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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Does the “differential” of a unitary representation give continuous operators on the space of smooth vectors?

Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which ...
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41 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
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Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
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27 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
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1answer
40 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
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What are the modes of vibration of a genus-2 surface?

So it's spherical harmonics for a sphere. The vibrations of a torus presumably are just ordinary string harmonics around each loop. But what are the harmonics on a genus-2 surface (a donut with 2 ...
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57 views

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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33 views

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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Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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54 views

estimate of fourier transform

I am reading a paper and I don't understand one thing in the paper.Consider the convolution operator $Tf=f*\mu$ acting on $f\in L^p(\mathbb{R}^n)$, where $\mu$ is a measure defined by ...
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1answer
30 views

Norm triangle inequality for convolutions proof

I'm trying to prove that $$\|f*g\|_{L_1}\le{\|f\|_{L_1}\|g\|_{L_1}}$$ with respect to a Haar measure over a group G. Using Fubini's theorem, I'm up to ...
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73 views

How to show that $\int_G f(t) dt = \int_G f(t^{-1}) dt$?

I am reading the lecture notes. On page 34, line 13, it is said that $\int_G f(t) dt = \int_G f(t^{-1}) dt$. How to prove this identity? I think that if we let $s=t^{-1}$, then ...
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1answer
32 views

The continuous embedding of weighted $L^1$ space.

Take $\omega_0$: $\mathbb R^N\to \mathbb R^+$ such that $\omega$ l.s.c. and $\omega_0\geq 1$ and satisfies $$ \frac{1}{|{B}|}\int_{B(x,r)} \omega_0(y)\,dy\leq C\omega_0(x) \tag 1 $$ for any ball ...
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13 views

positive definite character

‎‎We know each character on dual group of ‎$‎Z‎; ‎‎\widehat{Z}‎$‎‎, is positive definite and if‎ ‎$‎‎\chi‎‎ ‎\in‎ \widehat{Z}‎$ then ‎$‎‎\left\| ‎‎‎\chi‎‎ ‎\right\|‎_{‎\infty‎}‎‎=‎‎\chi(1)‎$‎. But I ...
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1answer
25 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
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10 views

Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
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157 views

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator ...
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1answer
61 views

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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Topology on dual of an abelian discrete topological group.

We define the compact-open topology on the dual of an abelian topological group. Please describe compact open topology more explicitly in the case where G is equipped with the discrete topology, for ...
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52 views

Dual group of $\mathbb Z$

We know $\hat{\mathbb Z}=\mathbb T$ and the map $\alpha\longmapsto\chi_{\alpha}$ is an isomorphism of $\mathbb T$ on to the character group of $\mathbb Z$, but I can't prove this map is continuous? ...
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42 views

Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
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18 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
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24 views

Do all continuous piecewise affine functions belongs to the class ($A_1$) of Muckenhoupt functions?

Given $\Omega\subset\mathbb R^N$ is open, we say $\omega$: $\Omega\to [0,+\infty)$ belongs to Muckenhoupt class $A_1$ if there exists some $C>0$ such that $$ \frac{1}{|{B}|}\int_{B(x,r)} ...
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22 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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Behaviour of Fourier Inverse Transform after non-linear modulation

Suppose $\phi$ is a continuous nowhere differentiable function. $g$ some function in Schwartz space such that $\hat{g}$ has compact support. Define $f(x) = \int_{-\infty}^\infty ...
3
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3answers
60 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
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110 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
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22 views

The Riesz transform kernel satisfies Hörmander's condition

Define the kernel $$K_i(x) = \frac{x_i}{\lvert x\rvert^{d+1}}$$ as a function $\mathbb{R}^d \setminus \{ 0 \} \to \mathbb{C}$ for $d \geq 3$ (say). I want to prove that this kernel satisfies ...
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Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
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1answer
31 views

Inverse-Fourier transform of a function after non-linear frequency modulation

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is ...
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1answer
39 views

Image of a function with small BMO norm

This is a question related with the regularity of harmonic maps. Let $N\geq 1$ and $f:\mathbb{R}^N\to \mathbb{S}^2$, where $\mathbb{S}^2=\{x\in \mathbb{R}^3 : \|x\|=1\}$. Assume that the BMO ...
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63 views

Integrability of Maximal Convolution Operator

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent ...
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35 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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1answer
38 views

How does this conjecture correspond to Carleson's theorem for the case of $d=1$?

From this source, on page 36 (bottom) there is a conjecture stated and it was said that the case of $d = 1$ corresponds to Carleson's theorem. A picture included here : But when I look at wiki ...
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45 views

Locally constant property

Suppose f is positive and Schwartz function. Fix $N>0$ and $A>0$. Suppose that for any $x \in [-N,N]$, $$A \leq \int_{-N}^{N}f(x-z)dz$$ Then do the inequality $$A \leq C_{r} ...
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1answer
18 views

Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
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32 views

Harmonic functions and Brownian motion

How can I prove that harmonic functions have the mean-value property using Brownian motion ${B_t}$? I know that I need to use the fact that $B_{t\wedge\tau}$ is a martingale where $\tau$ is a ...
4
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1answer
58 views

$\int x J_0(k x)e^{-x^2/2}dx$ Bessel function decomposition of a gaussian

$$\int ^\infty _0 x J_0(k x)e^{-x^2/2}dx$$ The integral above corresponds to fourier transform in radial coordinates. The fourier transform of a 2D gaussian is still a 2D gaussian. So the integral ...
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Importance of boundedness of classical operators of harmonic analysis

Boundedness of classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals have been extensively investigated in various function spaces. For ...
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64 views

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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Radial Limits of Singular Inner Functions

Given a positive singular measure $\mu$ on $[-\pi,\pi]$, we define a singular inner function by $$S(z)=\exp\left(-\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right).$$ It is ...
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Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
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The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is non-negative. We can observe that the function $|f|^2$ has a nice property : ...
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1answer
15 views

Jacobian of the Kelvin transform

The Kelvin transform of the circle in $\mathbb{R}^n$ with centre $\textbf{u}$ and radius $r$ is defined by $$\textbf{y} \mapsto \textbf{u} + r^2|\textbf{y} - \textbf{u}|^{-2}(\textbf{y}-\textbf{u}).$$ ...
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38 views

Is $L^{1}(\Omega,\mu)$ only an algebra when $\Omega$ is a group?

Let $G$ be a locally compact group. Then we may snap our fingers, mention some measure theory results, and the group algebra $L^{1}(G)$ instantly appears. Is there an example of the more generalized ...
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53 views

Fourier transform of exponent

I need to count the Fourier transform of the following function but it does not seem so obvious for me. $f(x)=(e^{-ab})-1$ for $x\ge0$ and $f(x)=0$ for $x<0$ where: $a=1$ and $b=-1$ I don't ...
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43 views

Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
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1answer
32 views

Sunrise Lemma, left and right maximal functions

Would anyone be able to provide a proof of the following lemma: Where $| \cdot |$ is the Lebesque measure and $M_L$ and $M_R$ are the left and right maximal functions defined on $\mathbb{R}$ ...
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23 views

A convolution equation with two unknowns

I consider the following convolution-type equation with two unknowns $f_1$, $f_2$: $$ a_1 * f_1 + a_2 * f_2 = 0, $$ where $a_1$, $a_2 \in L^1(\mathbb R)$ and $*$ is the ordinary convolution. This ...
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39 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...