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 series expansion of $x(t) = \sum\nolimits_{z \in \mathbb{Z}} (-1)^z \delta(t - 2z)$

Find the Fourier series expansion of $x(t) = \sum\nolimits_{z \in \mathbb{Z}} (-1)^z \delta(t - 2z)$, where $\delta(\cdot)$ denotes the Dirac delta function (unit impulse). I can infer that the ...
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257 views

Please recommend good text on complex Fourier series/analysis

I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular ...
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1answer
365 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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55 views

Fourier transform to find an harmonic function (Strauss)

I am trying to solve one of the problems of section 12.4 of the book "Partial Differential Equations" by Strauss. The problem says: Use the Fourier transfor in the $x$ variable to find the harmonic ...
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75 views

Existence of a function

I need some help: I am thinking about this problem. Any advice would be appreciated. Let's fix $\epsilon>0$. Does there exists some $f\in C^0([0,\pi])$ such that: ...
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1answer
48 views

$p$-stable Random Variables for $p>2$?

I will preface this by saying I am certainly no expert in Probability theory. My actual problem is an interpolation one, in which I am considering interpolation of bandlimited functions with shifts ...
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Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
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29 views

Product of an $L^\infty$ function and an $L^1$ function is integrable

For every $f \in L^1(\mathbb{R^n})$ let $$ \hat{f} : \mathbb{R}^n \to \mathbb{C}, \quad \hat{f}(\xi) := \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) \exp{(-i \langle x, \xi \rangle)} ...
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What functions stem from Fourier Series with rational-only coefficients?

Given the Fourier series $$f(z) = \sum_{k=-\infty}^\infty c_k e^{ikz}$$ but with $c_k\in(\mathbb Q+ i\mathbb Q)$ instead of $\mathbb C$ (or even purely real), are the functions obtained this way in ...
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Fourier transform of a triangular pulse

So I've been practicing some fourier transform questions and stumbled on this one; To start off, i defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as ...
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6answers
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Fourier Analysis textbook recommendation

I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the ...
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What are some good Fourier analysis books?

I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
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Can an arbitrary real function be written in terms of quadratures of an arbitrary frequency with time dependent coefficients?

Given a real function $f$, and a frequency $\Omega$, is it the case that there exist two other real functions $I$ and $Q$ such that $f$ can be written as $$f(t) = I(t) \cos(\Omega t) - Q(t) ...
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Deconvolution and Polynomial factoring using the FFT

I've been trying to implement a general N dimensional deconvolver for various engineering applications and some math curiosities. For speed and simplicity I've decided to try and do this with help of ...
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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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1answer
38 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
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3answers
3k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
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Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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Project a signal $S(t) = \sum_0^{\infty}A(k)e^{if(k)t}$ to 3d domain $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $

Definitions: $\vec{v}e^{\vec{w}x} = \vec{v}\cos(x) + \vec{w}\sin(x)$ $ \psi_0(t) = x\hat{i}t + y\hat{j}t + z\hat{k}t, \ x,y,z\in R$ $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $ ...
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DFT of subdomain of periodic domain

$f(t_i,x_j)$ is a solution of stochastic differential equation on grid. $j=[0,N+1]$, $i=[0,\infty]$ and boundary conditions are periodic: $f(t_i,x_0) = f(t_i,x_N)$ and $f(t_i,x_{N+1}) = f(t_i,x_1)$ ...
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138 views

$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
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Necessary to assume $f\in C^\infty$ in this Fourier transform problem?

Consider the following problem. Is the hypothesis that $f\in C^\infty$ necessary, or could we weaken it and assume just that $f$ is continuous? Let $\hat f$ denote the Fourier transform of the ...
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2answers
72 views

“Every function can be represented as a Fourier series”?

It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series. So this got me thinking about the ...
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24 views

Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
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2answers
56 views

A question on use of square integrable functions

I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function $f(x)$ satisfies the condition ...
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Behaviour of Fourier Inverse Transform after non-linear modulation

Suppose $\phi$ is a continuous nowhere differentiable function. $g$ some function in Schwartz space such that $\hat{g}$ has compact support. Define $f(x) = \int_{-\infty}^\infty ...
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1answer
37 views

Fourier transform of a pyramid

Has anyone calculated the Fourier coefficients for a pyramid function? Let us define the pyramid function as, $z = f(x,y)$. We are looking at 5 planes making up the pyramid. The 4 base points and apex ...
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42 views

zero distribution of the Fourier kernel $\Phi(u)$ for Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
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2answers
46 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
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1answer
35 views

A form of Nash's inequality, $\|f\|_2\le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta$

For $f\in \mathcal{S}(\mathbb{R})$ can anyone help me prove the following Nash inequality, $$\|f\|_2 \le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta.$$ I believe $\alpha$ and $\beta$ should be $2/3$ and ...
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Fourier transform of isotropic Laplace distribution (2D)

How would I evaluate the Fourier transform of an isotropic 2D Laplace distribution? $F(\omega_x,\omega_y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-b \sqrt{x^2+y^2})\exp(-j\omega_x ...
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1answer
17 views

How to prove these inequalities using Fourier analysis methods

I wondered if anyone could help me prove these inequalities for $f\in \mathcal{S}(\mathbb{R})$ and $\lambda>0$: $(1) ...
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1answer
195 views

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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1answer
22 views

conditions on Fourier Transform of derivative

At page 445 of Myint-U's Linear Partial Differential Equations (4th Ed), Fourier Tranform of derivative is defined as: Let $f$ be a continuous and piecewise smooth in $(-\infty, \infty)$. Let $f(x)$ ...
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122 views

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
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113 views

Reconciling two intuitions about convolution

There are two intuitive things convolution does. In the time domain, it represents the distribution of the sum of two independent random variables. In the frequency domain, it's just multiplication. ...
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209 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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2answers
51 views

Use Fourier Transform to Show that $f=0$ a.e.

I was working through an old qualifier on my own when I ran across this following question that I was unable to crack. Here it is verbatim: "Let $f\in L^2(\mathbb{R}, \mathcal{L}, m)$ and suppose ...
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1answer
45 views

If the Fourier Transform of $f(x)$ is known, can one deduce the Fourier Transform of $|x|f(x)$? [closed]

If the Fourier Transform of $f(x)$ is known, can one deduce the Fourier Transform of $|x|f(x)$ ? I've been trying to find the Fourier Transform of $|x|^{7/6}K_{-1/6}(x)$. I know the transform of ...
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Fourier Transform of $|x|^\frac{7}{6} K_{-\frac{1}{6}}(|x|)$ [closed]

What is the Fourier Transform of $|x|^{\frac{7}{6}} K_{-\frac{1}{6}}(|x|)$ with $K_{-\frac{1}{6}}$ the modified bessel function of the second kind?
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2answers
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How is the Fourier transform a geneeralization the the Fourier series?

I have taken a self-tought course on the subject of Fourier series and Fourier transform and I got the message the the latter is a genaralization of the first. I know that the idea that the Fourier ...
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Inverse Fourier Transform gives a complex function when it shouldn't

Last I had an exam and there was the following question: Find $f: \mathbb{R} \rightarrow \mathbb{R}$ such that ...
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1answer
30 views

What is Fourier transform of space variable? on the similar grounds what is the Laplace transform of the same?

I understand that the transform of time domain is frequency domain and the transformation of time to frequency domain is done by Fourier/Laplace transforms. I am confused about the transformation of ...
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27 views

Fourier Transform on $L^1(\mathbb{R})$

For $f,g\in L^1(\mathbb{R})$, prove or disprove: $\hat{f}(\xi)+e^{i\pi \xi^2}\hat{g}(\xi) = 0$ for all $\xi\in\mathbb{R}$ implies $\hat{f} = \hat{g} = 0$.
2
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1answer
30 views

Inverse-Fourier transform of a function after non-linear frequency modulation

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is ...
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31 views

Fourier series & Fourier transformation [closed]

Tell me where we use Fourier series & transform in real life? Please mention an example problem which helps me to understand easily about Fourier series &Fourier transformation?
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2answers
105 views

What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
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2answers
58 views

Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$ I get all the steps of the proof, except ...
3
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1answer
33 views

How to understand the mapping between a periodic function to its Fourier coefficients?

For a periodic function $f(x)=f(x+T)$, its Fourier transform can be written as an infinite sum: $$ f(x)=\sum_{-\infty}^{\infty}c_n e^{2\pi i x/T}. $$ This seems to suggest that the information ...