Articles of fourier analysis

Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) dx. $$ I can’t figure out how to evaluate this integral. Am I trying the wrong approach to calculate the transform or should I be able the integral. Note the integral […]

$\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 Riemann-Lebesgue lemma about Fourier series, but $\frac{f(t)}{t}$ is not an integrable function. I tried to define $g(t) = \frac{2f(t)\sin(\frac{t}{2})}{t} $ which tends to $f(0)$ as $t\to0$ and $g(t)$ […]

Why the difference between definitions of the discrete/continuous Fourier transforms?

I should preface this question with the fact that I’m not familiar with the meaning/utility of the Fourier transform. Perhaps more accurately: I may have learned them, but have since forgotten; in any case, I’m just looking into some info about it to help a student out with its calculation, as I figured that would […]

number of zeros of the superposition/interference of sine oscillations

There is a tricky problem to solve and we ask for your kind help. In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases and frequencies. They all start at $f(x_0=0)=0$. Is there a way to estimate or precise calculate the […]

Question about proof of Fourier transform of derivative

If $f\in L^1(\mathbb{R})$, $f'(x)$ exists and is continuous, and $f’\in L^1(\mathbb{R})$, then $\widehat{f’}(t)=2\pi i \widehat{f}(t)$. I’ve stated the above theorem from a textbook that I’m reading. The author uses the Sobolev inequality in the proof to show that $f(x)\to 0$ as $x\to 0$. (And Sobolev inequality, as stated in the textbook requires continuity of $f’$.) […]

Complex Measure Agreeing on Certain Balls

I came across this problem and am lost as to how to solve it. Let $r>0$ be fixed. Suppose $\mu, \nu$ are complex Borel measures on $\mathbb{R}^d$ such that for each open ball B of radius $r$, $\mu(B)=\nu(B)$. Then $\mu=\nu$. I thought this might be an application of $\pi-\lambda$ theorem, but I realized we wouldn’t […]

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$ The answer given by Wolfram Alpha is $\sqrt{2\pi}\xi\exp(-\xi^2/2)$. Observe how this is related to the Fourier transform of $x\exp(-x^2/2)$: the part $\int_{-\infty}^{\infty}x\exp(-x^2/2)\cos\xi x \ \mathrm dx=0$ since the integrand is odd. In addition, what are the Fourier transforms of $x^k\exp(-x^2/2)$ for $k=2,3$? Related: How do I compute $\int_{-\infty}^\infty e^{-\frac{x^2}{2t}} […]

Identity for all uniformly bdd functions follows “by basic Fourier (transform) analysis”?

I found an interesting identity in a paper in a paper I am reading (page $9$). The statement is as follows: $\mathcal F$ is the Fourier transform, $P$ the law of a random variable $X$ and $\varphi$ is the characteristic function of $X$ $$ \int_{\mathbb R}f\ast\mathcal F^{-1}[\frac 1{\varphi}(-\bullet)](x)P(dx)=f(0) $$ for any uniformly bounded function $f$. […]

Fourier transform is real if $f$

I want to prove that the Fourier transform $F(\xi)$ of a function $f$ will be a real function when, and only when, $f(x)$ is an even function. I’m using the following definition of Fourier transform: $F(\xi) = \int_{-\infty}^{\infty} \! e^{-2x\pi i \xi}f(x) \, \mathrm{d}x$. I have problems trying to prove that $f$ is even. Can […]

Writing function as infinite Fourier sum with sine kernel

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Also, $f$ is continuous and goes to $0$ at $\pm \infty$. Let $K(y)=\dfrac{1}{\pi y}\sin(\pi y)$. Show that $f(x)=\sum_{n=-\infty}^\infty f(n)K(x-n)$ for every $x$. This looks like a Fourier sum of some sort, but it’s rather strange that the index $n$ goes from $-\infty$ to $\infty$. […]