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

0
votes
1answer
44 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
1
vote
1answer
132 views

Fourier transformation on a torus and the definition of fractional Laplacian

as we know, in $R^n$, for a function $f$, we can define its Fourier transform as $$\hat f(\xi)=\int_{R^3}f(x)e^{-ix\cdot \xi}d x,$$ with this, the Laplacian of $f$ can be elegently defined by ...
2
votes
0answers
67 views

Fourier's Method Question

I've been asked to use Fourier's method to obtain the following solution; $$u(x,t) = \sum_{n=1}^{\infty} B_n e^{-(n \pi C / L)^2 t} \sin(\frac{n \pi x}{L})$$ $$B_n = \frac{2}{L} \int_0^L \sin(\frac{n ...
3
votes
1answer
78 views

Inverse FT of $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ (Contour integration)

Given is the Fourier transform of some function $z(t)$: $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ I now want to invert the tranform using contour integrals. How can I do that? I ...
3
votes
1answer
248 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
1
vote
0answers
56 views

The Fourier-Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
3
votes
0answers
54 views

$L^1$ norms of Short Time Fourier Transforms

Fix $f \in L^2(\mathbb{R})$ s.t. $||f||_2=1.$ When will $$V_f f (x, \omega)=\int_{\mathbb{R}} f(t)\overline{f(t-x)}e^{-2 \pi i t\omega}dt$$ the STFT of $f$ with respect to the window $f$ be in ...
1
vote
2answers
556 views

Why output of FFT is same as input data size ?

What I understand from DFT formula below I can decide the N my self. I can try to use just 16 bins to describe a function or I may even use 4 , it won't be very accurate but I can do it right? The ...
2
votes
0answers
164 views

How to solve this recurrence relation (related to discrete Fourier transform)?

I am having trouble with the following recurrence relation: $$c_{n+1} - c_{n-1} = 2\alpha \sin \frac{(2n-1)\pi}{N} c_n, \quad\forall n \in \mathbb{Z},$$ where $N$ is odd and the initial condition is ...
4
votes
2answers
2k views

Fourier transform of a function of compact support

My professor occasionally assigns optional difficult problems which we do not turn in from Stein and Shakarchi's Complex Analysis. I am currently studying for a test in that class and try to get all ...
3
votes
1answer
585 views

Discrete Fourier Transform: Understand Negative Frequencies

I am trying to learn DFT on my own. I have been struggling for a while now around understanding the concept of negative frequencies and notably what happens when $k$ is greater than $N/2$ in the ...
1
vote
2answers
136 views

Very narrow FFT window functions

What is the flat-top window function that provides the narrowest possible lobe width? I'm doing FFT analysis and I need the resulting main lobe of a sine wave to be as narrow as possible but avoiding ...
3
votes
2answers
124 views

Bound on signal after passing through 2 pole lowpass filter

I am trying to implement some simple digital filters for a software synthesizer. This link seemed like a good start, and many things reference it: http://www.musicdsp.org/files/Audio-EQ-Cookbook.txt ...
1
vote
1answer
699 views

Question on Proof of Fourier's Uncertainty Principle

The Fourier Uncertainty Principle: If $f\in \mathscr L^2(\mathbb R)$ and $xf(x),\xi\hat f(\xi)\in \mathscr L^2(\mathbb R)$ then \begin{equation}\left\|xf(x)\right\|_2\big\|\xi\hat f(\xi)\big\|_2\ge ...
2
votes
1answer
565 views

3D fourier series

I wonder how I can write a function $f(\textbf{r})$ as a fourier series, when $f$ is periodic, in the sense that there exists a $ \textbf{T}_i \neq \textbf{0} $ so that $f(\textbf{r} + \textbf{T}_i) = ...
17
votes
3answers
1k views

Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using ...
1
vote
0answers
33 views

Is it possible to calculate a single frequency bin in $O(\log N)$ time (considering the $N\log N$ performance of FFT algorithms)?

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...
0
votes
1answer
65 views

Boundedness of continuous summable function

Let $f\colon\mathbb{R}\to\mathbb{C}$ be a continuous function. If we suppose that $f$ is a $L^1(\mathbb{R;C})$ function too, then can we conclude that $f$ is bounded? ADD: I asked the preceding ...
0
votes
2answers
130 views

Using a function in Matlab

Very new to MATLAB and Im trying to use the FFT function. I got a video which showed me that a function and a normal m file is needed. Created that but now dont know how to call the function from the ...
2
votes
0answers
129 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
1
vote
1answer
477 views

Whose basis is {1,sin(x),cos(x),sin(2x),cos(2x),…}?

Whenever $f(x)$ is a (Riemann) integrable function on $[-\pi,\pi]$ we can define its Fourier series $f=a_0/2+\sum a_nsin(nx)+b_ncos(nx)$.But we give arbitrary sequence {$a_n$} and {$b_n$},I think ...
0
votes
1answer
121 views

Discrete Fourier Series: What Happens After N/2?

I am really confused! I started to study Fourier series. I think I understand the theory approximately (I am still new to it). I was curious so started to read about DFT which I thought would be ...
1
vote
0answers
60 views

Question on Fourier Transform

Fourier transform on $f$: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ $\xi\in\mathbb{R}^d$. How to show that $$\hat{f}(\xi)=\frac{1}{2}\int_{\mathbb{R}^d}[f(x)-f(x-\xi')]e^{-2\pi ...
0
votes
2answers
578 views

If $f$ is a function of moderate decrease then $\delta \int f(\delta x) dx = \int f(x) dx$

A function of moderate decrease is a map from $\mathbb{R}$ into $\mathbb{C}$ such that there exists $A \in \mathbb{R}$ such that $\forall x\in \mathbb{R}, \ |f(x)| \lt \frac{A}{1 + |x|^{1+\epsilon}}$. ...
0
votes
1answer
60 views

Is there a countable Fourier transform for infinite sequences?

There's the discrete Fourier transform and the continuous one, but where's the one for infinite sequences. Let $(a_i) \subset \mathbb{C}$ be a sequence of complex numbers. The naive ways of defining ...
6
votes
1answer
111 views

Looking for guidance on a Fourier integral

Working with a Fourier transform problem, I've encountered the following integral: $$ \int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx $$ where $a$, $b$, and $c$ are real ...
2
votes
0answers
97 views

Harmonic Oscillator and Fourier Series

I am currently studying Fourier Series (on my own). I am using a few different references/sources. Some are more trying to give an intuition about Fourier Series and others are more rigorous. ...
3
votes
1answer
206 views

Interpreting the sign of fourier coefficients

I am studying the Fourier series right now. Hopefully it's going okay. Now I have been playing a little bit with taking the product of a wave function (a sine or cosine with some phase) with a sine ...
5
votes
1answer
123 views

Fourier Series: going from $a_n$ and $b_n$ to $c_n$

I sort of understand the principle of the Fourier series, but when I watch the wiki page I don't understand how to get from: ${a_0 \over 2} + \sum_{n=1}^N[a_n cos({2\pi n x \over P}) + b_n sin({2\pi ...
3
votes
1answer
336 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
4
votes
3answers
182 views

What is $\int_{0}^{\infty}e^{ixw}dw$?

We know that $$\int_{-\infty}^{\infty}e^{ixw}dw=\delta(x)$$ More details, see http://en.wikipedia.org/wiki/Dirac_delta_function Now my question is $$\int_{0}^{\infty}e^{ixw}dw=?$$ Be grateful with ...
1
vote
2answers
60 views

How do they do this w.l.o.g. so freely (Fourier series).

Theorem 2.1. Suppose that $f$ is an integrable function on the circle with $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$. Then $f(\theta_0) = 0$ whenever $f$ is continuous at the point ...
0
votes
0answers
173 views

How do I interpret continuous time fourier transform plot?

Suppose I have a fourier transform $X(f)$ of an energy signal $x(t)$. Now how do I interpret that continuous fourier transform plot. For example if the input signal is a rectangular pulse the fourier ...
3
votes
1answer
597 views

Wave equation 1D inhomogeneous Laplace/Fourier Transforms vs Green's Function

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
2
votes
2answers
837 views

1D Fourier Transform of Piece-wise function

I have the following piecewise function: $$ x(t) = \begin{cases} 1 & |t| \le T_0, \\[6pt] 0 & |t| > T_0. \end{cases} $$ I apologize for the formatting. I need to compute the Fourier ...
1
vote
1answer
259 views

Asymptotic behavior of Fourier transform

Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$, $f(x) = |x|^{-1}$. It is locally integrable, and its distributional Fourier transform is $F(f)(k) = g(k) = 4\pi/|k|^2$. Intuitively, the ...
5
votes
1answer
261 views

Integral approximation of Fourier series.

Consider a function $f$ defined on the real line. Consider the restriction of the function to the interval $[-L,L]$ and periodically extend the function using a Fourier series $$f_L(x) = ...
7
votes
1answer
96 views

percentage of numbers starting with $2$ in $\{2^n\}$

I have once heard a professor telling (during a course on Fourier theory) that there is a way to determine the numbers starting with a $2$ in the sequence $\{2^n\colon n\in\mathbb{N}\}$. I asked him ...
0
votes
0answers
70 views

laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
1
vote
1answer
150 views

Fourier Series and Solving Differential Equations

I am getting stuck on how to use Fourier Series to solve ODE's. Take the problem where \begin{equation} E(t)=200t(\pi^2-t^2), \end{equation} for $t$ between $-\pi$ and $\pi$ (period of $2\pi$), ...
0
votes
1answer
43 views

Is it possible to show that the Gaussian is a fixed point of the Fourier transform using a fixed point theorem?

We know that $\text{exp}(-\alpha |x|^2)$ is a fixed point for the unitary Fourier transform if $\text{Re } \alpha > 0$. Is it possible to show this using a fixed point theorem?
0
votes
1answer
134 views

Solve integral equation using convolution

I'm trying to solve an integral equation by identify the convolution and then transforming, but I'm getting to a really confusing expression, where I'm not sure how to continue: $$ ...
2
votes
2answers
102 views

about Fourier transformation on zero-padded vector

I have a vector $x$ of n elements. I did a fft on it and return another vector of n elements also (i.e.$X = \text{fft}(x)$). Now I am trying to pad the $x$ vector by n zeros so to get $y$ $$ y = [x ...
1
vote
1answer
213 views

l2 norms, rapidly decreasing functions and fourier transforms

Let $f\colon \mathbb{R} \to \mathbb{C}$ be a rapidly decreasing (rd) function. Let $\mathcal{F}(f)$ be the Fourier transform of $f$. It is known that 1) $\| \mathcal{F}(f) \|_2 = \| f \|_2 $ ...
2
votes
3answers
227 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
3
votes
3answers
184 views

The inverse Fourier transform of $1$ is Dirac's Delta

From the definition of the Dirac delta $\delta_0$ one can infer that its Fourier transform is identically equal to $1$. But going in the other direction is not as straightforward. How can one show ...
2
votes
0answers
140 views

Fourier transforms of Marcum Q function

I cannot find in the literature the following: $$\int_0^{\infty}\,Q_1(a,bx)\,\cos(\omega x)\,\,dx$$ and $$\int_0^{\infty}\,Q_1(a,bx)\,\sin(\omega x)\,\,dx$$ with $a,\omega>0$. $Q_1(a,x)$ is ...
2
votes
1answer
59 views

Fourier Series of $f(x) = x^n$ - Fast Method?

Is there a fast way to compute the real Fourier series of $$f(x) = x^n \ ?$$ How about the complex fourier series? If there isn't a fast way for arbitrary $n$, how about $n = 5$ or something at ...
4
votes
1answer
197 views

A Cauchy principal value integral, using contour integration and Plemjel.

I came across the following integral $$lim_{\epsilon->0+}\int_\mathbb{R}\frac{e^{-ax^2+ibx}}{x+i\epsilon}dx$$ with a,b>0. Using Plemjel's formula led me to evaluating ...
0
votes
1answer
64 views

Amplitude versus time producing unexpected patterns.

I am writing a program to generate audio frequencies in multi-channel PCM format. This question may be more suited on an audio forum but I would like to know what is going on mathematically. My ...