Geodesic
Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion
Matematičeskie zametki, Tome 78 (2005) no. 5, pp. 658-675 
Yu. K. Dem'yanovich. Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion. Matematičeskie zametki, Tome 78 (2005) no. 5, pp. 658-675. http://geodesic.mathdoc.fr/item/MZM_2005_78_5_a2/
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     author = {Yu. K. Dem'yanovich},
     title = {Embedded {Spaces} of {Trigonometric} {Splines} and {Their} {Wavelet} {Expansion}},
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     pages = {658--675},
     year = {2005},
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     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_5_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

With each infinite grid $X:\dots we associate the system of trigonometric splines $\{\mathfrak T_j^B\}$ of class $C^1(\alpha,\beta)$, the linear space $\mathscr T^B(X)\overset{\textrm{def}}=\{\tilde u\mid\tilde u=\sum_jc_j\mathfrak T_j^B\ \forall\,c_j\in\mathbb R^1\}$, and the functionals $g^{(i)}\in(C^1(\alpha,\beta))^*$ with the biorthogonality property: $\langle g^{(i)},\mathfrak T_j^B\rangle=\delta_{i,j}$ (here $\alpha\overset{\textrm{def}}=\lim_{j\to-\infty}x_j$, $\beta\overset{\textrm{def}}=\lim_{j\to+\infty}x_j$). For nested grids $\overline X\subset X$, we show that the corresponding spaces $\mathscr T^B(\overline X)\subset\mathscr T^B(X)$ are embedded in $\mathscr T^B(X)$ and obtain decomposition and reconstruction formulas for the spline-wavelet expansion $\mathscr T^B(X)=\mathscr T^B(\overline X)\dotplus W$ derived with the help of the system of functionals indicated above.

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