# Questions tagged [fourier-analysis]

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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### Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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### Fourier transform for dummies

What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit at Mathoverflow. ...
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### How to calculate the Fourier transform of a Gaussian function?

I would like to work out the Fourier transform of the Gaussian function $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$ It seems likely that I will need to use differentiation and the shift rule at some ...
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### Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
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### Derivative of convolution

Assume that $f(x),g(x)$ are positive and are in $L^1$. Moreover, they are differentiable and their derivative is integrable. Let $h(x)=f(x)*g(x)$, the convolution of $f$ and $g$. Does the derivative ...
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### Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
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### A log improper integral

Evaluate : $$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$ I found it can be simplified to $$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$ I found the exact value in the ...
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### Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$A_f(g) = \int f(\tau)g(t-\tau)d\tau$$ and apply it to $g(t)=e^{ikt}$. ...
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### Compactly supported function whose Fourier transform decays exponentially?

It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential ...
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### Integration (Fourier transform)

$\mathrm{Re}(i) = 0$, but the fourier transform of $f(x) = e^{-ix^2}$ is $g(\alpha) = \sqrt{\pi\over i}\times e^{i\alpha^2 \over 4}$, is it not? Is there an easy to show that it is so, knowing the ...
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### 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 ...
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### Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed. But I'...
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I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ a_0=\... 2answers 3k views ### Applications of Pseudodifferential Operators I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ... 1answer 915 views ### Limit of \lim_{t \to \infty} \frac{ \int_0^\infty \cos(x t) e^{-x^k}dx}{\int_0^\infty \cos(x t) e^{-x^p}dx} Let \begin{align} f(t,k,p)= \frac{ \int_0^\infty \cos(x t) e^{-x^k}dx}{\int_0^\infty \cos(x t) e^{-x^p}dx}, \end{align} My question: How to find the following limit of the function f(t,k,p) \... 1answer 3k views ### A function and its Fourier transform cannot both be compactly supported I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know f is a trigonometric polynomial, I see how to finish the ... 1answer 863 views ### Convolution of an L_{p}(\mathbb{T}) function f with a term of a summability kernel \{\phi_n\} ... is the result in L_{p}? A remark in my notes says yes but I can't see how to verify it. As was pointed out to me in a previous question I asked last night, I need to show that the following ... 3answers 7k views ### Fourier transform of 1/cosh How do you take the Fourier transform of f(x) = \frac{1}{\cosh x} This is for a complex class so I tried expanding the denominator and calculating a residue by using the rectangular contour ... 1answer 5k views ### Inverse Fourier transform of Gaussian I need to calculate the Inverse Fourier Transform of this Gaussian function: \frac{1}{\sqrt{2\pi}} exp(\frac{-k^2 \sigma^2}{2}) where \sigma > 0, namely I have to calculate the following ... 2answers 405 views ### Bounds on f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx} Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \... 1answer 2k views ### Derivative of Fourier transform: F[f]'=F[-ixf(x)] Let us define the Fourier transform of the Lebesgue-summable function f\in L_1(\mathbb{R},\mu_x) as F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x, where \mu_x is the Lebesgue linear ... 3answers 3k views ### Known proofs of Wirtinger's Inequality? I am looking for proofs of the (Poincare-) Wirtinger inequality which states that if f:[0,\pi]\to \mathbb{C} is C^1 and f(0)=f(\pi)=0 then \int_0^\pi |f(t)|^2 dt \leq \int_0^\pi ... 1answer 1k views ### Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ... 1answer 446 views ### On the Fourier transform of f(x)=\ln(x^2+a^2) I would like to derive the Fourier transform of f(x)=\ln(x^2+a^2), where a\in \mathbb{R}^+ by making use of the properties: \mathcal{F}[f'(x)]=(ik)\hat{f}(k)\\ \mathcal{F}[-ixf(x)... 1answer 218 views ### Selfadjoint Restrictions of Legendre Operator -\frac{d}{dx}(1-x^{2})\frac{d}{dx} Problem: Let Lf =-((1-x^{2})f')' be the Legendre differential operator defined on the domain \mathcal{D}(L) consisting of twice absolutely continuous functions on (-1,1) for which f, Lf \in L^{... 1answer 936 views ### L_{p} distance between a function and its translation I'm working through a proof and one of the comments is that for a function f\in L_p (\mathbb{T}):\lim_{t\to 0}\;\|f(\cdot + t) - f\|_p = 0.$$How do I prove it? I think it is intuitively ... 7answers 50k views ### Difference between Fourier transform and Wavelets While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas ... 4answers 6k views ### What is Fourier Analysis on Groups and does it have “applications” to physics? I am trying to be as specific as possible, but I am extremely unclear about this topic (Fourier Analysis on Groups). In Reed-Simon Vol II (Fourier Analysis, Self-Adjointness) there is some ... 1answer 4k views ### Does rapid decay of Fourier coefficients imply smoothness? Under the isomorphism of Hilbert spaces L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n, smooth functions on the circle are mapped to rapidly decaying sequences (see wikipedia). Is the ... 2answers 3k views ### Which functions are tempered distributions? Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function$$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$where \... 8answers 2k views ### Conceptual/Graphical understanding of the Fourier Series. I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ... 1answer 4k views ### Fourier transform of \text{sinc}^3 {\pi t}$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$I want to calculate the Fourier transform. I can't calculate this integral:$$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t
What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.