Abstract
The aim of the paper is construction of calibration relations in the case of class of coordinate non-polynomial splines connected with refinement of grids. An embed\-ding of spline spaces is established for arbitrary refinement of grids. The reconstruction matrixes in the case of a grid on an open interval and a grid on a segment are constructed. The system of biorthogonal linear functionals to splines is constructed. The decomposition matrixes in the case of a grid on an open interval and a grid on a segment are constructed.
Highlights
Splines and wavelets are widely used in information theory
Wavelet decompositions are connected with constructing eective algorithms for processing
one can apply the powerful tools of harmonic analysis
Summary
È ñåòêà X ∈ X (K0, α, β) äëÿ íåêîòîðîãî K0 1, òî ïðè äîñòàòî÷íî ìàëîì hX ïðîñòðàíñòâî S(X, A∗, φ) ëåæèò â ïðîñòðàíñòâå Cm−1(α, β). Ðàññìîòðèì ïîñëåäîâàòåëüíî ïðåäñòàâëåíèå ôóíêöèè ωk−1(t) íà ïðîìåæóòêàõ (xk−1, xk), (xk, ξ), (ξ, xk+1). 1. Çàïèñûâàÿ àïïðîêñèìàöèîííûå ñîîòíîøåíèÿ (2) ïðè t ∈ (xk−1, xk) = (xk−1, xk) äëÿ ôóíêöèé ωj(t) è ωj (t) (íà ñåòêàõ X è X ñîîòâåòñòâåííî), íàõîäèì a∗k−2 ωk−2(t) + a∗k−1 ωk−1(t) ≡ φ(t),. 3. Ïðè t ∈ (ξ, xk+1) = (xk+1, xk+2) èç àïïðîêñèìàöèîííûõ ñîîòíîøåíèé èìååì a∗k ωk (t) a∗k+1 ωk+1(t). Ωk+1(t) dTk−1 a∗k ωk (t) a∗k+1 ωk+1(t), îòêóäà ïîëó÷àåì òîæäåñòâî (15), èáî íà ðàññìàòðèâàåìîì ïðîìåæóòêå ωk−1(t) ≡ 0. Òîæäåñòâî (15) óñòàíîâëåíî äëÿ âñåõ ðàññìàòðèâàåìûõ ïðîìåæóòêîâ; ó÷èòûâàÿ, ÷òî ôóíêöèè ωk−1 ëåæàò â ïðîñòðàíñòâå C(α, β), ïðèõîäèì ê âûâîäó, ÷òî ñîîòíîøåíèå (15) ñïðàâåäëèâî íà èíòåðâàëå (α, β). 1. Èç àïïðîêñèìàöèîííûõ ñîîòíîøåíèé (2) ïðè t ∈ (xk, ξ) = (xk, xk+1) íàõîäèì a∗k ωk (t), îòêóäà, óìíîæàÿ âåêòîðñòðîêó è èñïîëüçóÿ a∗k+1 è.
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