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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Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that $\beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\}$. The Fourier ...
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2answers
193 views

Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
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1answer
160 views

Bounds on integral

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where ...
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128 views

Series of Maximal Operator

Let $p\in(1,\infty)$. Assume that we have a sequence of functions $\{f_i:i\in\mathbb{N}\}\subset L^p(\mathbb{R}^n)$ such that $$ \left(\sum\limits_{i=1}^\infty|f_i|^2\right)^{\frac{1}{2}}\in ...
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1answer
126 views

Fejer kernel applied to a measure

Let $\mu$ be a positive finite measure on $\mathbb R$. Is it true that $$\int_{\mathbb R} T \text{sinc}^2(Tx) d\mu(x) \sim\frac{\mu([-1/T,1/T])}{1/T}, \text{ as } T\to\infty?$$ Here ...
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1answer
84 views

$L^p$ bounds for linear operators

A typical way to show $L^{p}$ boundedness of a linear or sub-linear operator is to show a weak type $(1,1)$ bound and an $L^{p}$ bound for some $p$, and then combine these two using interpolation to ...
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56 views

$\Lambda_p$-set for compact abelian group

We denote by $|A|$ the cardinal of a set $A$. Let $S$ be a subset of $\mathbb{Z}$. Denote $S_N=S\cap [0,N]$ where $N$ is an integer. Suppose $2<p<\infty$. There is well-known that if $S$ is a ...
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193 views

Left Haar Measure on the Borel subgroup of the general linear group

If we consider the group of upper triangular matrices $B=\bigl(\begin{smallmatrix} a&b\\ 0&a^{-1} \end{smallmatrix} \bigr)$ where $a$ and $b$ are either real or complex and $a\neq1$, then the ...
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1answer
124 views

Truncation in Lorentz spaces

I am reading a paper, whose author state the following: if $f \in L^{(q,\infty)}(\mathbb{R}^N)$, then $f_\delta \in L^p(\mathbb{R}^N)$ for every $p \in [1,q)$, where $\delta > 0$ and $$ f_\delta = ...
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1answer
173 views

Convergence in mean of differentiated Fourier series

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be integrable on $[-\pi,\pi]$ and $2\pi$-periodic. Let $$ \frac{a_0}{2}+\sum\limits_{n=1}^\infty (a_n \cos nx+b_n \sin nx) $$ be the Fourier ...
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54 views

Wiener's Lemma on Locally Compact Abelian Groups

The version of Wiener's lemma that I know, from Katznelson's Introduction to Harmonic Analysis, is $\lim_{N\to\infty}\frac{1}{2N}\sum_{-N}^{N} |\hat{\mu}(n)|^2 =\sum_{t\in\mathbb{T}}|\mu(\{t\})|^2$. ...
3
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1answer
783 views

Properties of subharmonic functions

A function $f$ is called subharmonic if $f:U\rightarrow\mathbb R$ (with $U\subset\mathbb R^n$) is upper semi-continuous and $$\forall\space \mathbb B_r(x)\subset ...
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1answer
117 views

Problems on Schwartz Functions

(1) What are all positive Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ? (2) What are all Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ? (3) What ...
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178 views

A Schwartz function problem

Let f be a strictly positive Schwartz function on $\mathbb R$. Does it imply $\sqrt f$ is a Schwartz function on $\mathbb R$?
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1answer
785 views

properties of a real analytic function

If there are a radius $r>0$ and constants $M,C\in\mathbb R$ for all $y\in U$ with $$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb ...
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512 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
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73 views

About regularity of the maximal operator of Hardy-Littlewood

What difficulties arise when you consider the centered maximal operator, such that you can't prove that it maps BV into BV? Does someone have some reference which that maps BV into BV? PS: The ...
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2answers
613 views

amplitude of sine wave with multiple frequencies

I'm having some troubles determining the amplitude/magnitude of the following equation. $$ A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3) $$ Since each part is at a ...
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3answers
696 views

Solution of Laplace's equation in an annulus with constant Dirichlet conditions?

What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
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1answer
156 views

Question from Stein's Singular Integrals and Differentiability Properties of Functions.

My question is in regards of Stein's proof that Hilbert transform is of weak $(1,1)$ property, on page 30 of the textbook I mentioned in my title. On page 32 he writes that because $|\nabla K| \leq B ...
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96 views

Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
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1answer
267 views

Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
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219 views

A function in BMO space

Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ...
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1answer
105 views

Showing this particular function is harmonic

Just a quick question that I hope nobody would mind answering. I'm reading through a page on this book http://books.google.com/books?id=dslVpWam-4QC&pg=PA252#v=onepage&q&f=false, and I'm a ...
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2answers
264 views

Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
4
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2answers
724 views

Approximation of the identity and Hardy-Littlewood maximal function

The inequality seems to be simple but I could not find the right limits of integration. $$\sup_{\delta>0} |f*K_{\delta}|(x)\leq c f^*(x)$$ Where is some positive constant, $f$ is integrable, ...
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1answer
297 views

Question about support of distributions

I'm reading "Pseudo-differential Operators and the Nash-Moser Theorem" and at the top of on page 8 they write: "Finally, we note that if $u \in C^0(\Omega)$, then the support of $u$ defined above ...
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2answers
81 views

The integral of a function over $S^1$

Let $S^1=\mathbb R/\mathbb Z,$ I was wondering how to calculate the integral of a function over $S^1$ and why. Like, $\int_{S^1}1 dx=?$ Given an "appropriate" function $f$, what is $\int_{S^1}f(x)dx?$ ...
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107 views

Harmonic measure or harmonic kernel

In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr ...
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112 views

Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
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5answers
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What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
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463 views

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
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1answer
219 views

On covering lemma and Calderón–Zygmund decomposition

I am working on something which needs to understand covering lemmas and Calderón–Zygmund decomposition. These type of lemmas are as in the following link ...
3
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1answer
276 views

Applications of Young's convolution inequality

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided ...
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1answer
155 views

On a duality Fefferman-Stein's inequality

Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following: *For ...
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337 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
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1answer
611 views

Spherical harmonics give all the irreducible representations of $SO(3)$?

It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? ...
4
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1answer
143 views

Product of Sidon sets

Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition). Consider the set ...
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1answer
559 views

Properties of Haar measure

Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
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87 views

Preduals of Banach spaces and in particular of $\text{BMO}(\mathbf R^d)$

In general the predual of a Banach space is not unique. If there are multiple ones must they be isomorphic? More specifically is $H^1(\mathbf R^d)$ the only predual of $\text{BMO}(\mathbf R^d)$ or ...
3
votes
1answer
266 views

Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
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Proving the maximum principle for harmonic real valued functions in $\mathbb{R}^n$

Assuming the principle is stated as such: Let $U\subset\mathbb{R}^n$ be a bounded domain and $u$ harmonic in $U$ such that $\sup_{x\in U}u(x)\leq A$ for some $A\in\mathbb{R}$. Then either $\forall ...
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481 views

Rate of divergence for the series $\sum |\sin(n\theta) / n|$

In the following we consider the series $$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$ parametrized by $\theta$. It is well known that this series (taking the limit ...
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1answer
662 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
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1answer
87 views

Restriction and completion of Haar measure on $\mathbf{R} \times \mathbf{R_d}$ to Borel $\sigma$-algebra

Let's consider the measure space $(G, \mathfrak{M}, \mu)$, where $\mu$ is the Haar measure on topological group $G:=\mathbf{R} \times \mathbf{R_d}$, ($\mathbf{R}$ is the group of reals with the ...
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1answer
345 views

Stone-Weierstrass implies Fourier expansion

To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem: Let $G$ be a compact abelian topological group ...
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243 views

A Van der Corput style inequality for highly oscillatory integrals

Suppose $f$ has at least two continuous derivatives, $f'$ is monotonically increasing, and $f' \geq \lambda$ for some $\lambda > 0$. How might one find the upper bound $|\int_a^b ...
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1answer
187 views

Surjective endomorphism preserves Haar measure

How to prove the following statement: Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ ...
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1answer
256 views

Example of a locally compact connected Abelian group with non-$\sigma$-finite measure

I look for an example of an Abelian locally compact topological group $G$ such that: $G$ is connected and Haar measure on $G$ is not $\sigma$-finite and $\{0\} \times G \subset \mathbf{R} \times G$ ...
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2answers
607 views

Bounded linear operators that commute with translation

I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ ...