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
1answer
62 views

delaying signal

What does delaying a signal mean? Graphically? Mathematically? Is it, advancing to the next numbers, or using the previous numbers? Suppose i have $x[n] = \{0,1,2,3,4,5\}$ and i use $x[n-m]$ (example ...
6
votes
1answer
838 views

How smooth is a smooth function?

Let's say a smooth function is a $\mathcal{C}^\infty$ function on $\mathbb{R}$. Some smooth functions are not analytic, the most notorious example being the bump functions. A non-analytic ...
11
votes
2answers
289 views

Fundamental role of the Fourier Transform

I am currently learning about the Fourier Transform and the associated Fourier Analysis. So far I realize that it has a number of applications, but more than that it seems to be central to Functional ...
1
vote
1answer
72 views

Condition for differentiablility

Suppose I have two functions that are Schwartz class, say $f,g \in S(\mathbb{R})$, and suppose I have another function $\psi(x)$ such that \begin{equation} g(x) = \psi(x)f(x) \end{equation} I would ...
2
votes
0answers
148 views

Help with understanding a proof in Fourier Analysis

I have a stack of lecture notes that I am currently going through to teach myself a little bit about Fourier Analysis. Now I struggle with the following Lemma, which is needed to talk about the ...
3
votes
1answer
106 views

Showing a function is Schwartz

This question is related to this post, for which I received a really good answer that gave a beautifull solution, yet I am still trying to understand one thing on the side that is not covered by the ...
1
vote
1answer
502 views

Chirp Transform and Convolution

I was reading about the discrete fourier transform from the CLR algorithms book and I came upon an exercise whose hint confuses me. The exercise reads as follows: The chirp transform of a vector ...
2
votes
1answer
318 views

Question on Schwartz function

I am seeing the Schwartz Space for the first time today and I have trouble understanding the following argument: Given that $f \in S({\mathbb{R}})$ with $f(x_0) = 0$ Taylor's theorem tells us that ...
3
votes
3answers
178 views

Quick question on Fourier Transform of $\exp(-\frac{x^2}{2})$

I am currently looking at an example of how to calculate the Fourier Transform for the function \begin{equation} f(x) = \exp\left({-\frac{x^2}{2}}\right) \end{equation} Now $f$ solves the differential ...
3
votes
1answer
76 views

Showing $e^{-x^2} \in \mathcal{S}(\mathbb{R})$

I am currently studying Fourier Analysis on my own and have just been started to look at the Schwartz Space of rapidly decaying functions. One example of such functions is given in the notes that I ...
3
votes
0answers
333 views

What is the Fourier transform of a phase modulated signal?

I'm studying some Fourier analysis and have for a while been trying to figure out how to apply the Fourier transform to a phase modulated signal. More rigorously stated, what is $$ ...
11
votes
3answers
13k views

What does the Fourier Transform mean in the context of images?

This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is ...
1
vote
0answers
491 views

Fourier transform independent of kernel?

I've tried computing a windowed Fourier transform using various kernels that were all made from periodic signals of the form a + bi with b being a shifted version of a. I used square waves, sin waves, ...
3
votes
1answer
1k views

Uniform convergence of Fourier Series

I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand: Given that we know the series $f(x) = \sum c_k e^{ikx}$ ...
1
vote
0answers
826 views

Square wave transform

One can define an analogous transform to the Fourier transform that uses square waves as the basis instead of sinusoids. Everything seems to work out in parallel and I imagine one can even come up ...
4
votes
0answers
73 views

Why are linear functions the natural analogue of exponential functions in a tropical semiring?

I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says It's not obvious how to interpret the ...
8
votes
1answer
265 views

Expectation of a Random Subset of the Roots of Unity.

Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...
0
votes
1answer
134 views

Concatenation within DFT

The Discrete Fourier Transform (DFT) is given by $X_k = \sum_{n=0}^{N-1} x_n e^{2\pi i k n/N}$ for $k=0,1,\ldots N$. I store the variables $X_k$ and would like to add $m$ zeros to the variables ...
4
votes
1answer
314 views

Interchanging integration and differentiation in heat equation

I'm trying to solve the following problem: Assume that we are given $f(t,x) \in C^{\infty}(\mathbb R \times \mathbb R^n)$ such that $f(t, \cdot) \in \mathcal S(\mathbb R^n)$ for all $t>0$ ...
0
votes
1answer
43 views

Sequence of complex $L_1$ functions with $L_1$-divergent limit of the fourier operator

In a task I should first show that $\mathcal{F}(e^{-\frac{x^2}{2 k}})=\sqrt{k} e^{-\frac{k \xi ^2}{2}}$ if $\text{Re}(k)>0$. Then they say that one may conclude that there is a sequence $(f_k)_{k ...
1
vote
1answer
321 views

Fourier transform of Kronecker deltas

I have a binary 2D image that consists of 95% black pixels with a few white pixels scattered about, and I want to convolve it with a 2D gaussian kernel. I'm hoping to exploit its sparsity to improve ...
2
votes
1answer
319 views

Fourier transform of function involving $\log$

I found the following problem which I am unable to solve. Calculate the following integral $$\int_{\mathbb{R}} \frac{d\omega}{2\pi} \log (1 + i a/\omega ) e^{-i \omega t}$$ for $a>0$ and ...
0
votes
2answers
122 views

Proving the locus of a Fourier series is a system of perpendicular lines [closed]

From "Fourier's series and integrals" by H.S. Carslaw, there is the following question: Prove the zero locus of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} \sin(n x) \sin(n y) = 0$ is ...
2
votes
1answer
938 views

FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions

In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ arising from a charge distribution $\rho(\mathbf{r})$ is $$ \Delta ...
0
votes
2answers
155 views

$f,g: \mathbb{R} \to \mathbb{C}, 2\pi $ periodic. Prove: if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$

Let $f,g: \mathbb{R} \to \mathbb{C}$ $2\pi$ periodic , Riemann integrable in $[0, 2\pi]$. I need to prove that if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$. We ...
6
votes
2answers
1k views

Prove: Fourier series of $e^{\cos x} \sin (\sin x)$ is $\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$

I'd love your help with proving that the following series $$\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$$ is the Fourier series of $e^{\cos x} \sin (\sin x)$. I tried to find $\hat f(n)$ using ...
1
vote
1answer
561 views

Relation between Fourier transform and Fourier series

Let $f$ be a function on $\mathbb R^n$ whose Fourier transform $\hat f$ exists. Is there any relation between integrability of $\hat f$ and summability of the series $\sum_{n \in \mathbb Z^n} \hat ...
3
votes
2answers
1k views

On rate of decay of integrals (fourier transform)

Given, $f(x)$ is a monotonically decreasing function, and $f(x) \geq 0$ for all $x \in [a,b]$. Suppose that for $f(x)$ the following holds (Riemann-Lebesgue Lemma): $$\lim_{n\to\infty} \int_a^b ...
2
votes
3answers
268 views

How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
1
vote
2answers
204 views

$f,g$ are two continuous functions with period$=1$, are the Fourier coefficients $f*g=f(n)g(n)$?

Let $f,g$ be two continuous functions with period$=1$. Are the Fourier coefficients of $f*g$ are given by the products $f(n)g(n)$ (of the $n$-th coefficient in each series)? Thanks!
2
votes
1answer
81 views

question on fourier transform.

I ask myself what $$ {\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)'(\xi) )(s) $$ is. If it was just about $$ {\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)(\xi) )(s) $$ it would ...
3
votes
1answer
296 views

Conditions for a finite Fourier series

Under what circumstances is the Fourier series of a function guaranteed to have a finite number of coefficients?
5
votes
1answer
193 views

Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$?

Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $$\tilde f(k) = \frac{\mathcal ...
3
votes
1answer
181 views

How many terms in a series expansion

General: If $f \in C^1$ is a periodic function defined over some multi-dimensional space, then it should be possible to express $f$ as a FINITE fourier series. is this true of any periodic basis? is ...
2
votes
1answer
478 views

Approximation of smooth periodic functions by trigonometric polynomials

I know that there is a following version of the Weierstrass approximation theorem for functions on $[a,b]$. For every $f:[a,b] \rightarrow \mathbb{R}$ there is a sequence $(P_n)$ of polynomials such ...
2
votes
1answer
94 views

A condition for $\hat f$ to be integrable

Let $f \in L^1 (\mathbb R^n)$. Suppose that $f$ is continuous at zero and that the fourier transform $\hat f$ of $f$ is non-negative. Does this imply that $\hat f \in L^1$ (and hence, by the inversion ...
1
vote
0answers
109 views

Dealing with integrals and Fourier transforms.

I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int ...
2
votes
1answer
1k views

n-dimensional Fourier integral

let be $ \int_{R^{n}}dVf(r)e^{i(k.r)} $ the n-dimensional Fourier integral. $ dV=dxdydz.... $ the volume and $ (k.r)= \sum_{n} k_{n}.x_{n} $ is the scalar product of the position vector 'r' and the ...
4
votes
2answers
523 views

Pointwise, uniformly and absolute convergence of Fourier series

I'd really love your help with this one: I got this Fourier series for $f(x)=x$ in $[-\pi,\pi]$: $$\sum_{1}^{\infty}\frac{2(-1)^{n+1}\sin(nx)}{n}$$ and I need to check if it's (i) pointwise ...
2
votes
2answers
163 views

Why does the following Fourier series does not converge for $x \in R$, and does for $x \in [0,2\pi]$?

I would really love your help with the following facts that I can't understand. I can't understand why the following Fourier series does not converge: $$\sum_{0}^{\infty}\frac{e^{inx}}{n^2}.$$ 1.If ...
7
votes
1answer
1k views

Solving the heat equation with Fourier Transformations

Can anyone help me with this IVP heat equation problem? I have $$u_t-u_{xx}=g(x,t)$$ where $x \in \mathbb{R}$, $t>0$, $u(x,0)=0$ So i've found by taking a Fourier transformation that ...
5
votes
3answers
515 views

Fourier series - what is the difference between the Fourier series of $f(x)=x$ in $x \in [0,2\pi]$ and in $x\in [-\pi,\pi]$?

I was asked to compute the fourier series of $f(x)=x$ in two different intervals: $x \in [0,2\pi]$ and $x\in [-\pi,\pi]$. What is the real difference between the two series, cause we know that for ...
2
votes
1answer
155 views

Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component. Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get ...
3
votes
1answer
362 views

Fourier transform (logarithm) question

Can we think, at least in the sense of distribution, about the Fourier transform of $\log(s+x^{2})$? Here '$s$' is a real and positive parameter However ...
3
votes
1answer
1k views

Intuition behind decay of Fourier coefficients

Many other posts have discussed the standard result that the smoothness of a function is related to the rate at which its Fourier coefficients decay. For example, there are proofs that show that if ...
1
vote
2answers
427 views

When is the convolution with a tempered distribution again a tempered distribution?

If $f$ is a Schwartz function on $\mathbb R^n$ and $g \in L^1(\mathbb R^n)$, then if $g$ is the Poisson kernel, is $f\ast g$ a Schwartz function? are there any known sufficient conditions on $g$ to ...
1
vote
1answer
273 views

On covering lemma and Calderón–Zygmund decomposition

I am working on something which needs to understand covering lemmas and Calderón–Zygmund decomposition. These type of lemmas are as in the following link ...
2
votes
0answers
82 views

Deduce the global differential equation from the pointwisely defined equation in Fourier space

Let $G\in \mathcal{F}(\mathbb{R}^{n+1})'$ be a distribution on the space of spatial Fourier transform'able function, ie an $L^1_{\mathrm{loc}}(\mathbb{R^{n+1}})$ function, $G = G(t,\xi)$. Assume ...
2
votes
0answers
613 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 ...
2
votes
3answers
233 views

How can the FT be used to find the FS of the periodized signal?

Let's say I know the Fourier transform of a function that is $0$ outside some interval, for example a triangle wave. How can this be used to find the Fourier series of the related periodic function, ...