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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Can a lower semi-continuous function blow up to $+\infty$?

I was confused in lecture today that the professor said a l.s.c function can not blow up to $+\infty$... In my point of view, a l.s.c function can blow up to $+\infty$... for example, we take ...
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39 views

What is the Fourier Transform of an absolute function?

I would like express that the Fourier transform of the function $$ |f(x)| $$ as $$ \widehat{|f|}(\xi) = \text{a function of } \widehat{f}(\xi) $$ In fact, I want to know the relation of ...
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98 views

Derive Hausdorff-Young inequality from Paley's inequality

Given a sequence $(c_j)_{j\in\mathbb{Z}}$ of complex numbers with $\lim_{|j|\to\infty}c_j=0$. Define the rearrangement $c_j^*$ as follows: for $j\geq 0$, $c_j^*$ is the $j+1-$th largest element of the ...
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Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
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Left invariance of a differential operator

Given $x, a\in G$ with G a group and x and a fixed, does the left invariance of a differential operator D on G imply that $D[f(a^{-1}x)]=D[a^{-1}f(x)]$?
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Are singular integral operators bounded on $L\log L$?

My question is regarding singular integrals of Calderon Zygmund type. It is known that the maximal function is bounded on $L\log L \mapsto L^1$ (but not on $L^1$) and satisfies the same operator ...
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59 views

Structure of $\Bbb Q_p/\Bbb Q$

The group $\Bbb A_\Bbb Q/\Bbb Q$ of adeles mod rationals is: Isomorphic to the solenoid Isomorphic to the group $\Bbb A_\Bbb Z/\Bbb Z$ Dual to the rational numbers This raises the question of the ...
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proof that the Fourier series of $ f\ast g $ uniformly converge.

Let   $f,g$  be   $2\pi$-periodic piecewise continuous functions. proof that the Fourier series of $ f\ast g $ uniformly converge. Where $ f\ast g $ denotes the convolution operator ...
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The equivalent of “tempered distributions” for the Mellin transform?

The Fourier transform is defined for tempered distributions. For these distributions, the test functions are those functions decreasing more quickly at $\pm \infty$ than $|x|^{-n}$ for all n. In ...
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Do combined waves with non-rational frequencies have a common period?

I am facing a problem where I have two waves combined: \begin{equation} y = A\sin(b_1x)+B\cos(b_2x) \end{equation} Where $ b_1 $ and $ b_2 $ are non-rationals. i.e. \begin{align} & b_1 = ...
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37 views

Verification of a weighted inequality calculation

I was reading Fourier analysis by J. Duoandikoetxea, and checking out the proof of the $(L^p,L^p)$ inequality $$ \left| \left| \left( \sum_j|T_jf_j|^2 \right)^{\frac{1}{2}} \right| \right|_p\leq ...
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22 views

What do we call the harmonics in a discrete Fourier series representation?

In harmonic analysis using discrete Fourier series, if I'm using the 0f, 1f, 2f, 3f and 4f for representation where f = frequency, what is the correct way to say how many harmonics I'm using for ...
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42 views

Decide if the improper integral of a Fourier transform converges

I have the function: $$f(x)=\left\{\begin{matrix} e^{-x^{10}} & ,x>0\\ -e^{-x^{10}} & ,x<0 \end{matrix}\right.$$ I need to answer: ...
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65 views

estimating a convolution type maximal function

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}_{+}$ be a $C^1$ function with $supp(\phi) \subset B(0,1)$ and $\int \phi = 1$. Define $$\phi_t(x) := t^{-n} \phi({x/t})$$ and set $$ M_{\phi} f(x) := ...
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52 views

Convergence of improper integral over Fourier transform.

So I have the Fourier transform $$ \widehat{f}(\omega)=\frac{1}{1+|\omega|} $$ of some function $f(x)$. I need to know if the two integrals below converge or not. $$ ...
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54 views

Solving integral equation using convolution and Fourier transform.

So I have the integral equation : $$\int_{-\pi}^{\pi} f(t)f(x-t) dt = -\cos (x).$$ I know that I should use Fourier transform or Laplace transform and to use the convolution theorem, but I'm not ...
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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. ...
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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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question about property of $L^p$ Lipschitz space

$f\in L^p$ is said to satisfy $L^p$ Lipschitz condition of order $\alpha$ if there exists $C>0$ such that $\displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p ...
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Argument for extenstion of the Fourier transform

Could anyone please point out if there is any mistakes in the following arguments for the extension of the Fourier transform from $\mathcal{S}(\mathbb{R})$ to $L^2(\mathbb{R})$: "Since the compactly ...
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39 views

$\||u|\|_{H^s(\mathbb R^n)} \le C \| u \|_{H^s(\mathbb R^n)}$ holds for even if $s$ is not an integer?

Let $u: \mathbb R^n \ni x\mapsto u(x) \in \mathbb C.$ I would like to know that the inequality $$ \||u|\|_{\dot H^s(\mathbb R^n)} \le C \| u \|_{\dot H^s(\mathbb R^n)} $$ or $$ \||u|\|_{H^s(\mathbb ...
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29 views

$L^2$ norm for difference of translation of disjoint interval

I just need some hint since I've been stuck in this for several hours.. Let $I_1,I_2,\ldots,I_K$ bne disjoint intervals in $[-1/2,1/2)$,and $f(x)=\sum_{j=1}^K\chi_{I_j}$, where $\chi_I(x)$ is the ...
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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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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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what is the difference between Pseudo-differential operators and Pseudo-differential calculus?

What is difference between Pseudo-differential operators and Pseudo-differential calculus? (Or is it the same field just the different name) [I want just rough idea for motivation; (I am familiar ...
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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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1answer
56 views

Sunrise Lemma, left and right maximal functions

Would anyone be able to provide a proof of the following lemma: Where $| \cdot |$ is the Lebesgue measure and $M_L$ and $M_R$ are the left and right maximal functions defined on $\mathbb{R}$ ...
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38 views

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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Does a Plancherel-style theorem for the Hardy space $\mathcal{H}^2(\mathbb{T})$ exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathcal{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
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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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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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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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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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1answer
44 views

Assumptions on the Borel measure in Stein's Harmonic Analysis

I am currently reading the proof of Theorem 1 on Page 13 of Stein's Harmonic Analysis which proves that if $f \in L^{1}(\mathbb{R}^{n})$, then for every $\alpha > 0$, $$\mu(\{x: (Mf)(x) > ...
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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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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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199 views

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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57 views

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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117 views

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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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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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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223 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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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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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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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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1answer
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 ...