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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.

4 votes
2 answers
504 views

Maximal function related to the Ornstein-Uhlenbeck operator.

On $\mathbf R^d$ the Ornstein-Uhlenbeck operator is defined as ($\partial_i = \frac{\partial}{\partial x_i}$). $$L = \frac12 \sum_i \partial_i^* \partial_i$$ where $\partial_i^* = -\partial_i + 2 x …
Jonas T's user avatar
  • 455
3 votes
2 answers
3k views

Eigenvalues convolution-type operator

Let $J_1$ be the Bessel function of the first kind and let $H_1(x) = \frac{J_1(|x|)}{|x|}$ for $n = 1$. Define the operator $Tf(x) = (f * H_1)(x)$ from $L^2$ to $L^2$. Since the $H_1$-function is the …
Jonas T's user avatar
  • 455