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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Limit of a convolution product

For $\gamma \in \mathbb{R} \setminus \{ 0 \} $ let $k : \mathbb{R}^n \to \mathbb{R}$ be defined by $k(x) = |x|^{-n + i \gamma}$; I've been asked to prove that $\text{p.v. } k * f = \lim_{\epsilon \to ...
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25 views

References for distances in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
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22 views

If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?

if $\{H_n\}$ fourier sequence of a function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
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28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
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21 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
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Planar sets in R^{2} with bounded Fourier transforms

I have a question which I'd be greatly happy to hear an answer for (because it sounds really cool!). Say I'm given some "nice" region $\Omega$ in $[0,1]^{2}$, how can one determine all the $p>0$ ...
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20 views

Proving $\|[b,T](f)\|_{p}\le C\|b\|_{BMO(\mathbb{R}^{n})}\|f\|_{p}$ using the Fefferman-Stein inequality

Let $1<p<\infty$ and $1<r<\infty$ and let $\mathcal{M}_{r}(g)$ denote $\mathcal{M}(|g|^{r})^{\frac{1}{r}}$, where $\mathcal{M}$ is the Hardy-Littlewood maximal function. Also let $T\in ...
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+500

Triangle inequality fails in $L^{1,\infty}$

It can be proved that $\forall\varepsilon>0$ there exists $C(\epsilon)>0$ such that for all $f,g\in L^{1,\infty}(\Bbb R^n)$ we have that $$ ...
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50 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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20 views

Equivalent conditions for weak $L^p$ spaces for $p\leq 1$

I have difficulty doing the following exercise from Tao's real analysis book: Let $X$ be $\sigma$-finite measure space and $0<p\leq 1$. Then show that the following are equivalent: $f$ is in ...
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24 views

Fundamental solution for the p-harmonic and p-biharmonic equation

I am working on $p$-Laplace equation. that is $$\tag{1} -\text{div}(|\nabla u|^{p-2}\nabla u)=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$ and the $p$-bilaplace equation, that is $$\tag{2} ...
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41 views

Sequences $(U_n)$ of neighborhoods of $0$ in a LCA group with $m(U_n)\to 0$

Let $G$ denote an infinite compact abelian group with Haar measure $m$ (so $m(G)=1$). Given a neighborhood $U_1$ of the unit $0$ in $G$ we can find a symmetric neighborhood $U_2$ of $0$ such that ...
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1answer
27 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
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11 views

Riesz basis of Paley-Wiener space.

Let us consider the Paley-Wiener space: $$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$ Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real ...
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28 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of ...
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1answer
36 views

Does the Hilbert transform of Schwarz function decay far away

The Hilbert transform $H$ of Schwarz functions can be defined as \begin{equation} Hf(x)=\int_{|y|<1}\frac{f(x-y)-f(x)}{y}dy + \int_{|y|>1} \frac{f(x-y)}{y}dy. \end{equation} I would like to ...
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1answer
58 views

Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...
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1answer
24 views

Are linearly independent harmonic polynomials orthogonal upon integration over the sphere?

There is a theorem that states that the vector space of homogeneous polynomials decomposes into an orthogonal direct sum of vector spaces of harmonic polynomials as $$ \mathcal{P}_n = \mathcal{H}_n ...
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1answer
16 views

Can we relax the hypothesis of Uniqueness theorem for Fourier series?

I know this fact: "Suppose that $f\in L^{1}(\mathbb T)$ and $\hat{f}(n)=0$ for all $n\in \mathbb Z,$ then $f=0 $ all most everywhere on $\mathbb T$." My Question is: Suppose that $f\in ...
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52 views

When $Q(\alpha)=\int_B |f(x) - \alpha| \, dx$ is minimal?

I've been asked to prove (or find a counterexample) that the quantity $$Q(\alpha)=\int_B |f(x) - \alpha| \, dx$$ is always minimal when $\alpha=\bar{\alpha} = \frac{1}{|B|} \int_B f(x) \, dx$ ...
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21 views

Is $A(D)$ a complemented subspace of $C(T)$?

Let $T$ be the unit circle and $D$ the open unit disk. A function $f$ belongs to $C(T)$ if it is continuous at $T$. A function $g$ belongs to $A(D)$ if it is continuous at $\overline{D}$ and ...
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18 views

Formula connecting the resolvent opeartor andthe spectral density?

I want to know if it is a formula connecting the resolvent opeartor $(\lambda - T)^{-1}$ for a selft-adjoint operator $T$ and its spectral density $e_{\lambda}$. Thank you in advance
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Formula connecting the resolvent and the heat kernels

Using the well known formula connecting the resolvent and the heat operators associated to a selft-adjoint opeartor $A$ \begin{align} (\zeta - A)^{-1} = \int_{0}^{\infty} e^{-\xi t} \, e^{t A} dt; ...
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8 views

Prove that $|Hf(x)|\le C|x|^{-\alpha}$ for $-1<x<0$ where $ Hf(x):=\frac1{\pi}\operatorname{p.v.}\int_{\Bbb R}\frac1sf(x-s)\,ds $

Let $f(x):=|x|^{-\alpha}\chi_{]-1,0[}(x)$, for some $\alpha\in]0,1[$; let's prove that $|Hf(x)|\le C|x|^{-\alpha}$ for $-1<x<0$, where the Hilbert transform of a measurable function $f:\Bbb ...
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44 views

Show that $\iint_{S^1\times S^2}\frac{|f(x)-f(y)|^2}{\sin(\pi(x-y))}dxdy<\infty $

1) Let $f\in H^{1/2}(S^1)\cap C^0(S^1)$. Show that $$\iint_{S^1\times S^1}\frac{|f(x)-f(y)|^2}{\sin^2(\pi(x-y))}dxdy<\infty .$$ 2) Conversely, if $f\in C^0(S^1)$ and $$\iint_{S^1\times ...
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Invariant subspace shift operator $\psi H^2$

$H^2$ - Hardy space analytics function on unit disk; $M_z:H^2 \to H^2$, $M_z f = zf(z)$. We know Beurling theorem: Every invariant subspace shift operator other than $\{0\}$ has the form $\phi H^2$, ...
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23 views

Bounding Fourier coefficients using translates of a function

Having real trouble solving this problem, I know there's probably something obvious that I'm missing but it's driving me mad! Exercise 1.3. Let $f \in C(\mathbb{R} / \mathbb{Z})$. For $y > 0$ ...
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1answer
76 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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332 views

Prove that $|k(x)|\le C|x|^{-n}$ under suitable hypothesis on $k\in\mathcal{C}^1(\Bbb R^n\setminus\{0\})$

DON'T BE AFRAID FROM THE +500 BOUNTY: it doesn't matter that I KNOW this problem is really hard, I put it only because I need to solve the problem really URGENTLY! Let $n\ge2$; given a kernel ...
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1answer
70 views

Do we have the pointwise bound $\left|\tilde{f}\right| \lesssim_d Mf$?

Edit. I know I am missing a mean value on all integrals, but unfortunately I do not know if it is possible to make an integral sign with a horizontal slash through it with MathJax. For any locally ...
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8 views

Estimation with Hölder condition

Do you have any hints about how to prove (or find a counterexample) that, given $f \in \mathcal{C}^1 ( \mathbb{R}^n \smallsetminus \{ 0 \}) $ such that $$\int_{|x|=r} f(x) \, dS(x) = 0$$ for all ...
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1answer
44 views

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

I have some problems with my notes: my teacher wrote that if a sequence $\{g_j\}_j\subseteq L^{1,\infty}(\Bbb R^n)$ (which is the weak $L^1$ space, endowed with the quasinorm $||\cdot||_{1,\infty}$) ...
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22 views

A Hilbert transform that takes several functions

While playing with some PDE I came across a singular integral that looks something like ...
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36 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists ...
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70 views

Why is this theorem equivalent to the informal explanation given by Tao?

I will copy-paste the statement and the theorem from this paper by Tao about an uncertainty principle for groups of prime order. http://arxiv.org/pdf/math/0308286.pdf Theorem 1.1: Let $p$ be a prime ...
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1answer
42 views

Completeness of Schwartz space

I wanna to prove the completeness of Schwartz space $\mathscr{S}(R^{n})$ equipped with the induced topology from a set of seminnorms $$\|f(x)\|_{\alpha,\beta}=\sup_{x\in ...
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1answer
36 views

About Hardy-Littlewood maximal function from Grafakos' “Classical Fourier Analysis”

The Hardy-Littlewood maximal operator $M$ is defined by $$ M f(x)=\sup_{r>0}|B(x,r)|^{-1}\int_{B(x,r)} |f(y)|dy. $$ There is the following example in Grafakos's "Classical Fourier Analysis" book ...
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The eigenvalue for mollified function

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $E:=\{u_k\}_{k=1}^\infty$ forms a basis for $L^2$ and we ...
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1answer
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Hodge decomposition thm. Why $\Delta(E^p)=d\delta(E^p)\oplus\delta d(E^p)=d(E^{p-1})\oplus\delta(E^{p+1})?$

$\DeclareMathOperator{Img}{Im}$ In Warner's "Foundations of differentiable manifolds and Lie groups" we read that $$E^p\stackrel{(1)}{=}\Delta(E^p)\oplus H^p\stackrel{(2)}{=}d\delta(E^p)\oplus\delta ...
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1answer
40 views

Compostion on $H^2(U)$

Below is a question that I'm attempting to do but so far have made no progress. Any suggestions would be helpful. Show that whenever $0 < \alpha < \frac{1}{2}$, then $\left( \frac{1+z}{1-z} ...
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1answer
36 views

Proof that $\|S_N\|_p < \infty $ is equivalent to $\|S_N f - f\|_p \to 0$ as $N \to \infty$

I am having difficulties with the proof of proposition 1.9 in the book "Classical and multilinear harmonic analysis, Vol. 1" by C. Muscalu and W. Schlag. The following statements are equivalent ...
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1answer
46 views

Calderón-Zygmund theorem doesn't seem has correct hypotesis

This theorem states that, given ANY $f\in L^1(\Bbb R^n)$ and ANY $\alpha>0$, there exists a sequence of (mutual disjoint open with sides parallel to the axis) cubes $\{Q_k\}_{k\ge1}$ such that $$ ...
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18 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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29 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
22 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
34 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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79 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
18 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 ...
2
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20 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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140 views

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 ...