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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Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
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Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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Why real variable methods can take place of complex variable methods in harmonic analysis' research?

We know that one complex variable methods have been largely used in the early research of harmonic analysis. But, as is known to me, there is much difficulty when mathematicians attempt to generalize ...
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12 views

Hardy-Littlewood maximal operator

Consider the centered Hardy_littlewood maximal operator $$ \mathcal{M}f(x):= \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| \text{d}y $$ and the uncentered $$ Mf(x):= \sup_{r>0, ...
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Prove $U$ is subharmonic?

My attempt: Integration by parts says $\int u \triangle \varphi=\int\triangle u \varphi$. We know the left hand side is always $\ge 0$, and hence $\int\triangle u \varphi \ge 0$, since $\varphi \ge ...
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Assumptions on the Borel measure in Stein's Harmonic Analysis

I am currently reading the proof of Theorem 1 on Page 13 of Stein's Harmonic Analysis which proves that if $f \in L^{1}(\mathbb{R}^{n})$, then for every $\alpha > 0$, $$\mu(\{x: (Mf)(x) > ...
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A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
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27 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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31 views

Proof using convolution?

there. I am a novice in graduate school. This is the first time I learn PDE in graduate level. I found it so hard. I am going to have a test next week and I am so worried about it. Since I always ...
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Estimation of trignometric polynomial and lipshitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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$\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$

I am trying to show the following: $\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$ I tried to use Fubini's theorem and the fact that ...
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28 views

Convergence of Fourier series in Sobolev space

So the problem is if $f\in H^{\frac{1}{2}}([0,1])\cap C([0,1])$, then $S_Nf$, the partial sum of fourier series converges uniformly to $f$. How would you show this by considering the quantity ...
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27 views

Help with biharmonic functions

I tried to use using Green's formula to prove $\triangle u=0$. But it turned out to be wrong because $u=0$ doesn't imply $\triangle u=0$ on the boundary. I am a novice in graduate pde, so I didn't ...
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32 views

Local barrier implies barrier?

there. This is part of the textbook of Gibarg's PDE: My question is that how to verify the part in red? How to know $\overline w$ is continous in $\overline \Omega$? Thanks so much! Your help ...
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Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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33 views

Characterization of the Haar measure in terms of the integrals of characters

I was reading a paper and I think that they used the following theorem: Let $G$ compact group and $\mu$ a probability measure on $G$. If $$\hat{\mu}(\xi)= \int_G \overline{\xi(x)} d\mu(x) = ...
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39 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
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1answer
35 views

Find a harmonic function on two concentric balls?

My attempt: I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial ...
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40 views

_How to prove $\triangle h=f$??

there! The following problem has been really bothering me for hours. My problem: How to prove $h(x)=\int_{R^n}\Gamma (x-y)f(y)dy$ is bounded and $\triangle h=f$ in $R^n$. My attempt: I found the ...
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21 views

Robust Uncertainty Principles: Exact Signal. Reconstruction From Highly Incomplete. Frequency Information — Help Understanding

In the above paper the authors claim that a sparse signal in the time domain which is supported on set T of cardinality |T| can be recovered from its partial Fourier coefficients if the number of the ...
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1answer
23 views

Harmonic functions that uniformly convergent?

Let $u_k$ be continuous on $\overline\Omega$, $u_k$ harmonic in $\Omega$. Suppose $u_k|\partial\Omega$ converge uniformly. Then $u_k$ converge uniformly in $\Omega$. The hint is using Maximum ...
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32 views

Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
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37 views

Proof of reflection principle for harmonic functions

** My attempt: Hi, there! I have known how to prove the above statement when $u\in C^2(U)$, however, I have question about proving the above statement. Because it is $u\in C^2(U^{+}) \cap ...
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44 views

Fourier transform of $f(z)=z^n$ on the unit disk in the complex plane

What is the Fourier transform of the function $f(z)=z^n$, $|z|<1$, and $f(z)=0$ if $z$ is outside of the unit disk.
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31 views

integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
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20 views

Locally Compact Groups - Reference Request

I start reading an article about locally compact groups $G$ and the group algebra $L^1(G)$,and I need a good book to introduce myself to these concepts. Can you help please? Thanx
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1answer
26 views

How to use second derivative test?

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
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1answer
32 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
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31 views

Convergence of Fourier series in $L^\infty$

So if $f\in L^1(\mathbb{T})$ and $S_Nf\rightarrow f$ in $L^\infty(\mathbb{T})$ ($S_Nf$ is the partial sum of the fourier series of $f$), then $f$ is continuous. How do we show that this is true? In ...
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1answer
31 views

Estimating the rate of convergence of $|S_Nf-f|$ given that $\|f\|_{H^s}\leq 1$

Given that the Soloblev space norm $$\|f\|_{H^s}^2=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ and the inequality $$\|f(\cdot +\theta)-f\|_{L^2}\leq 2\pi ...
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37 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
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29 views

Wiener Algebra, absolute convergence of fourier series

So how do you prove if $f, g\in L^2(\mathbb{T})$, then $f*g\in \mathbb{A}(\mathbb{T})$. $\mathbb{T}$ denote $[0,1)$ and $\mathbb{A}(\mathbb{T})$ denote the Wiener algebra such that if $f\in ...
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1answer
44 views

Fourier coefficients of a (finite, regular, positive) measure are absolutely summable => the measure has a density

Let $\mu$ be a finite, regular, positive measure on $[0,1)$ such that $\sum_{n\in\mathbb{Z}} |\hat{\mu}(n)| < \infty$. How can I prove that there exists $f(x)$ such that $\mu(dx) = f(x)dx$? ...
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31 views

Question about Hilbert transform(applying plancherel theorem)

Let $f\in S(\mathbb{R})$(Schwartz function on real line). Then Hilbert transform $H$ of $f$ is defined by $\displaystyle Hf(x)=\lim\limits_{t\rightarrow0}\int_{|y|>t}\frac{1}{y}f(x-y)\,dy$ One ...
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20 views

Integral of harmonic function in a ball

Let $f\in C^2(\Omega)$ an harmonic function in $\Omega$, and: $$ \phi(r) = \frac{1}{2\alpha_2r} \int_{\partial B_r(x)} f(y) d \sigma(y) $$ Prove that $\phi '(r)=0$ by calculating the line integral. ...
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1answer
26 views

$L^2$ boundedness of Calderon Zygmund operator

A Calderon Zygmund kernel $K$ is a function $K :\mathbb{R}^d-\{0\}\longrightarrow \mathbb{C}$ satisfying, for some constant $B$, 1)$|K(x)|\leq B|x|^{-d}$ for all $x\in\mathbb{R}^d$ ...
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1answer
24 views

Determine an operator is weak type (1,1) by its Fourier multiplier

I have an interest on associated Fourier multipler $m$ for a given operator $T$ defined for $f\in L^p(R^d)$, $1\le p<\infty$ by $$ \hat{Tf}(\xi)=m(\xi)\tilde{f}(\xi),\ \ \xi\in R^d $$ where ...
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130 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
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27 views

How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
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1answer
18 views

Reference concerning weak-type (1,1) operator in $\mathbb{R}^d$

I wish someone give me some reference on weak-type (1,1) operator in the d-dimensional Euclidean space. Thanks for any kind help.
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1answer
43 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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Calderón reproducing formula : $\int_{0}^{\infty}\int_{R^d}|\phi_{t}(x-y)||(\phi_t*f)(y)|\frac{dt}{t}dy<\infty$

Suppose that $\int f=0$, $f \in L^2$ and $f$ has a compact support. Let $\phi$ be radial, and such that $\mathrm{supp}(\phi) \in B(0,1)$. Plus, assume that $\int_{R^+} ...
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23 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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34 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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Sufficient condition for an operator to be compact in Hilbert space of holomorphic function with respect to Gaussion weight (Fock space).

What I read in a book I could not understand, some one please help. Let $\mathcal{F}=\{f:\mathbb{C^n}\rightarrow\mathbb{C}: \text{$f$ is holomorphic and}\int_{\mathbb{C}^n}\lvert ...
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99 views

Odd form of controlling derivatves

In Muscalu, Schlag - Classical and Multilinear Harmonic Analysis (Cambridge Universitv Press 2013), page 299 there is a rather odd estimate for wich I cannot find any justification: Functions used: ...
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33 views

Is the action on $L^2$ arising from a measure preserving action continuous?

Let $G$ be a locally compact topological group, $X, \mu$ a probability space, and $G\times X \rightarrow X$ a measurable group action which preserves $\mu$ (i.e. $\mu (gA)=\mu(A)$) . Does it follow ...
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1answer
22 views

Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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12 views

The variation of Calderon reproducing formula

I'm reading the book 'Classical and multilinear harmonic analysis, Muscalu'. I fail to understand the page 261. Actually, I doubt that the proof is right. Let $f \in BMO(\mathbb{R^d}$) have compact ...
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1answer
40 views

Statement regarding mean value theorem for harmonic functions

Suppose $u$ is a function that is harmonic on a domain $D$. Could someone offer a proof of the following statement? $$ u(z_0) = \frac{1}{\pi r^2}\iint_{\{z-z_0\}<r} u(x+iy) \, dx\, dy $$ ...