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, Littlewood-...

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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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83 views

Upper Bound for Operator Norm in Marcinkiewicz Interpolation Theorem

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and $\|T(f)\|_{...
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204 views

Carleson embedding theorem

The Carleson embedding theorem gives a criterium by which to decide when for a measure $\mu$ the operator that takes a function on the real line to its harmonic extension (by convolution with Poisson ...
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140 views

Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem

I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
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Computing the modularity function of upper triangular matrices

Put $B_p := \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \in GL_2(Q_p) : a, b, c \in Q_p \right\}$ the subgroup of upper triangular matrices in $GL_2(Q_p)$, $Q_p$ denoting the $p$-adic ...
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137 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 $\text{sinc}(x)=\...
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141 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 X)\...
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130 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 z^...
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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 ...
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170 views

Steinhaus theorem in topological groups

Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq \...
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333 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 $d(...
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5k views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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200 views

Proof that $dx/|x|$ is a Haar measure on non-zero reals?

Most importantly, what is the meaning of this notation $\lambda = dx/|x|$? How do I compute say $\lambda(0,1)$ for example?
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109 views

Decomposition and harmonic analysis of $L^2(S^n)$

Deitmar and Echterhoff write in their book Principles of Harmonic Analysis that `It follows from the Peter-Weyl theorem that the $SU(2)$ representation on $L^2(S^3)$ is isomorphic to the orthogonal ...
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359 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family $\...
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92 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
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145 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

We know that a Fourier series for signal $x(t)$ is given as $$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$ So my question is what ...
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89 views

Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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45 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball $...
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124 views

Sublinearity of Hardy-Littlewood Maximal Function on Sobolev Spaces

Define the (centered) Hardy-Littlewood maximal function by $$\mathcal{M}f(x)=\sup_{r>0}\dfrac{1}{m(B(x,r))}\int_{B(x,r)}\left|f(y)\right|dy,\ f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$$ We say ...
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115 views

Equivalence of Schwartz Space Definition

I've come across two definitions for what it means for a function to be in $\mathfrak S$, the Schwartz space. A function $f \in \mathfrak S$ if $f \in C^\infty$ and for all $j, k \geq 0$ integers, $\...
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90 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ {L^{1}...
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121 views

Sobolev spaces vs. Hardy Spaces

I have seen sobolev spaces (the ones with the p norms of the derivatives of a multivariable function) and Hardy spaces (the objects investigated in harmonic analysis when one asks about tangential and ...
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300 views

Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...
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455 views

Nirenberg-Gagliardo- Sobolev inequalities

I need a small help in understanding the following that how "Nirenberg -Gagliardo-Sobolev inequalities" were used. This is a part of the paper. Denote $$ H^1=W^{1, 2}(\Omega)\\ V_1=\{ f\in H^2 (\...
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1answer
109 views

Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.

I search the reference for the proof of the following theorem: Let $G$ be a locally compact group. Then the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable. The ...
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331 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 ...
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335 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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147 views

Convolution on noncommutative group algebras

If $G$ is a non-Abelian locally compact group, and $f$ is in $L^1{(G)}$ and $u$ is in $L^{\infty}(G)$, and $f\ast u=0$ can it be concluded that $u\ast f=0$?
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118 views

Replicating Kolmogorov's Counterexample for Fourier Series in Context of Fourier Transforms

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More ...
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64 views

Showing the range of Fourier transform on $L^1(\mathbb{R})$ is in $L^1(\mathbb{R})$ for a particular scenario

This is a problem from Katznelson's Harmonic Analysis book(Page 143, Problem 8): Suppose $f\in L^1(\mathbb{R})$ and is continuous at $0$. Also suppose $\hat{f}\geq 0$, then show that $\hat{f}$ is in $...
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52 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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1k views

Laplace operator and Fourier transform

Let $f$ be a function with compact support in $D\subset \mathbf{R}^2$. Is is known that $$-\Delta f=F^{-1}|x|^2 F f,$$ where $F$ is the Fourier transform. Which is the $n-$dimensional analogous ...
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157 views

Identity involving partial sums of Fourier series

Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$ How can I show that $$\sum_{j=0}^{N-1}...
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1answer
657 views

Prove partial derivatives of uniformly convergent harmonic functions converge to the partial derivative of the limit of the sequence.

I think the title says it all. If you have a sequence of harmonic functions from a bounded complex domain to the real numbers, show that on a subset at a positive distance from the boundary of the ...
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333 views

non tangential maximal function and Hardy-Littlewood maximal function

I'm studying harmonic analysis and found that we can bound non-tangential maximal function by Hardy-Littlewood maximal function. Most books don't give the proof of it. How can I see that? Is there a ...
3
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1answer
375 views

Decaying Fourier transform and smoothness

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ I want to show ...
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283 views

What is a “domain” in the maximum-minimum principle?

The maximum-minimum principle says that A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant. Here is my question: If we restrict our attention in ${...
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35 views

Question on Schwarz space why is it $\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty $

Why Schwarz space is given by the set of $f\in \mathcal C^\infty (\mathbb R^n)$ s.t. $$\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty ,$$ where $\alpha \in \mathbb N^n$ and $N\in\...
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53 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
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75 views

Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t f\|_{\...
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55 views

The decay conditions of the Poisson summation formula

The Poisson summation formula states that for a function $f$ satisfying $$ |f(x)|\leq A(1+|x|)^{-n-\delta},~|\hat{f}(\xi)|\leq A(1+|\xi|)^{-n-\delta} $$ for $\delta>0,$ then the equality $$ \sum_{...
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1answer
63 views

Maximal Function Estimate

Suppose $\psi$ is a rapidly decreasing function; i.e. for all $N>0$ there exists a constant $C_{N}$ such that $\left|\psi(x)\right|\leq C_{N}(1+\left|x\right|)^{-N}$. Define a family of functions $\...
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165 views

A questions about the schur's lemma

Schur's lemma is this: If (ρ1,V1) and (ρ2,V2) are irreducible representations of a group G, then any nonzero homomorphism ϕ:V1↦V2 is an isomorphism. or Schur's Lemma. a. A unitary ...
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1answer
56 views

Covolution (space) over compact Lie groups

Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual $\psi^*(x)=\...
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1answer
73 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
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87 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
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180 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 ...
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422 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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1answer
72 views

Are the continuous functions on $G$ dense in $L^{1}(G)$?

If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?