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Results tagged with harmonic-analysis
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user 153260
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.
5
votes
1
answer
266
views
A domination property for the Hardy space $H^1$
In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)|, \, …
3
votes
Accepted
Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?
Yes I think this is true.
Let $\mathcal{G} := \{ \times_{i=1}^n[k_i,k_i+1] : k_i\in \mathbb{Z} \} $ be the grid of cubes of side length $1$ and vertices in $ \mathbb{Z}^n$. For $Q\in \mathcal{G} $ le …
3
votes
Accepted
Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions
I asked the polydisc experts for a reference and in fact it is known. It was proved by by J.P. Rosay in this paper. It is in french but it shouldn't be difficult to understand.
3
votes
0
answers
88
views
Upcrossing lemma and subharmonic functions
I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $
\lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-upcro …
2
votes
1
answer
132
views
On a density property of signed singular measures
Suppose that $\mu$ is a signed finite Borel measure which is singular with respect to the Lebesgue measure in $[0,1]$. Is it true that there always exist a point $x\in [0,1]$ such that
\begin{equation …
1
vote
0
answers
63
views
Hilbert like transform on the circle
Suppose that $u $ is a smooth function real valued function defined on an open neighborhood of the unit disc $ \mathbb{D} $, which satisfies a second order elliptic partial differential equation;
\beg …
1
vote
Accepted
Orthonormal bases in RKHSs via interpolating sequences
It is true that if a sequence $(k_n)$ is interpolating for $M(\mathcal{H})$, then the normalized Kernel vectors $g_n:=K_{k_n}/\Vert K_{k_n} \Vert_\mathcal{H} $ form a Riesz system in $\mathcal{H}$. O …
1
vote
Fourier series but different waveform
Partial answer: About linear independence, it is true that if $f$ is non constant then the dilations $f_n(x)=f(nx), n\in \mathbb{N}$ are linearly independent. In fact suppose for a finite sum we hav …