Localized Polynomial Frames on the Interval with Jacobi Weights

By Pencho Petrushev, Yuan Xu. 2005

65 citations recorded by the source corpus.

AbstractAs is well known the kernel of the orthogonal projector onto the polynomials of degree n in L2(wα,β, [−1, 1]), wα,β(t) = (1 − t)α(1 + t)β, can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling a compactly supported C∞ function then the resulting function has nearly 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 L2(wα,β) with elements having almost exponential localization.

Subfields: Applied Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bc15f336d36f3906d81335cd95fd55a26f75dfae

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In Applied Mathematics

1. Gerard Kerkyacharian
2. T. Coulhon
3. Pencho Petrushev
4. Yuan Xu
5. Feng Dai
6. Dominique Picard
7. P. Auscher
8. Daryl Geller

In Statistical and Nonlinear Physics

1. Pencho Petrushev
2. Gerard Kerkyacharian
3. T. Coulhon
4. Yuan Xu

Citations in the corpus

• Localized Polynomial Frames on the Ball (2006, background): “If σ > d/p+ 2µ|1/p− 1/2|, µ ≥ 0, 0 < p <∞, then (4.19) Jp := ∫ Bd Wµ(y)dy Wµ(n; y)p/2(1 + nd(x, y))σp ≤ c n−dWµ(n;x)1−p/2, where c > 0 depends only on p, µ, and d. Proof.”
• Decomposition of Spaces of Distributions Induced by Hermite Expansions (2007, methodology): “Our development here is a part of a bigger project for needlet characterization of Triebel-Lizorkin and Besov spaces on nonclassical domains such as the unit sphere [11], the interval with Jacobi weights [8, 13], and the unit ball [9, 14].”
• Heat Kernel Generated Frames in the Setting of Dirichlet Spaces (2012, methodology): “Another consequence of this is that our theory covers completely the construction of frames and the development of Besov and Triebel-Lizorkin spaces on [−1,1] with Jacobi weights from [35, 47].”
• Thomas Bayes’ walk on manifolds (2012, unknown): “The projector on Vk is given by the following formula (see [10], [23]), with λ = μ+ d−1 2 , Pn(x, y) = c(d, μ) n+ λ λ ∫ 1 −1 Gn ( 〈x, y〉+ u √ 1− |x|2 √ 1− |y|2 ) (1 − u2)μ−1du.”