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, Littlewood-...

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21 views

Application of Givens rotation to two matrices

I've been reading the following paper , and have come across something that seems like an error to me. In Algorithm 2, the Givens rotation, $q_s$, is applied in the following way: $A \leftarrow q_s ...
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30 views

A question on Schwartz space on the integers

I have a question regarding Schwartz space on the integers. The definition (of the semi-norms which can generate such space) is given as follow: Can someone explain why we need a square on the norm ...
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1answer
24 views

Find $f\in C^0(S^1)$ that satisfy $\limsup_{n\to \infty }\|S_nf -f\|_{L^\infty }>0$

I have to construct a continuous function $f\in \mathcal C^0(S^1)$ (where $S^1=\mathbb R/\mathbb Z$) that satisfy $$\limsup_{n\to \infty }\|S_n f-f\|>0$$ where $S_nf$ is the $n-$th Fourier partial ...
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1answer
37 views

Notation in harmonic analysis

in the Paper "The multilinear restriction estimate: a shoort proof and a refinement" the author Ioan Bejenaru uses the brakets $\langle,\rangle$, for example in the inequality 2.5. What does this ...
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103 views

Is Plancherel's theorem true for tempered distribution?

Let $f, g\in L^{2},$ by Plancherel's theorem, we have $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle.$$ My Question is: Is it true that: $$\langle f, g \rangle= \langle \hat{f}, \...
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1answer
28 views

In what sense is the poisson kernel in higher dimensions an approximate identity

Given a function $g \in C^2(S^{n-1})$ the unit sphere then the unique solution to laplaces equation on the unit ball with this boundary data, is given by reflecting the green's function for the ...
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27 views

Decay of heat kernel on $\mathbb{T}$

I'm studying Muscalu and Schlag's Classical and Multilinear Harmonic Analysis, v. 1. One problem asks to study the heat equation on $\mathbb{T}$, i.e. $$u_{t} = u_{\theta \theta} \quad \text{on} \, \...
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If the Fourier serie $S_nf\longrightarrow g$ in $L^p$ then $f=g$.

Suppose that $f\in L^1(\mathbb S^1)$ where $\mathbb S^1=\mathbb R/\mathbb Z$. Suppose that the sequence of partial Fourier sums $\{S_nf\}_{n\geq 1}$ converge in $L^p(\mathbb S^1)$ toward some $g\in L^...
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110 views

Riesz transform does not preserve continuity

I've read somewhere that the Riesz operator $R_j$ defined by $$R_j f(t) := c(n) \, \text{pv} \int_{\mathbb{R}^n} \frac{x_j}{|x|^{n+1}} f(t-x) \, dx$$doesn't preserve the continuity, but I can't ...
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66 views

Is Riemann–Lebesgue lemma valuble in $L2(\mathbb{R})$

If $f\in L_1$ on $\mathbb{R}$, that is to say, if the Lebesgue integral of $|f|$ is finite, then the Fourier transform of $f$ satisfies $$\hat{f}(z):= \int_{\mathbb{R}} f(x)e^{-izx} dx \rightarrow 0, \...
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23 views

Convergence of Sobolev function in a domain with a curve removed

Let $B\subset \mathbb R^2$ be given as a unit ball. Let $\omega\subset W^{2,2}(B)\cap L^\infty$ be given. (So we are only in 2d space) Let $\Gamma\subset B$ be a closed Lipschitz curve such that $\...
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1answer
19 views

About the weighted norm for integralation [closed]

Let $w:\mathbb{R}^n\rightarrow \mathbb{R}$ be a non-negative, measurable function. And we define a weighted integrable space $L^1(w)$ whose norm is defined by \begin{equation} \|f\|_{L^1(w)}:=\int_{\...
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33 views

Is this statement true? (characterize elements of dual group)

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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17 views

theorem on representation of topological group

i am currently pursuing a course in basic abstract harmonic analysis.i am really stuck in the followinng theorem: let u be a unitary representation of a group G(locally compact hausdorff space) on a ...
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15 views

The regularity of harmonic function over the tour

Let $B_1,B_2\subset \mathbb R^2$ be the ball centered at $0$ with ridus $1$ and $2$, respectivily. Define $\Omega:=B_2\setminus B_1$. Let $\phi\in C_c^\infty(\mathbb R^2)$. We consider the following ...
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15 views

problem on finding representations

i am currently pursuing a course in basic harmonic analysis.i have gone through a lot of texts but i am really finding difficulty to proceed in the following problem: (1) find a representation of $\...
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52 views

Do compact convergence topology and w*-topology coincide on the Pontryagin dual group of a LCA group.

Given a locally compact group $G$, consider its Pontryagin dual $G^{'}$. Do compact convergence topology and weak*-topology on $G^{'}$ coincide?. When we talk about weak*-topology on $G^{'}$, it ...
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1answer
22 views

Unitary dual of $\{0\}$ and $\mathbb R$

How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively. Thank you ...
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1answer
18 views

Show that there is $C_1,C_2$ s.t. $C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$ where $D_N$ is the dirichelet kernel.

Let $D_N=\frac{\sin(\pi(2N+1)x)}{\sin(\pi x)}$ the dirichlet kernel. Show that there is $C_1,C_2$ s.t. $$C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$$ where $\mathbb S^1=\mathbb R/\mathbb ...
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1answer
59 views

Inequality regarding $g (x) \in S(\mathbb{R})$

A proof relies on a inequality I cant a mange to understand ; $\sup_{x} \mid x \mid ^{l} \mid g(x-y) \mid \le A_{l}(1+ \mid y \mid )^{l} $ given that $g \in S(\mathbb{R})$ i.e $\sup_{x} \mid x \mid ^{...
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1answer
57 views

Applying the Riesz-Thorin Interpolation theorem

Consider a linear operator $T$ given by $$T(f)(x)=\int_{Y}K(x,y)f(y)d\nu(y),\qquad x\in X.$$ Let $1\le p_{1}$, $q_{1}\le\infty$, $p_{0}=1$, $q_{1}=\infty$, $\frac{1}{p_{1}}+\frac{1}{p_{1}'}=\infty$ ...
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1answer
25 views

What is the harmonic conjugate of $u=4xy-3x+5y$?

What is the harmonic conjugate of $u=4xy-3x+5y$? I got $u'x=4-y=v'y$ then I integrated $v'y$ to get $v= 2y^2-3y+h(x)$. Then I did $-u'y=v'x$ so, $5= h'(x)$ then I integrated $5$ with respect to $x$ ...
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52 views

Matrix representation of Heisenberg group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3$ is given by $$ \begin{pmatrix} 1 & a & c\\ 0 & ...
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36 views

Fourier transform identity on $L^{p}(\mathbb{T})$

Let $p\in (1,2]$. I want to prove there exists $C$ such that for $f\in L^{p}(\mathbb{T})$ we have $$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{p}|n|^{p-2}\le C\|f\|^{p}_{L^{p}(\mathbb{T})}$$ If $p=1$, it's ...
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19 views

Nonlinear Maximum Principle estimate

Im interested in the in the 2D Boussinesq equations given by $$\begin{cases}\partial_{t}u+u\cdot\nabla u+\nabla p+\Lambda^{\alpha}u=\theta e_{2}\\ \nabla\cdot u=0\\ \partial_{t}\theta+u\cdot\nabla\...
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1answer
28 views

What the expression of a one-dimensional representation of $H$

Let $G= \{ g=(x,y,t); \quad x,y,t \in \mathbb R\}$ be the Heisenberg group and $H= \{ g=(x,y,t) \in G; \quad x=0\}= \{ h=(0,y,t); \quad y,t \in \mathbb R\}$ be a subgroup of $G$. I want to know why ...
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15 views

Show that $ \sum_{\mathbb{Z}/N\mathbb{Z}} f(n) g(n+r)h(n+2r) = \sum_{a \in \mathbb{Z}/N\mathbb{Z}} \hat{f}(a)\hat{g}(-2a)\hat{h}(a)$

I found this Fourier series identity in a book on Harmonic analysis but the proof is inclear. Maybe it makes more sense using bra-ket formalism. $$ \sum_{r, n \in \mathbb{Z}/N\mathbb{Z}} f(n) g(n+r)...
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25 views

General derivative of composition

We know that if $G\in \mathcal{C}^1(\mathbb{R})$ such that $G(0)=0$ and if $u \in W^{1,p}(I)$, then $G(u) \in W^{1,p}(I)$ and $(G(u))' = G'(u) u'$. Do we have a similar result for $W^{s,p}(I)$ with ...
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26 views

Proof of existence of a non-trivial character on a local field of positive characteristic

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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32 views

Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ \int_0^1\|f(...
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1answer
24 views

let $f(x)$ is real polynomial. Can we say that $f(\left| x \right|)$ is subharmonic?

A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}{{2\pi }...
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1answer
32 views

Is the product of two subharmonic function necessarily subharmonic?

Defin: A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}...
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1answer
47 views

If $G=\mathbb {R}$, how $\hat{f}(\chi)=\int_{G}f(x)\overline{\chi (x)}dx$ becomes $\hat{f}(\xi)=\int_{\mathbb{R}}f(x)e^{-ix\xi}dx$?

The dual group of a locally compact Abelian group is used as the underlying space for an abstract version of the Fourier transform. If a function $f$ is in $L^{1}(G)$, then the Fourier transform is ...
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1answer
28 views

Different methods to compute a unitary representation

Given a nilpotent Lie group $G$ (for example the Heisenberg group), what is the most effective method to calculate their unitary representation: The orbit method due to Kirillov; or The induction ...
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1answer
51 views

Does this function grows faster than any exponential function?

I need following result in understanding of a result given in a paper.Any ideas to prove this? Consider the function $\hat {M}: \bf R \to R$ defined as $$\hat{M}(y)=\int_{\bf R}\vert f(x) \vert e^{\...
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65 views

where can I find the proof of this theorem of wavelet frames?

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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1answer
29 views

How can we prove that the discrete Fourier transforms preserves the inner product up to a constant factor?

Let $d\in\mathbb N$, $\omega\in\mathbb C$ be a primitive $n$-th root of unity and $$\operatorname{DFT}_\omega:\mathbb C^d\to\mathbb C^d\;,\;\;\;z\mapsto\left(f_z\left(\omega^0\right),\ldots,f_z\left(\...
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1answer
31 views

Trying to understand the proof of Beurling's theorem for Fourier transform pairs from Lars Hormander's Paper

Let $f \in L^1(\bf R)$ and assume that $\int \int_{\bf R^2}\vert f(x) \hat{f}(y) \vert e^{\vert xy \vert}dxdy< \infty$ where $\hat f$ denotes Fourier transform of $f$. Then $f=0$ Proof: Set $\...
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1answer
45 views

How do you prove that a function $f \in L^1( \bf R)$ and its Fourier transform cannot simultaneously be very small at infinity?

In a Research Paper it is stated that(Without any proof): It is well known that a function and its Fourier transform cannot simultaneously be very small at infinity? In my First course of ...
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92 views

Are Sobolev spaces $W^{k,1}(\mathbb R^d)$ and $H^{k,1}(\mathbb R^d)$ the same?

We consider the following spaces $H^{k,p}(\mathbb R^d)$, $k \geq 1$ is integer, $p \geq 1$ (Bessel potential spaces): $$ H^{k,p}(\mathbb R^d) = \bigl\{ f \in L^p(\mathbb R^d) \colon \mathcal F^{-1}[...
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55 views

Prove that the Pontryagin dual of $\mathbb{R}$ is $\mathbb{R}$.

From Wikipedia: the group of real numbers $\mathbb{R}$, is isomorphic to its own dual; the characters on $\mathbb{R}$ are of the form $r \to e^{i\theta r}$. How can I prove this assertion?
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46 views

Direct sum decomposition of $L^2(\mathbb{R})$ using Fourier Transform

Let $L_+^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^+}\}$ and $L_-^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^-}\}$, where $\hat{f}$ denotes the Fourier ...
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45 views

Proof of Howe-Moore Property for SL(n,R)

On page 210 of Howe and Tan's Non-Abelian Harmonic Analysis there is the following proposition and proof: Let $(\rho, V) $ be a unitary representation of $SL(n,\mathbb{R})$. The following are ...
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25 views

Coadjoint representation $Ad^*$ of the Heisenberg group

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
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21 views

The $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group)

I did not understand why, the $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group and $\mathcal{g}*$ is the dual space of its Lie algebra $\mathcal g$) are given by i) The ...
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1answer
55 views

Bound a complex exponential sum when we can only estimate its argument

Suppose we have a function $f : \mathbb{N_0} \to \mathbb{R}$ for which we can give an estimate of its values, and say its values $f(n)$ are roughly uniformly distributed for $n$ in some range $[1,N]$. ...
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17 views

references for basic course in abstract harmonic analysis

I am currently studying a basic course in abstract harmonic analysis. I am really finding difficulty in understanding the concepts of representation of algebras, groups, etc. Moreover, any of the ...
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1answer
46 views

Example of a linear functional, but not a distribution

I'm looking for an example of a linear functional $u: C_c^\infty(\Omega) \to \mathbb C$ ($\Omega \subset \mathbb R^n$ open), which is not a distribution. I could not find anything... I thought of ...
2
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1answer
29 views

Decay property of oscillatory integrals in $\mathbb{R}^n$

We know that an oscillatory integral in $n$ dimensions is an integral of the form \begin{equation*} I(\lambda)=\int_{\mathbb{R}^n}e^{i\lambda\phi(x)}f(x)dx \end{equation*} where $\phi\in C^{\infty}$ ...
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1answer
37 views

Concluding that if $f\in L^{1}_{\text{loc}}(\mathbb{R}^{n})$ and $\mathcal{M}(f)\in L^{1}(\mathbb{R}^{n})$ then $f=0$ a.e.

Let $\mu$ be a Lebesgue measure, $\mathcal{M}$ be the Hardy-Littlewood Centered Maximal Function, and the rest as in the title. Then to this end I define $f_{R}(x):=f(x)\chi_{|x|\le R}$ where $\chi$ ...