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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Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
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1answer
272 views

Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the ...
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78 views

Function supported on [-1,1] with arbitrary prescribed sub-exponential Fourier decay?

Given $g:[1,\infty) \rightarrow (0,\infty)$ with $g(t) = o(t)$, does there exist $f:\mathbb{R} \rightarrow [0,\infty)$ with support contained in $[-1,1]$ such that $$ \widehat{f}(y) = ...
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Fourier coefficient of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x})$ for $\nu \in (0,\frac{1}{2})$.

In Zygmund's Trigonometric Series, vol I, on page 19 section 2.22 they write that Riemann showed that the Fourier coeff of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x}))$ for $\nu \in (0,\frac{1}{2})$ ...
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1answer
59 views

To partition the unity by translating a single function

I am trying to show that there exists a (real or complex-valued) function $\psi \in C^\infty(\mathbb{R}^n)$ having the following properties: The support of $\psi$ is contained in the unit ball ...
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92 views

Prove that $\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |)) \in MPSH(\Omega)$

This's an example: For $u(z_1,z_1,\ldots,z_n)=\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |))$, where $z=(z_1,z_1,\ldots,z_n) \in \Omega=\mathbb{C}^n \setminus\{0\} ...
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1answer
155 views

Subharmonic, Plurisubharmonic

Can you give me two examples of Subharmonic, Plurisubharmonic? (and not Subharmonic, not Plurisubharmonic) . Then prove that your examples. I'm looking forward to your help. Thanks.
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1answer
65 views

Harmonic function 1

I have had trouble when I try to prove the following: For $u: \Omega \to \mathbb{R}$ be a harmonic function iff $u \in C^2(\Omega)$ and $\Delta u=\dfrac{\partial ^2u}{\partial ...
2
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1answer
72 views

Book searching in Pluripotential theory

Can anyone recommend me a book on pluripotential theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why ...
3
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1answer
104 views

Research paper in harmonic analysis that can be read in parallel to studying the subject.

In my idle hours I started to learn some math I touched only superficially in academia in former times. Among others I am working through the books of A. Deitmar on harmonic analysis. I've almost ...
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1answer
339 views

Integrating $\sin(n\theta(x))/\sin(\theta(x))$ for some function $\theta(x)$

I have an indefinite integral of the form: $$ \int \frac{\sin(n\theta(x)))}{\sin(\theta(x))} dx. $$ $\theta$ is a function of $x$ (and actually a complicated one). Is it possible to integrate it ...
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51 views

Definite positive measure and GNS representation

Let $G$ be a locally compact group. Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$. In [D, p ...
12
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2answers
423 views

Are Fourier Analysis and Harmonic Analysis the same subject?

Are Fourier Analysis and Harmonic Analysis the same subject? I believe that they are not the same. Maybe there is big difference between those subjects but I need to know what is the main difference ...
3
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2answers
127 views

Fourier-restriction on Hyperplane

In Muscalu, Schlag "Classical and multilinear harmonic analysis, Volume 1" (2013), Excercise 11.1 is to prove, basically, that there exists a function $f\in L^p \quad \forall\ p>1$ such, that the ...
2
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2answers
524 views

strong maximum principle - harmonic function

Consider the following the theorem in the classical PDE book of Evans( chapter 2 ) : (part of the strong maximum principle) Let $U$ a open set in $R^n$ and $u \in C^2 (U) \cap C(\overline{U})$, with ...
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1answer
69 views

question regarding Fourier restriction estimates

Thanks for reading my post. I am trying to prove the following claim: If we have \begin{equation*} \left\|\hat{f}\right\|_{L^q(N_{1/R}(S))}\lesssim R^{\alpha-1/q}\left\|f\right\|_{L^p(B(0,R))} ...
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112 views

the spectrum and determinant of the Laplacian on $S^3$

I came across the following statement in a paper: On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell ...
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73 views

fourier series-integration of fourier series [closed]

Can anyone please help me top solve this question. Integrate from $s=0$ to $s=x$ ($–\pi≤x≤\pi$)the basic fourier series $2∑_{n=1}^\infty\frac{(-1)^(n+1)}n \sin(ns)$, and ...
3
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92 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
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83 views

A problem about the convergence of convolution

Let $G$ be a topological group. Let $f_{n}$ and $g_{n}$ be two sequences in $L^{p}(G)$ that are convergent to $f$ and $g$, respectively. Let $f * g \in L^{p}$. Is $f_{n} * g_{n}$ convergent to $f * ...
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84 views

Equidistribution of sequence $an^b$, 0<b<1, a>0

This is Exercise number 8, Chapter 4 in Stein's book on Fourier Analysis. I'm supposed to solve it by giving an asymptotic bound to the exponential sum created from Weyl's Criterion, but so far have ...
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1answer
316 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
3
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1answer
238 views

Harmonic analysis in number theory

When I was reading Folland's A course in abstract harmonic analysis, I was told these materials have wonderful applications to number theory. However, I do not see really a lot of examples there. Can ...
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161 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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A question about the Littlewood-Paley decomposition. [duplicate]

Let $\{f_k(x)\}_{k=0}^\infty$ be a Littlewood-Paley decompositon, that is, $$ f_k \in C_c^\infty $$ $$ \sum_{k=0}^\infty f_k (x) = 1,$$ $$ \text{supp} f_0 \subset \{ |x| \leq 2 \},$$ $$ \exists f ...
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1answer
77 views

A property of Littlewood-Paley decomposition

Let $f_j \in C_c^\infty$ and assume there exists $M>0$ such that $\text{supp}f_0 \subset \{ |x| \le M \}$ and $\text{supp} f \subset \{ 1/M \le |x| \le M \}$. Define $f_j$ by $$ f_j (y) = f(y/2^j) ...
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1answer
71 views

A simple question of Littlewood-Paley decomposition.

Let $\{f_k(x)\}_{k=0}^\infty$ be a Littlewood-Paley decompositon, that is, $$ f_k \in C_c^\infty $$ $$ \sum_{k=0}^\infty f_k (x) = 1,$$ $$ \text{supp} f_0 \subset \{ |x| \leq 2 \},$$ $$ \exists f ...
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55 views

How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.
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Zero-crossings of periodic function

Given a periodic function of the form $$ s(\theta) = \sum_{h=1}^H a_h \cdot \cos(h\cdot \theta + \phi_h),$$ how can I determine its zero-crossings $\theta_0$?
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108 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
3
votes
1answer
132 views

Uncertainty principle in harmonic analysis

Given a function $f$ (in some suitable (EDIT: nice) function space) supported on a ball $B\subset\mathbb{R}^n$ of radius $R>0$, I have commonly heard people, who understand these things, say stuff ...
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1answer
155 views

Why are square functions important in analysis?

I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship ...
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1answer
86 views

Question which involves Riesz potential

Let $f :S^{n-1}\longrightarrow R_+$ , $P_{j,k}. j,k \in N$ be a spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonics) and $\displaystyle{f\left(\frac{x}{|x|}\right)|x|^{1-\frac ...
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1answer
142 views

Corollary from Khintchine's inequality

Let $z_1,\dots,z_n\in\mathbb{C}$ and $\epsilon_j\in\{-1,1\}$ for $j=1,\dots,n$ independent random variables with $P(\epsilon_j=\pm 1)=1/2$. Khintchine's inequality states that ...
4
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900 views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
2
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1answer
81 views

Application of Green's theorem to probability

I encountered this problem while reading a statistic text. Since I am not quite familar with the background knowledge. Wonder can someone help me to explain the details of the following proof? ...
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1answer
159 views

Harmonic mean of absolute value squared discrete Fourier transform

Let $X[k] \in \mathbb{C}$ be the discrete Fourier transform of $x[n] \in \mathbb{R}$, where $k,n = 0,1,2,...,N-1$. Parseval's theorem relates the arithmetic mean (AM) of absolute value squared ...
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1answer
60 views

Hilbert transform and maximal function

What is the relation between the Hilbert transform (in $\mathbb{R}$) and maximal functions?
3
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1answer
121 views

Use of the Littlewood-Paley decomposition to recover the $H^s$ norm

Let $\phi\in C^{\infty}_0(\mathbb{R}^n)$ be such that $$\{\lvert \xi\rvert \le 1\} \prec \phi \prec \{\lvert \xi \rvert < 2\}^{[1]} $$ and define the Littlewood-Paley projectors as ...
5
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1answer
84 views

Function invariant under Hilbert transform

Let $f\in C_0^\infty(\mathbb{R}^n)$ and $$Hf(x)=p.v.\int_{\mathbb{R}}\frac{f(x-y)}{y} dx$$ the Hilbert transform of $f$. Is it possible that $Hf=f$ (a.e. and possibly after extending $H$ to $L^p$ ...
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1answer
122 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
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55 views

approximate Fourier transform

Let $\mathcal{F}$ stand for the Fourier transform. Suppose $f : [-\delta/2,\delta/2] \to \mathbb{C}$ is a "nice" function. Is it true that $$\left|\mathcal{F} \left(e^{imx} \left(e^{ix^2}-1 ...
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2answers
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Estimating time in harmonic signal

I hope someone can help me with the following problem: Assume a periodic signal of the form $$\begin{align} s(t) &= \sum\limits_{p=1}^P \sin(p\Omega_0t)\\ &= \sum\limits_{p=1}^P ...
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1answer
51 views

Uniform Rectifiability

What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$? The precise context is this paper by Mattila, Mernikov and Verdera.
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231 views

Interpolation using trigonometric polynomials of bounded modulus

Consider a grid of points $T=\{t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to derive conditions on $t_1,\ldots,t_m$ (interpolation points) under which for any sequence of complex numbers ...
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1answer
166 views

Where should the 2$\pi$ go in the Fourier Transform?

In some lecture notes on Harmonic Analysis from Terence Tao here, he defines the fourier transform by $$\hat{f}(\tau)=\int_{\mathbb{R}}e^{-2\pi i t\tau}f(t)dt$$ and then says This is really the ...
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1answer
115 views

A question of positively homogeneous functions

Let $f$ be a positively homogeneous function of degree $k$, i.e., $$ f(x, \lambda y) = \lambda^k f(x, y)$$ for any $(x,y) \in \mathbb R^n \times \mathbb R^n, \lambda >0$. Then how can I show that $ ...
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80 views

Proof of Caratheodory's theorem about the unique determination of a linear combination of sinusoids

Following is a statement of Caratheodory's Theorem about a positivelinear combination of sinusoids :- Any positive linear combination of k sinusoids is uniquely determined by its value at time t ...
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116 views

Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
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1answer
63 views

Proof that Muckenhoupt's $A_q$ Condition Implies $A_p$ for $p<q$?

It is said $f\in A_p$ if it satisfies the following (Muckenhoupt's $A_p$) condition: ...