arXiv subjects

George Kyriazis

Publications and source records attributed to George Kyriazis.

4 recordsLinked to original sources

arXiv2023

Spaces of distributions on product metric spaces associated with operators

We lay down the foundation of the theory of spaces of distributions on the product $X_1\times X_2$ of doubling metric measure spaces $X_1$, $X_2$ in the presence of non-negative self-adjoint operators $L_1$, $L_2$, whose heat kernels have Gaussian localization and the Markov property. This theory includes the development of two-parameter functional calculus induced by $L_1, L_2$, integral operators with highly localized kernels, test functions and distributions associated to $L_1, L_2$, spectral spaces accompanied by maximal Peetre and Nikolski type inequalities. Hardy spaces are developed in this two-parameter product setup. Two types of Besov and Triebel-Lizorkin spaces are introduced and studied: ordinary spaces and spaces with dominating mixed smoothness, with emphasis on the latter. Embedding results are obtained and spectral multiplies are developed.

math.FA↗

arXiv2018

Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators

We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel-Lizorkin spaces. Spectral multipliers for these spaces are established as well.

math.CA↗

arXiv2009

Inversion of noisy Radon transform by SVD based needlet

A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in $L^p$ ($1\le p\le \infty$) norms for functions with Besov space smoothness. A practical implementation of the method is given and several examples are discussed.

math.ST↗

arXiv2006

Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

The Littlewood-Paley theory is extended to weighted spaces of distributions on $[-1,1]$ with Jacobi weights $ \w(t)=(1-t)^α(1+t)^β. $ Almost exponentially localized polynomial elements (needlets) $\{ϕ_ξ\}$, $\{ψ_ξ\}$ are constructed and, in complete analogy with the classical case on $\RR^n$, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\{\ip{f,ϕ_ξ}\}$ in respective sequence spaces.

math.CA↗