Abstract
Piecewise continuous spline of Lagrange type are constructed. An embedding of spline spaces is established for arbitrary refinement of grids. The system of biorthogonal linear functionals to splines is constructed. Wavelet decompositions and decomposition and reconstruction algorithms in the case of an infinite flow for a grid on an open interval and a finite flow for a grid on a segment are constructed.
Highlights
Splines and wavelets are widely used in information theory
Wavelet decompositions are connected with constructing effective algorithms for processing large digital information flows
If (α, β) = R1 and a grid is uniform, one can apply the powerful tools of harmonic analysis (in the space of functions L2(R1) and in the space of sequences l2), lifting scheme or wavelet scheme
Summary
Система {ωj(t)}j∈Z линейно независима, благодаря локальности носителей функций ωj(t) и отделенности от нуля функции φ на интервале (α, β). Линейная оболочка функций ωj(t), получаемых для функции φ ∈ C −1(α, β), φ(t) = 0 ∀ t ∈ (α, β) по формуле (2), называется пространством кусочно-непрерывных сплайнов на сетке X и обозначается через. 1. Записывая аппроксимационные соотношения (2) при t ∈ [xk, ξ) для функций ωj(t) и ωj (t) (на сетках X и X соответственно) ввиду равенства [xk, ξ) = [xk, xk+1) находим ωk(t) ≡ φ(t), ωk(t) ≡ φ(t); отсюда ∀t ∈ [xk, ξ) получаем тождество ωk(t) ≡ ωk(t), фактически совпадающее с тождеством (5) на рассматриваемом промежутке, ибо на нем ωk+1(t) ≡ 0.
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