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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Intersection of subspace of cyclical rotations with orthant
In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector
$\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
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Trigonometric polynomial derivative upper bound
Let $P$ be a trig poly of degree $N$ on the torus $\mathbb T$. Prove that
$$\Vert P' \Vert_\infty \lesssim N\Vert P \Vert_\infty.$$
I'm not sure how to approach this problem, though I feel like some ...
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Pre-factors and bounds for sine transform
Given
$$G(\omega)\sin(\omega t) = f(t)$$
Why does the following equation take the sine transform from $0$ to $2\pi / \omega$? And why is the pre-factor given as $\frac{\omega}{\pi}$?
$$G(\omega)=\frac{...
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Values of Lebesgue integrable function from the integral of the product of a known function.
Let's say we have three complex absolutely integrable and square integrable functions $A,B,C\in \mathbb{L}^1(\mathbb{C})\cup \mathbb{L}^2(\mathbb{C})$ such that the following holds:
$$A(y) = \int_{-\...
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How does this definition of Fourier transform in Fulton and Harris 3.32 relate to the usual notion of Fourier transform?
This is exercise 3.32 in Fulton and Harris' Representation Theory: A First Course.
It defines Fourier transform in a form unfamiliar to me, and I could not find any definition of Fourier transform ...
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All DFT of binary numbers subsets of prime length are nonzero
Let $p$ be a prime. Consider a sequence $S$ of $p$ binary numbers $x_n \in \{ 0, 1 \}$, i.e. $S = \{x_1, x_2, \cdots, x_p\}$, where the number of zeroes in $S$ is neither $0$ nor $p$. Then the ...
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An estimate using a variant of Holder's inequality in Fourier space
I have a problem of an estimate of $(1.3)$ in the paper https://arxiv.org/abs/2010.10460, which says that
Assume
$$\mathcal{F}\{Q_m[f,g]\}(\xi)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbb{R}^3}m(\xi,\...
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Fourier multipliers on $L^2(\mu)$
On $L^2(\mathbb{R}^d)$, we have $T_m$ defined $\widehat{T_m f} = m \widehat{f}$ is a bounded operator on $L^2$ if and only if $m \in L^\infty$.
What can be said about the same problem for more general ...
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Step in the proof that the function is nowhere differentiable in the Fourier Analysis textbook
I'm studying a Stein-Shakarchi Fourier Analysis textbook and I'm stuck on the proof on page 116.
Help me understand why differentiability at one point allows us to establish such an inequality. It ...
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Sharp constant in the $L^p$ regularity estimate?
Problem: Let us denote $\mathbb{W}^{2,p}(\mathbb{R}^2)$ the space of Sobolev functions in the plane. Let us denote with $\Delta$ the classic Laplacian operator. We know that there exists a constant $C&...
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Proving an identity related to Short-Time Quaternion Fourier Transform.
I begin with some simple context knowledge that is needed. Given a function $f \in L^1(\Bbb R^2, Cl_{0,2})$ we define the quaternion fourier transform of $f$ as the integral
$$ w \to Ff(w) = \int_{\...
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persavel identity
i want to prove parseval identity
$||f||_2^2 = S_N(f)^2$ ($S_N(f)$ partial sum of Fourier series)
in my proof i know bessel identity:
$||f||_2^2 \geq S_N(f)^2 $
and i know that giving f function ...
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Fourier Transform Method for Solving Fredholm Translation-Invariant Covariance-Kernel Integral Equations
I'm examining a problem involving the Fredholm integral equations of the second kind and trying to apply Fourier transform techniques. Particularly, the challenge arises when seeking to express the ...
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Semigroup property between pseudodifferential operators and differential operators
Given a positive integer $n$ and a>0. Let consider the operators $\nabla^n (\cdot)= \sum_{i=1}^d \partial_i^{n}(\cdot) $, and $(1- \Delta)^{\frac a 2} $ defined at the Fourier level as (modulus ...
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Do mollifiers exist in dimensions higher than 1?
A mollifier is defined as a function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ such that
$$\int_{\mathbb{R}^n} \varphi(x) dx = 1$$
$\varphi$ has compact support
$$\lim_{\epsilon \rightarrow 0} \...
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An identity for trigonometric Polynomial of degree $N$
For any trigonometric polynomial $P$ of degree $N$, which is of form $\sum\limits_{|k|\leqslant N}c_ke^{2\pi i kx}$, then how to establish the following identity?
$$P^{\prime}(x)=\sum_{k\in \mathbb{Z}}...
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Do there exist mathematical transforms other than the Fourier Transform for which there exists some sort of a fast convolution theorem?
One nice property of the Fourier transform is it's famous convolution theorem :
$$f*g = \mathcal{F}^{-1} \left\{ \mathcal{F}\left\{ f \right\} \cdot \mathcal{F}\left\{g\right\} \right\}$$
If we want ...
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Asymptotics of a multi-dimensional Fourier series
Let $\alpha\in(0,2]$ and $$f_t(u)=\sum_{m\neq 0}\frac{1}{m^\alpha}\exp(-\|m\|^2 t^2)\exp(im\cdot u),u\in \mathbb R^d.$$
Would anyone know a closed form expression for $f_t(u)$, in all dimension $d$? I ...
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Is the convolution of a tempered distribution and a Schwartz function also a function?
Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as
$$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$
where $\tilde{f}(x) = f(-x)$. The above ...
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double doubt working with fourier inverse theorem and a calculation of a sup
the first one is this :
I am working with Fourier inverse theorem and I have $f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega$ with $f \in L^1(\mathbb{R})$
...
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Rate of convergence of fourier series for non periodic functions
Consider
Suppose that an analytic function $f$ on $[-1,1]$ is not periodic, yet $f(-1)=f(+1)$ and $f^{\prime}(-1)=f^{\prime}(1)$. Integrating by parts the Fourier coefficients $\hat{f}_n$ show that $\...
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FFT Basic Question
I've followed a basic tutorial on FFTs and trying to get my head around what's going on here. As I understand it fourier transform converts from time domain to frequency domain. So in a hypothetical ...
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A Littlewood-Paley question
For a suitable function smooth function $\psi$ supported on $\{x\in\mathbb{R}^d: 2^{-1} \le |x| \le 2\}$, define $P_jf(x)$ as
\begin{aligned}
P_jf(x)=\mathcal{F}^{-1}\left(\psi(2^{-j}\xi)\...
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Fourier transform of $\frac{1}{|x|}$ on $\mathbb{T}^{n}$
Let $|x|=\sqrt{x_1^2+\cdots+x_n^2} ~$ for $~x\in\mathbb{T}^n=[-\frac{\pi}{2},\frac{\pi}{2}]^n$, $~n\geq 2$.
We can define the Fourier transform of $f(x)=\frac{1}{|x|}$ as
$$ \hat{f}(k)= \int_{\mathbb{...
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L_1 norm of spherical harmonics
Let $Y_{k,j}:{\mathbb S}^{n-1}\to {\mathbb R}$ be spherical harmonics on $n-1$ -dimensional sphere.
We know that $\|Y_{k,j}\|_{L_2({\mathbb S}^{n-1})}^2 = \int_{{\mathbb S}^{n-1}}Y^2_{k,j}(x)d\sigma(x)...
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Inverse Fourier Transform of $F(\omega) = \frac{1}{1+i(\omega-3)}$
I am having trouble finding the inverse Fourier transform of:
$$F(\omega) = \frac{1}{1+i(\omega-3)}$$
Currently i am assuming its $${e^{i3t} * e^{-t}}$$
A confirmation that i am at least heading in ...
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Extremely rigorous (research level) treatment of the laplace transform
I am looking for an extremely rigorous treatment of the laplace transform.
Stating the usual formulas for convenience:
$$
\mathcal{L}: ?\to?\quad\mathcal{L}(f)(s) = \int_{\mathbb{R}^n}f(x)\...
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Derivative of the Dirac delta function. The Fourier transform of this function is the a-th derivative of the Dirac delta function. How to interprete?
Evaluating this integral
$$\int_{-\infty}^{\infty} x^{a} e^{-i(k-k')x}dx=\sqrt{2\pi}(-i)^a\delta^{(a)}(k-k')$$
I got this result that lead to the definition of $$\delta^{(a)}(k-k')=\sqrt{2\pi}(i)^a\...
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Period and Fourier series
For signal x(t) I have to show it is periodic by finding its period. After that, I need to find Fourier coefficient $$ a_n $$ and use them to reconstruct signal $$ x(t) $$ taking a finite number of ...
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Conections between Parseval's Identities
I know how to prove that if $f,g \in \mathcal{S}$ (Schwartz space - or even ${L}^2$), then $$f*g \in \mathcal{S}$$ and $$\|f\|_2^2=\|\widehat{f}\|_2^2.$$ Here, $$\|f\|_2=\sqrt{<f,f>}= \left[ \...
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How do I prove that $\left(\frac{1}{10^5} \sum_{n=-\infty}^{\infty} e^{\frac{-n^2}{10^{10}}}\right)^2 \approx \pi? $
How do I prove that
$$\left(\frac{1}{10^5} \sum_{n=-\infty}^{\infty} e^{\frac{-n^2}{10^{10}}}\right)^2 \approx \pi?$$
I am thinking of start with gaussian function and using the poison summation ...
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Fourier Transform of $\exp\left[-e^{-k^2}\right]$
I want to calculate the inverse Fourier transform of $\exp\left[-e^{-k^2}\right]$. One of the ways I can imagine is to expand the exponential in a series
$$
\exp\left[-e^{-k^2}\right] = \sum_{n=0}^\...
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Convergence of Power Series to Its Cesaro Sum sentence in Fourier proof
Well, I am learning about Fourier sum, and I encountered Cesaro sum in the proof of convergent uniformly of Fourier sum,
I know that Fejér sentence says that:
$\|f(x) - \sigma_n(f))\| < \epsilon .$ ...
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Is RKHS of Laplace kernel on a subset of sphere a Sobolev space?
Consider $\mathcal{X}\subset \mathbb{S}^n$ and Laplace kernel $k(x,y)=\exp(-\|x-y\|)$.
Is the RKHS $H(\mathcal{X})$ given by $k(x,y)=\exp(-\|x-y\|), x,y \in \mathbb{S}^n$ equivalent to Sobolev space $...
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Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$
I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
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Notion of a basis that allows infinite sums
So with no other structure it is clear that a vector space has no notion of convergence and thus infinite sums make no sense. However, suppose $V$ is an inner product space, then we have a norm which ...
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Why do characters fail to characterize non-abelian LCH-groups?
For any locally compact Hausdorff abelian group (LCA group) $A$, a character $\xi\colon A\mapsto\mathbb{T} $ is definined as a continuous group homomorphism to the unit circle $\mathbb{T}\subseteq\...
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Riemann-Lebesgue Lemma and trigonometric polynomial
from my understanding Riemann-Lebesgue Lemma says that every continuous and 2π periodic function, the Fourier coefficient go to zero as n approach to infinity
$\lim_{|n|\to+\infty}\hat{f}(n)=0,$
but i ...
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Fourier Transform of $\sin(\sin(x))$ [closed]
How can I calculate the Fourier Transformation or Fourier series of $\sin(\sin(x))$?
I know that the function is odd and the Fourier series should look something like this
$$
\sin(\sin(x)) = \sum\...
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A question about norm of weighted Sobolev space $e^{\gamma t} H_\gamma^s$
This question really bothers me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic equation.
I am reading Multidimensional Hyperbolic ...
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Upper bound on "time-continuous Zadoff–Chu signals"
I'm trying to find a tight (as tight as possible) upper bound for $\lvert x_u(t) \rvert$, where $x_u(t)$ is the $T$-periodic signal defined by the Fourier sum
$$
x_u(t) = \frac{1}{N}\sum_{k=-N_0}^{N_0}...
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Some confusing basic inequalities on the Hardy-Littlewood Maximal Function
I am a beginner in harmonic analysis and I already know the definition of Hardy Littlewood Maximum Function. Our next considerations are all focused on $f\in L^1\left( \mathbb{R} ^n \right) $, but we ...
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Question on Rudin's Proof of the Fourier Transform Inversion Theorem (Theorem 9.11 in Real and Complex Analysis)
In Walter Rudin's Real and Complex Analysis, he gives a proof of the Fourier transform inversion theorem as follows:
Theorem 9.6 states that if $ f \space \epsilon \space L^1$, then $\hat{f}\space \...
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Question on Rudin's proof of Theorem 9.5 in Real and Complex Analysis
I'm working through Theorem 9.5 in Rudin's Real and Complex Analysis and I'm having trouble understanding one of the steps. The statement of the proof is as follows:
Rudin uses Theorem 3.14, which ...
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Taylor series for radial function
Denote by $\hat{J}$ the Fourier transform of $J\in C(\mathbb{R}^n,\mathbb{R})$, a nonnegative, radial function with
\begin{equation}
\displaystyle\int\limits_{\mathbb{R}^n}J(x)~\mathrm{d}x=1.
\end{...
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How do I compute the Fourier Coefficients of a Riesz Product?
Let $(\alpha_n)_{n\in \mathbb{N}}$ be a sequence of non-zero real numbers such that $\sum_{n=1}^\infty \lvert \alpha_n \rvert^2 < \infty$. We consider the Riesz Product
\begin{align*}
\prod_{n=1}^\...
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Fourier transform of $\frac{1-e^x}{1+e^x}$
I was trying to compute $\int_{-\infty}^{\infty}e^{ikx}\frac{1-e^x}{1+e^x}\, dx$ from a Mathematical trivium. I tried first with contour integration but finding the right shape was hard. I gave it a ...
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$A \subset \ell^2(\Bbb Z)$ as a collection of Fourier coefficients
Let $A$ be a closed subspace of $\ell^2(\Bbb Z)$. Suppose for every $\{a_n\}_n \in A$, we have $\{a_{n+m}\}_n \in A$ for each $m \in \Bbb Z$. Show that there exists a measurable set $E\subset \Bbb T$ ...
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Question about Theorem 9.2 in Rudin's Real and Complex Analysis
I am working through the chapter on Fourier Transforms in Rudin's Real and Complex Analysis, and I am having difficulty understanding a property of the Fourier Transform. Rudin defines the Fourier ...
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Zero set of double trigonometric polynomials.
Let $F\subseteq \mathbf{Z}^2$, ($\mathbf{Z}^2$ is the integer lattice), such that $|F| = \alpha < \infty$. For any $(m_1,m_2)\in \mathbf{Z}^2\setminus F$, we can define a double trigonometric ...
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