Questions tagged [harmonic-analysis]
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
1,584 questions
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Projection onto translation invariant subspaces
I'm currently looking at a subspace of $A \subset \ell^p(\mathbb{Z}^n)$ which is generated by some finitely supported elements and their translations. My question is an old one (but the answer is ...
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Lower bounds for $\|f*g\|_1$ with mean-zero Lipschitz functions on $[0,1]$
Let $f,g \in L^{1}([0,1])$ satisfy
$$
\|f\|_{1}=\|g\|_{1}=1, \qquad \int_{0}^{1} f(x)\,dx=\int_{0}^{1} g(x)\,dx=0,
$$
and assume
$$
f \in \mathrm{Lip}_{L_f}, \qquad g \in \mathrm{Lip}_{L_g}.
$$
...
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Besov spaces under multiplication
Let $ \sigma \geq \frac{n}{2} $. And consider the inhomogeneous Besov space $B^{\sigma}_{2,1}$ with the norm
$$ \Vert f \Vert_{B^{\sigma}_{2,1}}= \sum_{k=0}^{\infty} 2^{\sigma k} \Vert \Delta_{k} f \...
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Another question on large sieve inequality
Recently, I was interested in the large sieve inequalities. A few days ago, I came up with a question on the large sieve inequality involving 𝐺𝐿(2); see On the large sieve inequality involving $GL(2)...
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Cotlar type inequality for Riesz transforms
Let $R^s_\mu(x)= \int \frac{y-x}{|y-x|^{s+1}}d\mu(y), x,y \in \mathbb{R}^d, 0<s<d$ be the Riesz transform (of index $s$).
I would like to understand the proof of the following inequality. There ...
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Mapping properties of the strong maximal function and a question about the article of Cordoba Fefferman
I have been reading the article "A geometric proof of the strong maximal theorem" by A. Cordoba and R. Fefferman which can be found here. Right in the beginning of the article the authors ...
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Rank of tensor product of irreducible representations over finite symmetric group
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $\...
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Harmonic analysis on the non-trivial zeros of the Riemann zeta function?
Suppose I have some function $f(x)$ that satisfies constraints roughly as restrictive as those for Fourier series expansions, and I'm interested in alternative ways of expanding it between some bounds ...
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Sum in tensor product of irreducible representations on $S_n$
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $...
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A common used construction in Harmonic analysis
This may be rather elementary. How to construct such function $\eta$ as shown in the picture?
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On the large sieve inequality involving $GL(2)$ harmonics
I have a question on the large sieve inequality involving $GL(2)$ harmonics. Recall that one has the analog for $GL(1)$ harmonics that, for any complex numbers $\alpha_m,\beta_n$, one has
$$\sum_{q\le ...
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Traces mixing tensor products of Fourier coefficients on finite symmetric groups
Let $G$ be a finite (symmetric) group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$. Let $f:G\to \{0,1\}$ be a Boolean function on $G$ ...
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Weak L2 norm in proof of Carleson's theorem
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. Under Lemma 2.18 on page 10 it says: "...
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Bounding the largest Fourier coefficient of $f$ minus a class function on symmetric group $S_n$
Let $G=S_n$ be a symmetric group. A class function on $G$ is any $g:G\to\mathbb C$ that is constant on conjugacy classes, i.e. $g(xy)=g(yx)$ for any $x,y\in G$. Let $\mathcal C$ denotes the set of ...
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Quantitative differentiation via Littlewood-Paley theory
In an article by Cheeger, he presents an elementary method, which he refers to as quantitative differentiation, for obtaining a quantitative version of Rademacher's theorem. While these arguments use ...
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