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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Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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A property of a ideal of Banach algebras

Let $B$ be a Banach algebra and $A$ be a bi-ideal of it. Suppose that for any $b\in B$, $Ab=\{0\}$ implies $b=0$. Now could we say that for some $c\in B$ if $cA=\{0\}$ then $c=0?$
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Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...
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Intuition Behind the Riesz Transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. ...
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When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
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Please help me with Fourier series problem!

(a) Find Fourier series of $f(x)$ on $[-L,L]$ $$f(x)=\begin{cases} x(L-x) & 0\le x<L \\ x(L+x) & -L < x < 0 \end{cases} $$ (b) Find $f'(x)$ and $\int_{-L}^x f(x)\, dx$ and the ...
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41 views

fourier Series gone Bad

I understand why the series converges uniformly for $\lambda=\frac{2}{\pi}$ and i can get a very non elegant answer for a( as An=0 and Bn is a verly clumsy expression): the series as is: I ...
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28 views

proving a claim using discrete fourier transform of sequence

I get why the answer to 1 is: but I dont have a clue about b. Please help!
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34 views

Solve Intergrals Using Inverse Fourier Transform

a)Find f(x), the insverse fourier transform of F(ω) b) Does the fourier transform of f(x) equal to F(ω)? c)use your answers to calculate these Integrals: if I'm not mistaken the answer to a is: ...
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15 views

What does it mean for a subset of $L^{\infty}(G)$ to separate the points of $G$?

Let $G$ be a locally compact group, and $M\subset L^{\infty}(G)$. I would naively define $M$ to separate the points of $G$ if for all $s,t\in G$, there exists $f\in M$ such that $f(s)\neq f(t)$. But ...
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Pontryagin Dual of the Unit Circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
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70 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{r>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to show ...
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23 views

Convergence of $S_{n}(f;t)$

Let $f \in L^P(T)$ for some $p>1$. If $n$th Fourier partial sum $S_n(f;t)$ converges almost everywhere as $n \rightarrow \infty$, does the limit have to be $f(t)$ almost everywhere? I am trying to ...
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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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13 views

For Dirichlet's problem on the unit disk, show that the solution is harmonic.

I am trying to prove that $u(re^{i\theta}) = \sum\limits_{n=-\infty}^{\infty}\hat{f}(n)r^{|n|}e^{in\theta}$ is the solution to the Dirichlet problem on the unit disk if on the boundary of the unit ...
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Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=C$\ {0, 1}. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0log|z|−a_1log|z−1|$$ is the real ...
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47 views

Solutions of the constant coefficient Helmholtz equation via the Fourier transform

When $f$ is a rapidly decaying Schwartz function, $$ g(x) = \frac{1}{2\lambda} \int_{\mathbb{R}} \sin \left(2\lambda\left|x-y\right|\right) f(y)\ dy $$ is an element of ...
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A linear functional of $H^k$

Define $H^k$ as usual: $$H^k=\left\{f; \int (1+|\xi|^2)^k|\widehat{f}(\xi)|^2d\xi<\infty\right\}$$ Define $l(u)=\int (1+|\xi|^2)^k\widehat{u}(\xi)d\xi$, how to show that $l\in (H^k)^*$. I tried ...
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19 views

Norm convergence of approximations to the identity

Let $\varphi \in L^1(\mathbb{R}^d)$ be such that $\int_{\mathbb{R}^d} \varphi(x) \, dx = 1$. For each $\varepsilon>0$, let $\varphi_\varepsilon:= \varepsilon^{-d} \varphi\left( \dfrac x ...
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Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
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Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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Application of Plancherel/Parseval

Assuming $u,v\in L^1\cap L^2$, then how do you show that $$\int uv=\int \hat{u}\hat{v}$$ I tried using Plancherel, but didnt give any nice result. Any ideas/hints? Thanks
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Iterated Limits Along an Ultrafilter

Setting: Let $\mathfrak{U}$ be an ultrafilter on an index set $I$. Let $G$ be a compact group with identity $e$, and let $\mathbb{T}$ denote the unit circle in the complex plane. For each $i\in ...
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Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
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527 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
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Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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Inverse Fourier transform of cut off of Fourier transform

Suppose we have a function $f(x)$ such that $$|\frac{d^n}{dx^n}f(x)| \leq C(1 + |x|)^{-n}$$ Take the Fourier transform $\hat{f}(\xi)$ and consider the function $g(\xi) = \chi(\xi)\hat{f}(\xi)$, where ...
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A question about a proof in Lang's $SL_2(\mathbb{R})$

The following is a lemma in Lang's book $SL_2(\mathbb{R})$. It's the last line of the proof that I don't understand. Let $G=SL_2(\mathbb{R})$ , $E$ a Banach space, and let $\pi$ be an irreducible ...
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Why is boundedness of the ball multiplier equivalent to the convergence of Fourier transform in Lp?

Let $\mathcal{F}$ be the fourier transform operator and let $T_R$ = $\mathcal{F}^* \chi_R\mathcal{F}$ where $\chi_R$ is the indicator function on the ball of radius $R$. Hence $T_R$ is the fourier ...
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Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
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Lebesgue differentiation theorem for Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow ...
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Closedness of set in product topology

Let $A$ be an abelian group and let $\mathbb{T} \subset \mathbb{C}$ be the unit circle. For $a,b \in A$, let $M(a,b) \subset \mathbb{T}^A$ be the set of all functions $f: A \to \mathbb{T}$ such that ...
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1answer
31 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...
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A question from Stein's Harmonic Analysis - real variable methods' book.

In the book of Stein, on page 9: "Let us remark that these additional properties easily lead to the following conclusions among others. First note that $\mu (B) >0$ for any ball $B$, which is a ...
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44 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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What classical conditions give unique Laplace equation solutions on a half-plane?

Solutions of the Laplace equation on the upper half plane of $\mathbb{R}^{2}$ are not unique: $$ \frac{\partial^{2}}{\partial x^{2}}u+\frac{\partial^{2}}{\partial y^{2}}u = ...
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If $\{f_1,f_2,…\}$ a given frame, $T$ a bounded linear, prove that $(Tf,f)=\sum_{n=1}^\infty |(f,f_n)|^2$.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that $$A\|f\|^2\leq\sum_{n=1}^\infty ...
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support of function in topological group

Let $G$ be a compact Hausdorff topological group. Let $U$ be a neighbourhood of the identity $e$, and let $V = U \cap U^{-1}$ where $U^{-1} = \{x^{-1} : x \in U\}$. Apparently there always exists a ...
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Ideal in Matrix algebra

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space ...
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Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and ...
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57 views

Function decomposition in harmonic analysis

In the book "Weighted Norm Inequalities and Related Topics" by José García-Cuerva, J.-L. Rubio de Francia page 144 it was shown that for a measurable function $f$ and $t>0$ $$ |E_t|\leq ...
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Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
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Unitary representation on L_1(G)

If $u$ is a unitary representation of a locally compact group $G$ on a Hilbert Space $\mathcal{H}$ then $\pi_u :L^1(G)\rightarrow \mathcal{B}(\mathcal{H})$ given as $$\langle ...
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1answer
94 views

How can I prove that $\partial\varphi\neq0$ implies $\bar\partial\partial\varphi>0$?

Let $\Omega\subseteq\Bbb C$ open and $\varphi:\Omega\to\Bbb R$ strongly subharmonic, $\varphi\in\mathcal{C}^2$ such that $\partial\varphi\neq0$. My problem is to prove that ...
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1answer
36 views

Transpose in $ {SL}(2,\mathbb{R})$

Let $SL(2, \Bbb{R})$ denote the group of special invertible $2\times 2$ matrices over $\mathbb{R}$. As a locally compact group, the Haar measure of $SL(2, \Bbb{R})$ is computable through Iwasawa ...
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Existence of Inverse Laplace Transform

Let $F(s)$ the Laplace transform of a function $f(t)$ . Under which conditions on $f(t)$ there exist a unique $g(t)$ such that $g(t) = \mathcal{L}^{-1}\{e^{- F(s)}\}(t) \quad $ ? ...
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Can the 0 element of the Fourier Algebra be represented as a coefficient function of two non-zero vectors?

Consider an infinite discrete group $G$, and its associated Hilbert space $l^{2}(G)$. For $t\in G$, let $\lambda(t):l^{2}(G)\to l^{2}(G)$ denote the map $[\lambda(t)x](s) = x(t^{-1}s)$. That is, ...
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31 views

A proposition to prove the real interpolation of positive exponent

Let $p,A\in(0,\infty)$ $\|f\|_{L^{p,\infty}(X,\mu)} := \sup \{\lambda\mu(\{|f|\geq\lambda\})^{\frac{1}{p}}\}$ Show that the following are equivalent (1) $\|f\|_{L^{p,\infty}(X,\mu)}\leq C_pA$ for ...
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25 views

Prove $\hat{f}(\omega)\neq 0$ if $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete

Let $f\in L^1(\mathbb{R})$. s.t $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete. Prove that $\hat{f}(\omega)\neq 0$ for all $\omega\in\mathbb{R}$ Suppose the system is complete for any $g\in ...