Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion |
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Authors: | Yu K Dem'yanovich |
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Institution: | 1. St. Petersburg State University, St. Petersburg, Russia
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Abstract: | With each infinite grid X: ? < x ?1 < x 0 < x 1 < ? we associate the system of trigonometric splines $\{ \mathfrak{T}_j^B \}$ of class C 1(α, β), the linear space $$T^B (X)\mathop = \limits^{def} \{ \tilde u|\tilde u = \sum\limits_j {c_j \mathfrak{T}_j^B } \quad \forall c_j \in \mathbb{R}^1 \} ,$$ and the functionals g (i) ∈ (C 1(α, β))* with the biorthogonality property: $\left\langle {g(i),\mathfrak{T}_j^B } \right\rangle = \delta _{i,j}$ (here $\alpha \mathop = \limits^{def} \lim _{j \to - \infty } x_j ,\quad \beta \mathop = \limits^{def} \lim _{j \to + \infty } x_j$ ). For nested grids $\bar X \subset X$ , we show that the corresponding spaces $T^B (\bar X)$ are embedded in $T^B (X)$ and obtain decomposition and reconstruction formulas for the spline-wavelet expansion $T^B (X) = T^B (\bar X)\dot + W$ derived with the help of the system of functionals indicated above. |
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