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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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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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40 views

Regarding convolution of fejer kernel with a lipschitz function

Let $\mathbb{T}$ be the unit circle in the comlex plain. The Lipschitz class of function on $\mathbb{T}$ is defined here. And the fejer kernel is defined here. If $f\in Lip_{\alpha}(\mathbb{T}), 0<\...
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0answers
263 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 ...
3
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0answers
85 views

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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0answers
79 views

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 ...
4
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1answer
61 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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41 views

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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0answers
149 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 = \...
2
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3answers
241 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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2answers
164 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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155 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 ...
2
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1answer
150 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 ...
5
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1answer
104 views

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 $x \in D(A)$ ...
1
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1answer
121 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 ...
7
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2answers
160 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 ...
3
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0answers
40 views

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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97 views

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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1answer
42 views

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} \...
11
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1answer
237 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 ...
4
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1answer
153 views

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 $\leftrightarrow$ ...
3
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1answer
155 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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0answers
31 views

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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2answers
143 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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1answer
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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0answers
46 views

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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0answers
70 views

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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2answers
104 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 ...
2
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0answers
79 views

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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1answer
103 views

$L^p$ 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 ...
2
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1answer
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 ...
2
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1answer
77 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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0answers
51 views

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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85 views

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 ...
4
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108 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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0answers
118 views

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 ...
4
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1answer
103 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 $...
3
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2answers
80 views

Decay of bandlimited function

Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have. I read on "G. ...
2
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1answer
221 views

Differentiation on $[0,1]$

EDIT: Perhaps a more reasonable question after thinking about the answer I got would have been. Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
3
votes
2answers
117 views

Square-integrability of non-holomorphic Poincare series

In what follows, $\mathfrak{H}$ denotes the upper half plane, and $\Gamma=SL(2,\mathbb{Z})$. On the modular curve $\Gamma\backslash\mathfrak{H}$, consider the non-holomorphic Poincare series, defined ...
2
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1answer
144 views

Proof of $L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$

It is known that $L^1(\mathbb{R}) \ast f$ is dense in $L^1(\mathbb{R})$ for some $f\in L^1(\mathbb{R})$. So for such $f$ the closure of $L^1(\mathbb{R}) \ast f$ in the $L^1$ norm is $L^1(\mathbb{R})$....
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1answer
123 views

Are there any non-trivial convergent sequences in the maximal ideal space of the measure algebra?

Consider the measures on the circle, $M(\mathbb T)$, endowed with the convolution product which makes it a unital Banach algebra under the total variation norm. Denote by $\Delta$ the maximal ideal ...
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66 views

Does Novikov condition imply BMO martingale?

Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $...
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0answers
99 views

When is a minimal immersion holomorphic?

Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let: $\phi\colon X\to Y$ be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
2
votes
1answer
107 views

Solving classical parabolic equation by using Littlewood-Paley theory

Consider the following classical PDE in $R^n$: $$ \partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0. $$ Is there any references on solving the above equation by using the ...
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0answers
72 views

Is the product of two Sobolev functions in L^p?

Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$. Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$. Under what conditions on $\beta$ can we get ...
6
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1answer
246 views

What is the intuition behind Almgren's frequency function?

It is by now well-known that for a harmonic function $u : B_1^n(0) \to \mathbb{R}$, the ratio $$ N(r) := \frac{r\int_{B_r(0)}|\nabla u|^2}{\int_{\partial B_r(0)} u^2} $$ is a non-decreasing function ...
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0answers
106 views

Inclusion of Hardy spaces

It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality. It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
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0answers
31 views

Bound of analytic torsion for a line bundle

Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...
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1answer
98 views

Plancharel-Pólya inequality for functions of exponential type

If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$ then it can be proven that $$\int_{-\infty}^{+\...
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1answer
118 views

Is fractional Laplacian invariant under rotation?

If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...