Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic ...

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### A generalization of the character group

Let $G$ be a group.
We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$
where $Tor(\mathbb{T})$ is the group of torsion elements of the unit ...

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200 views

### Making the Fourier transform quantitative

I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...

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176 views

### The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...

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### Kernel of multiplication in noncommutative 2-torus

What is $\Omega^1 (A_{\theta})$, that is, what is the kernel of the multiplication map $m:A_{\theta} \otimes A_{\theta} \to A_{\theta}$ where $A_{\theta}$ is the noncommutative 2-torus with parameter $...

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### Extension of a derivation on w [duplicate]

Let I be a closed left ideal of a Banach algebra A such that A has a bounded approximate identity and I just has a right approximate identity. let $D:I\to I^*$ be a derivation. Does $D$ extend to a ...

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### connected set of sum of upper semi continuous function [closed]

Let $C(X)$:space of continuous functions on a compact space.Topology $C(X)$ is generated by sup-norm($||T||=sup_{v}\frac{||T(v)||}{||v||}$).
Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper ...

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### Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...

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367 views

### Proof of Minkowski theorem using harmonic analysis

I am trying to properly write a proof of Minkowski's theorem in a self-contained way and understandable by (good) undergraduates.
Theorem (Minkowski)
Let $L$ be a lattice of $\mathbb{R}^n$ and ...

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56 views

### Examples of unbounded pseudo-differential operators in $L^{\infty}$

During my graduate studies I've been told that pseudo-differential operators with symbols in $S^0=S^0_{1,0}$ (the simplest class) are bounded $L^2 \to L^2$, and also $L^p \to L^p$ for all $p \in (1, \...

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**1**answer

289 views

### Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?
This seems to be a well-known result, but I can ...

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### Interior of fundamental domains of lattices in locally compact groups

Let $G$ be a locally compact abelian group, and let $\Lambda$ be a lattice in $G$, i.e. a discrete subgroup such that the quotient group $G/\Lambda$ is compact.
A fundamental domain for $\Lambda$ in $...

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**1**answer

118 views

### Relationship between “Radial” Fourier transform and Fourier transform, especially at infinity

Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support.
What is the relationship between
$$
\widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...

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### Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...

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### Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...

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**1**answer

187 views

### Question on the definition of almost periodic function

According to Bohr, the definition of the almost periodic function is:
A function $f:\mathbb{R}\rightarrow \mathbb{C}$ is called almost periodic if it is continuous and if for every positive $\epsilon$,...

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235 views

### Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...

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292 views

### Is Serre duality related to Pontryagin duality?

I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...

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### gaussian upper bound on spherical heat kernel

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?

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### Convolution theorem on a non-abelian Lie group

Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...

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**1**answer

67 views

### Multiplicities in Plancherel theorem for SL2(R)

The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. ...

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### Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$

I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$.
In the ...

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157 views

### Is the Gelfand transform strictly continuous?

Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \...

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253 views

### Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation.
I encountered the following integral equation for functions $f:[...

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171 views

### Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?

Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...

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173 views

### Relation between Lie group characters and spherical functions on symmetric spaces

Setup: Let $G/K$ be an irreducible compact Riemannian symmetric space, where $G$ is a simply connected compact real Lie group, and $K$ is the maximal compact connected subgroup of some noncompact real ...

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**1**answer

161 views

### About Vitali covering theorem

In the paper by Nazarov, Treil and Volberg: Weak type estimates and Cotlar inequalities for Calderon-Zygmund ... /1998, Int. Math. Res. Not.,
www.math.brown.edu/~treil/papers/l1/l1-5.pdf
on the ...

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**1**answer

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### Invariant subspace in infinite dimensions

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...

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**1**answer

130 views

### $L^2$ function in Schwartz space?

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...

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167 views

### Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...

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### Decay of Fourier coefficients for Hölder functions on compact Lie groups

If $f$ is a complex-valued function on a compact Lie group $G$, we have a decomposition $f = \sum_\mu f_\mu$ corresponding to the Peter-Weyl decomposition $L^2(G) = \oplus_\mu (\dim \mu) V_\mu$.
For $...

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### A combinatorial proof of the Harrow--Kolla--Schulman theorem

Let $Q^n := \{0,1\}^n$ be the Hamming cube with the Hamming metric. (Recall that the Hamming is defined by the distance $d(x,y) := \# \{ i : x_i \neq y_i \}$.
For integers $0 \leq k \leq n$, define a ...

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**1**answer

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### Infinitely many independent functions that are only frequency localized?

A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...

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256 views

### Is there a physical/geometric proof for L^2 boundedness of Bourgain's maximal function along the squares?

One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function ...

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**1**answer

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### What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?

For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):
finite groups $\leftrightarrow$ ...

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**1**answer

158 views

### About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$

Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) ...

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### Spectral theory for Fuchsian groups of the first kind

There are tons of material on the spectral theory of $L^2(\Gamma\backslash G)$ for a lattice $\Gamma$ in $G=PSL_2({\mathbb R})$. There are also many papers on the case of $\Gamma$ being convex-...

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253 views

### Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?

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**1**answer

71 views

### Are the Prolate Spheroidal Wave Functions absolutely integrable?

I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability.
...

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### Hölder-Zygmund spaces of negative order

In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the ...

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### Is it possible to find an atlas for the set: $\{F:FE = I, E \text{ is a frame for } \mathbb{R}^n\}$

Let $E$ be the matrix whos rows are $ \{e_i^{\top}\}_{i=1}^m$. Let $E$ also be a frame of $m$ elements for $\mathbb{R}^n$, $m \geq n$. This means there exist two constants $A, B > 0$ such that:
$$
...

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108 views

### Is Weierstrass function nowhere pointwise Lipschitz?

Consider the classical Weierstrass function
$$
W(x)=\sum_{n=1}^\infty \frac{e^{i2^nx}}{2^n}.
$$
It is a well-known result that this function is nowhere differentiable (Hardy, TAMS 1916, Thm 1.31). In ...

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### A specific Schwartz function $f$ on $\mathbb C^2$

Choose a Schwartz function on $\mathbb C$ of the form $f(z)=f(r e^{i\theta})= f_0(r) e^{in\theta}$. Then $$(*) \quad f(e^{i\alpha} z)= e^{in\alpha} f(z), \quad \forall z\in \mathbb C.$$
Now, let $f$ ...

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### $L^p$ estimates and functions with positive Fourier transform

For $f\in\mathcal{S}$ a Schwartz function on $\mathbb{R}^n$ and $m$ a bounded function, define $Tf$ by $\widehat{T f}=m\cdot \widehat{f}$.
Fix $1<p<\infty$, $p\not=2$. Suppose we have proved ...

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70 views

### Determinant of a matrix involving the Prolate Spheroidal Wave Functions

The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation:
$$\int_{-T}^T\varphi_n(x) \text{sinc}(t-x) dx = \lambda_n \varphi_n(t)$$
where $\text{sinc}(t) = \sin(\pi ...

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83 views

### Differentiability of the logarithmic potential

Assume $\mu$ is a measure supported on a real finite interval $[a,b]$, and let
$$p_\mu(z)=\int\log|z-t|d\mu(t),$$
denote the logarithmic potential associated to $\mu$. Are there (possibly simple) ...

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### Sharp assumption on domain for elliptic boundary regularity

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded Lipschitz domain. Let $f\in H^{s}(\Omega)$, for $s\geq -1$. Then Lax-Milgram guarantees a unique solution to the Poisson problem
$$\begin{cases}\Delta u=...

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### Harmonic analysis on constant curvature hyperbolic spaces of arbitrary dimension

I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the ...

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110 views

### global estimate for biharmonic function

My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions
Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,...

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### On the bound of the Stein-Wainger oscillatory integral

Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...

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110 views

### Is the maximal function bounded on the Besov space?

The Hardy-Littlewood maximal function of a function $f$ is defined by
$$
M f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}|f(x+y)|dy,
$$
where $|B_r|$ denotes the Lebesgue measure of the ball $...