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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The importance of the Van der Corput lemma in analysis and beyond

The Van der Corput lemma states the following: Introduce the following oscillatory integral $$ I(a,b)=\int^{b}_{a}e^{ih(t)}dt. $$ Then $(1)$ if $|h'(t)|\geq \lambda>0$ and $h'$ is monotonic, then ...
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The $p=\infty$ case for an $L^2$ convolution operator on $\mathbb{R}^n$

Let $T$ be a convolution operator on $L^2(\mathbb{R}^n)$, suppose $K$ is a tempered distribution in $\mathbb{R}^n$ that coincides with a locally integrable function on $\mathbb{R}^n\setminus \{0\}$. ...
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Weighted Analogue of Mean Value Property

Let $f$ be a harmonic function defined on a planar disk $B \subset \mathbb{R}^2$ and continuous on $\partial B$. For which probability measures $\mu$ on $\partial B$ is it true that $$\int_{\partial B ...
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32 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 $$ ...
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37 views

Largest Interval Length for $1/3$ Translation Trick

On the Wikipedia page for dyadic cubes, the article claims that if $\Delta^{\alpha}$ denotes all the dyadic cubes in $\mathbb{R}^{n}$ translated by a vector $\alpha\in\mathbb{R}^{n}$, then There ...
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33 views

Connection between the Dini criterion and differentiation from first principles

The Dini criterion for the convergence of the partial sums states that if $~\exists ~ \delta>0$ for some point $x$ such that $$ \int_{|t|<\delta}\left| \frac{f(x+t)-f(x)}{t} \right|dt<\infty, ...
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Introductory books for ‎ ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
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When the inequality $ |x|^{-k} * |f(x)|^2 \le C |f(x)|^2 $ holds?

When can I expect that the inequality $$ |x|^{-k} * |f(x)|^2 \le C |f(x)|^2 $$ for some positive constant $C$? I would like to know the range of $k>0$. Here $*$ means the convolution on $\mathbb ...
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Paley-Wiener stability criterion.

On Young, Robert M. "An Introduction to Non-Harmonic Fourier Series", Revised Edition, 93. Academic Press, 2001., one can read a reformulation of so-called "Paley-Wiener criterion" for stability of ...
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A question on Riesz potential (Leibniz rule) : $ (-\partial_x^2)^{1/2} (fg) = f(-\partial_x^2)^{1/2} g + g(-\partial_x^2)^{1/2} f $?

I am wondering if I can regard the Riesz potential $$ (-\partial_x^2)^{1/2} = (-1)^{1/2}\partial_x $$, where the Riesz potential $(-\partial_x^2)^{1/2} = \mathscr{F}^{-1}|\xi|\mathscr F$ with the ...
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53 views

A Paley-Wiener theorem.

In Lund, John, and Kenneth L. Bowers. Sinc methods for quadrature and differential equations. SIAM, 1992. is recalled a Paley-Wiener theorem, as follows: Assume that f is entire and $f \in ...
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46 views

Fourier transform is unitary proof and other unitary integral operators

There is this old unanswered question: Proof the Fourier Transform is Unitary/Not Unitary What is the easiest way to see that the Fourier transform is unitary and why it is important to have constant ...
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23 views

Foundation Semigroups Examples

I recently started looking into the subject of foundation semi-groups and I'm trying to find simple (none-trivial) examples of foundations semi-groups. I know that every locally compact group is ...
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35 views

connection between integrablity on the locally compact group and compact subgroup of it

Let $G$ be an locally compact group with Haar measure $dx$ and $H$ is compact subgroup of it with normalize Haar measure $dh$. If $F$ belong to $L^1(G)$, the restriction of $F$ to $H$ belong to ...
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80 views

About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$

In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by: $$2\sum_{m\geq ...
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34 views

Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
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32 views

Does the “differential” of a unitary representation give continuous operators on the space of smooth vectors?

Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which ...
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56 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
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Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
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Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
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1answer
45 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
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What are the modes of vibration of a genus-2 surface?

So it's spherical harmonics for a sphere. The vibrations of a torus presumably are just ordinary string harmonics around each loop. But what are the harmonics on a genus-2 surface (a donut with 2 ...
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231 views

What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
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A question from Harmonic Analysis - real variable methods, orthogonality book by Elias Stein.

On page 73, it's written that $-\int_{|r|}^\infty s^{n-1} d_s \Phi(s\xi) = \Psi_\xi(r)$, and beneath that it's written that: $$(*)\int_{-\infty}^\infty \Psi_\xi(r)dr = 2\int_0^\infty r^n d\Phi(r\xi) ...
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Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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65 views

estimate of fourier transform

I am reading a paper and I don't understand one thing in the paper.Consider the convolution operator $Tf=f*\mu$ acting on $f\in L^p(\mathbb{R}^n)$, where $\mu$ is a measure defined by ...
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1answer
53 views

Norm triangle inequality for convolutions proof

I'm trying to prove that $$\|f*g\|_{L_1}\le{\|f\|_{L_1}\|g\|_{L_1}}$$ with respect to a Haar measure over a group G. Using Fubini's theorem, I'm up to ...
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How to show that $\int_G f(t) dt = \int_G f(t^{-1}) dt$?

I am reading the lecture notes. On page 34, line 13, it is said that $\int_G f(t) dt = \int_G f(t^{-1}) dt$. How to prove this identity? I think that if we let $s=t^{-1}$, then ...
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The continuous embedding of weighted $L^1$ space.

Take $\omega_0$: $\mathbb R^N\to \mathbb R^+$ such that $\omega$ l.s.c. and $\omega_0\geq 1$ and satisfies $$ \frac{1}{|{B}|}\int_{B(x,r)} \omega_0(y)\,dy\leq C\omega_0(x) \tag 1 $$ for any ball ...
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positive definite character

‎‎We know each character on dual group of ‎$‎Z‎; ‎‎\widehat{Z}‎$‎‎, is positive definite and if‎ ‎$‎‎\chi‎‎ ‎\in‎ \widehat{Z}‎$ then ‎$‎‎\left\| ‎‎‎\chi‎‎ ‎\right\|‎_{‎\infty‎}‎‎=‎‎\chi(1)‎$‎. But I ...
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1answer
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The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
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Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
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Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator ...
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1answer
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Upper bound on the integral $\int_{\mathbb R}\omega_I\omega_J$ with weights associated to intervals $I,J$

I'm currently studying Classical and Multilinear Harmonic Analysis. Vol. 1 by Camil Muscalu, Wilhelm Schlag. I need to verify following calculus inequality (Eq. 9.27, at page 255) ...
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Topology on dual of an abelian discrete topological group.

We define the compact-open topology on the dual of an abelian topological group. Please describe compact open topology more explicitly in the case where G is equipped with the discrete topology, for ...
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1answer
63 views

Dual group of $\mathbb Z$

We know $\hat{\mathbb Z}=\mathbb T$ and the map $\alpha\longmapsto\chi_{\alpha}$ is an isomorphism of $\mathbb T$ on to the character group of $\mathbb Z$, but I can't prove this map is continuous? ...
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Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
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30 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
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Do all continuous piecewise affine functions belongs to the class ($A_1$) of Muckenhoupt functions?

Given $\Omega\subset\mathbb R^N$ is open, we say $\omega$: $\Omega\to [0,+\infty)$ belongs to Muckenhoupt class $A_1$ if there exists some $C>0$ such that $$ \frac{1}{|{B}|}\int_{B(x,r)} ...
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Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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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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Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
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51 views

The Riesz transform kernel satisfies Hörmander's condition

Define the kernel $$K_i(x) = \frac{x_i}{\lvert x\rvert^{d+1}}$$ as a function $\mathbb{R}^d \setminus \{ 0 \} \to \mathbb{C}$ for $d \geq 3$ (say). I want to prove that this kernel satisfies ...
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Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
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1answer
42 views

Inverse-Fourier transform of a function after non-linear frequency modulation

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is ...
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1answer
48 views

Image of a function with small BMO norm

This is a question related with the regularity of harmonic maps. Let $N\geq 1$ and $f:\mathbb{R}^N\to \mathbb{S}^2$, where $\mathbb{S}^2=\{x\in \mathbb{R}^3 : \|x\|=1\}$. Assume that the BMO ...
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Integrability of Maximal Convolution Operator

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent ...
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Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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1answer
45 views

How does this conjecture correspond to Carleson's theorem for the case of $d=1$?

From this source, on page 36 (bottom) there is a conjecture stated and it was said that the case of $d = 1$ corresponds to Carleson's theorem. A picture included here : But when I look at wiki ...
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1answer
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Locally constant property

Suppose f is positive and Schwartz function. Fix $N>0$ and $A>0$. Suppose that for any $x \in [-N,N]$, $$A \leq \int_{-N}^{N}f(x-z)dz$$ Then do the inequality $$A \leq C_{r} ...