Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, ...

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Showing the range of Fourier transform on $L^1(\mathbb{R})$ is in $L^1(\mathbb{R})$ for a particular scenario

This is a problem from Katznelson's Harmonic Analysis book(Page 143, Problem 8): Suppose $f\in L^1(\mathbb{R})$ and is continuous at $0$. Also suppose $\hat{f}\geq 0$, then show that $\hat{f}$ is in ...
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Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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Laplace operator and Fourier transform

Let $f$ be a function with compact support in $D\subset \mathbf{R}^2$. Is is known that $$-\Delta f=F^{-1}|x|^2 F f,$$ where $F$ is the Fourier transform. Which is the $n-$dimensional analogous ...
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Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...
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Identity involving partial sums of Fourier series

Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$ How can I show that ...
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Prove partial derivatives of uniformly convergent harmonic functions converge to the partial derivative of the limit of the sequence.

I think the title says it all. If you have a sequence of harmonic functions from a bounded complex domain to the real numbers, show that on a subset at a positive distance from the boundary of the ...
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non tangential maximal function and Hardy-Littlewood maximal function

I'm studying harmonic analysis and found that we can bound non-tangential maximal function by Hardy-Littlewood maximal function. Most books don't give the proof of it. How can I see that? Is there a ...
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Decaying Fourier transform and smoothness

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ I want to show ...
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364 views

Asymptotic error of Fourier series partial sum of sawtooth function

In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement: $$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$ where $\{x\}$ denotes the ...
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What is a “domain” in the maximum-minimum principle?

The maximum-minimum principle says that A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant. Here is my question: If we restrict our attention in ...
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Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t ...
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The decay conditions of the Poisson summation formula

The Poisson summation formula states that for a function $f$ satisfying $$ |f(x)|\leq A(1+|x|)^{-n-\delta},~|\hat{f}(\xi)|\leq A(1+|\xi|)^{-n-\delta} $$ for $\delta>0,$ then the equality $$ ...
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Maximal Function Estimate

Suppose $\psi$ is a rapidly decreasing function; i.e. for all $N>0$ there exists a constant $C_{N}$ such that $\left|\psi(x)\right|\leq C_{N}(1+\left|x\right|)^{-N}$. Define a family of functions ...
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About p-adic numbers

I'm studying the dual group of the diadic rationals, $\widehat{\mathbb{Z}[1/2]}$, where the dual is the dual of Pontryagin of $\mathbb{Z}[1/2]$. In some papers says that $\widehat{\mathbb{Z}[1/2]}$, ...
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A questions about the schur's lemma

Schur's lemma is this: If (ρ1,V1) and (ρ2,V2) are irreducible representations of a group G, then any nonzero homomorphism ϕ:V1↦V2 is an isomorphism. or Schur's Lemma. a. A unitary ...
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Covolution (space) over compact Lie groups

Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual ...
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The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
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Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
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Fourier-restriction on Hyperplane

In Muscalu, Schlag "Classical and multilinear harmonic analysis, Volume 1" (2013), Excercise 11.1 is to prove, basically, that there exists a function $f\in L^p \quad \forall\ p>1$ such, that the ...
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374 views

Harmonic analysis in number theory

When I was reading Folland's A course in abstract harmonic analysis, I was told these materials have wonderful applications to number theory. However, I do not see really a lot of examples there. Can ...
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Are the continuous functions on $G$ dense in $L^{1}(G)$?

If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
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Fourier transform of a special Schwartz function

In Classical Fourier Analysis by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ ...
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$L^2$ norm for difference of translation of disjoint interval

I just need some hint since I've been stuck in this for several hours.. Let $I_1,I_2,\ldots,I_K$ bne disjoint intervals in $[-1/2,1/2)$,and $f(x)=\sum_{j=1}^K\chi_{I_j}$, where $\chi_I(x)$ is the ...
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What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
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A question from Harmonic Analysis - real variable methods, orthogonality book by Elias Stein.

On page 73, it's written that $-\int_{|r|}^\infty s^{n-1} d_s \Phi(s\xi) = \Psi_\xi(r)$, and beneath that it's written that: $$(*)\int_{-\infty}^\infty \Psi_\xi(r)dr = 2\int_0^\infty r^n d\Phi(r\xi) ...
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Banach algebra with left or right minimal ideal without minimal bi-ideal

Let $A$ be a Banach algebra( or a ring). A left ideal $I$ of $A$ is called a left minimal ideal, if $I\neq\{0\}$ and there is no any other non-zero left ideal of $A$ completely lies in $I$. With a ...
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Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: ...
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$L^{p}$ Boundedness of Fourier Multiplier without Littlewood-Paley

Suppose $\zeta\in\mathcal{S}(\mathbb{R}^{n})$ is a Schwartz function such that $\widehat{\zeta}$ is compactly supported away from the origin (say in the annulus ...
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Riesz projection as a Cauchy-type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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A question from Stein's Harmonic Analysis - real variable methods' book.

In the book of Stein, on page 9: "Let us remark that these additional properties easily lead to the following conclusions among others. First note that $\mu (B) >0$ for any ball $B$, which is a ...
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The Neumann Problem on a Half-space when dimension is $2$

Take $\Omega:=\{x=(x_1,x_2):\,-\infty<x_1<\infty,\,x_2>0\}$, i.e., the half-space, and I am interested in the Neumann problem \begin{cases} \Delta u=0&x\in \Omega\\ ...
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Left and Right Discrete Maximal Functions

Define the uncentered maximal function $$\widetilde{M}f(n)=\sup_{s,r\in\mathbb{Z}^{+}}\dfrac{1}{s+r+1}\sum_{k=-r}^{k=s}\left|f(n+k)\right|,$$ where $\mathbb{Z}^{+}=\left\{0,1,2,\ldots\right\}$. Define ...
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Application of weak $L^p$ estimate besides for proving boundedness of some linear operator

For all $1\leq p< \infty$, weak-$L^p(\mathbb{R}^d)$ space is defined as a set of all functions $f$ such that $$\gamma^p|\{x\in \mathbb{R}^d: |f(x)|>\gamma\}|<\infty$$ for every ...
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Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
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Fourier transform of a a function in the space

Which is the Fourier transform (in the sense of distributions) of the function $f(x)=x/\|x\|^n $, where $x$ belongs to the Euclidean space $ R^n$?
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A bound for the product of two functions in BMO

The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true $$ \|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}. $$ ...
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172 views

Uncertainty principle in harmonic analysis

Given a function $f$ (in some suitable (EDIT: nice) function space) supported on a ball $B\subset\mathbb{R}^n$ of radius $R>0$, I have commonly heard people, who understand these things, say stuff ...
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Harmonic function with condition on part of its boundary

Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
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Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
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The integral of a function over $S^1$

Let $S^1=\mathbb R/\mathbb Z,$ I was wondering how to calculate the integral of a function over $S^1$ and why. Like, $\int_{S^1}1 dx=?$ Given an "appropriate" function $f$, what is $\int_{S^1}f(x)dx?$ ...
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Applications of Young's convolution inequality

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided ...
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Restricted Direct Products in Koch's Number Theory

On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...
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Underdamped free vibration proof

I need to prove the solution form of: $$y''+2cwy'+wy=0$$ My book says, after assuming a solution of the form $Ce^pt$, you can show that: $$y=[A\sin(wt)+B\sin(sw)] \cdot e^pt$$ I tried using the ...
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The importance of the Van der Corput lemma in analysis and beyond

The Van der Corput lemma states the following: Introduce the following oscillatory integral $$ I(a,b)=\int^{b}_{a}e^{ih(t)}dt. $$ Then $(1)$ if $|h'(t)|\geq \lambda>0$ and $h'$ is monotonic, then ...
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Upper bound on the integral $\int_{\mathbb R}\omega_I\omega_J$ with weights associated to intervals $I,J$

I'm currently studying Classical and Multilinear Harmonic Analysis. Vol. 1 by Camil Muscalu, Wilhelm Schlag. I need to verify following calculus inequality (Eq. 9.27, at page 255) ...
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Gaussian is the only radial function which is separable

One way to characterize the Gaussian $ae^{b x^2}$ is that its a $C^1$ function $h$ that is radial $h(x,y) = h(\sqrt{x^2+y^2})$ and also separable, that is expressible as a product of one-dimensional ...
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Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
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Solutions of the constant coefficient Helmholtz equation via the Fourier transform

When $f$ is a rapidly decaying Schwartz function, $$ g(x) = \frac{1}{2\lambda} \int_{\mathbb{R}} \sin \left(2\lambda\left|x-y\right|\right) f(y)\ dy $$ is an element of ...