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).

learn more… | top users | synonyms

1
vote
0answers
52 views

Fourier Series and Coefficient Calculation Under two minutes

Example of one Question for preparing the entrance exam: Fourier series of function: $$ f(x)=f(x+2\pi), f(x) =\left\{ \begin{array}{rcr} 1 & & -\pi <x<0 \\ \sin x & &...
1
vote
0answers
20 views

Question about function with 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}\...
0
votes
1answer
53 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 ...
1
vote
3answers
54 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}\\ ...
0
votes
1answer
32 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 ...
0
votes
0answers
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 ...
0
votes
0answers
15 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 $x = t + \varepsilon g(t)$ ...
0
votes
2answers
41 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)...
2
votes
1answer
31 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 ...
0
votes
0answers
16 views

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 ...
0
votes
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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
20 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{...
0
votes
1answer
21 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 ...
0
votes
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
votes
2answers
140 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 ...
1
vote
1answer
40 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(\...
1
vote
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 $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
18 views

How to evaluate chirp transform in O(nlgn) time? [duplicate]

The question says to evaluate chirp transform in O(nlgn) time using the equation in the hint. But I'm unable to get any idea on how to prove the chirp transform from it. Any help is appreciated.
1
vote
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 ...
1
vote
0answers
34 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 ...
0
votes
2answers
30 views

Fourier series Coefficients and wolframalpha

1) Please can my answers be checked, including my final Fourier series. 2) Is it possible to use Wolframalpha to check my answers? If so, how will I go about doing this? Deduce the Fourier series ...
2
votes
1answer
42 views

Fourier Transform: Musical instruments

How do I Fourier Analyse the music produced by a musical instrument? What I mean is that what tools/applications are best suited to Fourier Analyse waves from musical instruments?
0
votes
1answer
42 views

Proving that a function belongs in the space of tempered distributions

Let $a>0$ and define $$g(\xi):=\frac{\sin a\xi}{\xi(1+\xi^{2})}$$ I want to prove that $g\in\mathscr{S}'(\mathbb{R})$ and consquently that $g\in L^{1}(\mathbb{R})$ (but this implication is ...
0
votes
0answers
39 views

Limit of $\lim_{t \to \infty} \frac{ \int_0^\infty \cos(x a t) e^{-x^p}dx}{\int_0^\infty \cos(x b t) e^{-x^p}dx}$

Let \begin{align} f(t)= \frac{ \int_0^\infty \cos(x a t) e^{-x^p}dx}{\int_0^\infty \cos(x b t) e^{-x^p}dx} \end{align} How to find a limit \begin{align} \lim_{t \to \infty} f(t) \end{align} for ...
0
votes
0answers
22 views

DFT of a sequence while ignoring some of its elements

I sample a signal at a certain frequency for a finite amount of time to get a sequence $$(x_n)_{n=1}^N = (x_1, x_2, ... , x_N)$$ with the intention of analyzing its power spectral density by ...
1
vote
1answer
22 views

Convergence of Sequence of Fourier Transforms

Let's say I have a sequence of functions $f_n\in L^1(\Bbb{R})$ such that $f_n\rightarrow f$ in $L^1(\Bbb{R})$, and $\hat{f}_n\in L^1(\Bbb{R})$ for all $n$ (where hat is the Fourier transform), and $...
2
votes
1answer
30 views

Approximation estimates in Sobolev spaces

Let's consider a bounded domain $\Omega \subset \mathbb{R}^d$, $d =2,3$, and let $\varphi$ be in $H^1(\Omega) \cap W^{1,\infty}(\Omega)$. Is it there a smooth (at least $W^{2,4}(\Omega)\cap W^{1,\...
0
votes
1answer
22 views

Show that there is $f\in \mathcal C^0(S^1)$ s.t. $\lim_{n\to \infty }\|S_nf-f\|_{L^\infty }\neq 0$.

I have to show that there is $f\in \mathcal C^0(S^1)$ s.t. $$\lim_{n\to \infty }\|S_nf-f\|_{L^\infty }\neq 0.$$ The proof goes as follow : we know that $\|D_n\|_{L^1}\geq c\log(n)$ where $D_n$ is the ...
2
votes
1answer
21 views

How can we show that $\int_{|\alpha |\leq N}\hat f(\alpha )e^{2i\pi x\alpha }d\alpha $ converges to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$.

I am a Ph.D. student in statistic, and I wanted to know if there is an easy way to show that the Fourier inversion converge to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$. In other word, that $$\lim_{N\...
2
votes
2answers
60 views

Is a function $f:\mathbb R/\mathbb Z\to \mathbb R$ bounded ?

Let $f:\mathbb R/\mathbb Z\to \mathbb R$ a function (1-periodic). Is such a function bounded ? (it's the fact that f is defined on the circle that disturb me). Indeed, for such a function (usually at ...
0
votes
0answers
18 views

Complex conjugation of Fourier transform in frequency domain

If I have some function (which is an array in Python or Matlab) $F(\nu)$ and want to derive $F^*(-\nu)$, I can do an inverse Fourier transform on $F(\nu)$ to get $f(t)$, then complex conjugation on $...
-1
votes
2answers
25 views

Integral, absolute value

Why did we get this identity? $$\frac{1}{\pi} \int_{0}^{2\pi} |x-\pi| dx = \frac{2}{\pi}\int_{0}^{\pi} \pi-x dx$$ And why do we integrate it to: $$-\frac{(\pi-x)^2}{2}$$ Why didn't we just ...
0
votes
0answers
27 views

How to calculate the integral of a fourier trasfom

I have to calculate this integral : $\int_{-\infty}^{+\infty} \hat G(\omega)e^{i\frac{\pi}{2}\omega}d\omega$ ($\hat G(\omega)$ is the Fourier trasform) with: $G:x\in \Bbb{R}\to \begin{cases} g(x), ...
4
votes
1answer
31 views

Pointwise convergence of Fourier series in two dimensions

By Carleson's Theorem, we know that for every $f\in L^2(\mathbb{T})$ $$ f(x)=\lim_{N\rightarrow\infty}\sum_{k=-N}^N\hat{f}(k)e^{2\pi ikx}\;\text{ a.e.} $$ Suppose now that $f\in L^2(\mathbb{T}^2)$. ...
0
votes
1answer
28 views

The Fourier transform of the Bartlett (triangular) window

I am trying to understand how to obtain the Fourier transform of the Bartlett (triangular) window. The Bartlett window is defined as $$ w_B(k)=\begin{cases}\frac{N-|k|}N,& |k|\le N;\\0,&|k|>...
1
vote
0answers
46 views

Is there a theorem that tells us that $\widehat{f\circ g}=\hat f(\hat g)$? [duplicate]

Is there a theorem that tells us that $\widehat{f\circ g}=\hat f(\hat g)?$ Here, $\hat f$ denotes the Fourier transform of $f$.
0
votes
0answers
26 views

Why $(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi}$?

Why $$(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi} ?$$ Indeed, to prove that the fourier transform is in the Schwarz ...
0
votes
1answer
45 views

How to calculate the Fourier trasform

I have to prove to what space $L ( \Bbb{R} )$ does not belong to the Fourier transform of : $G:x\in \Bbb{R}\to \begin{cases} g(x), & \text{if |x| $\leq$ $\pi$} \\ 0, & \text{if |x| $\gt$$\pi$...
0
votes
0answers
34 views

How to derive the Fourier Transform of the cosine function?

Given that the function $f(t) = A\cos(\omega t-\phi)$. I cannot get the results for the $f$ domain transform $F(f)$ and the $\omega$ domain transform $F(\omega)$ to be equivalent
4
votes
0answers
50 views

Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem

This question spanned from a previous interesting one. Let $k$ be a real number greater than $2$ and $$\varphi_k(\xi) = \int_{0}^{+\infty}\cos(\xi x) e^{-x^k}\,dx $$ the Fourier cosine transform of a ...
4
votes
1answer
57 views

How is the study of wavelets not just a special case of Fourier analysis?

As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$. How is wavelet analysis anything new ...
0
votes
1answer
30 views

Calculating integral value of Fourier series

Given fourier series: $$\mathrm{S}\left(x\right) = {3 \over \pi}\sum_{n = 0}^{\infty} {\sin\left(\left[2n + 1\right]x\right) \over 2n + 1}\,,\qquad \left\langle -\pi,\pi\right\rangle $$ Evaluate: ...
0
votes
1answer
45 views

Is Fourier series $L^2$?

Let $f\in L^2(0,1)$. I was wondering if the Fourier series of $f$ is a linear map $L^2(0,1)\to L^2(0,1)$. The linearity is obvious, but if $f\in L^2(0,1)$ does $S(f)\in L^2$ or not ? I tried as follow,...