define the periodic convolution of functions in L2([0; 1]). What theorem of convolution do I use to define this and how do I solve this?
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Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
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Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
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Is Fourier transform still writing a function as a series of sines and cosines?

In the Fourier series we write a function as a series of sines and cosines. Fourier transform seems to me to be totally different, we are not finding a series but rather a function $\hat f(w)$. So ...
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Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
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Elementary properties of Fourier transform on the space of tempered distributions.

I'm trying to prove that some of the basic facts of Fourier transform on $L^1(\mathbb{R}^n)$ also holds on the space of tempered distributions. For example: Suppose that $F\in \mathcal{S}^\prime$, ...
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Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
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Getting fourier coefficient by integrating over half the period?

In the book Schaum's Outlines of Analog and Digital Communications solved problem 1.2, the author calculates the fourier coeffecient $C_0$ for the rectangular pulse train: where $a$ is assumed to be ...
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Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
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How to show for $\alpha∈(0,1)$, any $f∈ C^\alpha([0,1]/{\sim})$ has a Fourier series $S_nf$ uniformly converging to $f$

Technically homework(a midterm) but its over and I'm itching to know the solution. I know how to show it for $\alpha>1/2$ (the Fourier series will converge absolutely), but apparently its true for ...
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Can $\sum_{a = -\infty}^{\infty} e^{i\omega aT_0}$ be represented by dirac delta functions?

The usual definition of dirac delta function says that $\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\ dp $. The appearance similarity makes me think that it may be possible ...