Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion

With each infinite grid $X:\dots<x_{-1}<x_0<x_1<\dotsb$ 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.