arXiv subjects

Pencho Petrushev

Publications and source records attributed to Pencho Petrushev.

21 records · Page 2Linked to original sources

arXiv2006

Localized polynomial frames on the ball

Almost exponentially localized polynomial kernels are constructed on the unit ball $B^d$ in $\RR^d$ with weights %functions $W_μ(x)= (1-|x|^2)^{μ-1/2}$, $μ\ge 0$, by smoothing out the coefficients of the corresponding orthogonal projectors. These kernels are utilized to the design of cubature formulae on $B^d$ with respect to $W_μ(x)$ and to the construction of polynomial tight frames in $L^2(B^d, W_μ)$ (called needlets) whose elements have nearly exponential localization.

math.CA↗

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↗

arXiv2005

Localized polynomial frames on the interval with Jacobi weights

As is well known the kernel of the orthogonal projector onto the polynomials of degree $n$ in $L^2(w_{\a,\b}, [-1, 1])$ with $w_{\a,\b}(t) = (1-t)^\a(1+t)^\b$ can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling a $C^\infty$ function then the resulting function has almost exponential (faster than any polynomial) rate of decay away from the main diagonal. This result is used for the construction of tight polynomial frames for $L^2(w_{\a,\b})$ with elements having almost exponential localization.

math.CA↗