Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion

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 $$T^B (X)mathop = limits^{def} { tilde u|tilde u = sumlimits_j {c_j mathfrak{T}_j^B } quad forall c_j in mathbb{R}^1 } ,$$ and the functionals g ⁽ⁱ⁾ ∈ (C ¹(α, β))* with the biorthogonality property: $leftlangle {g(i),mathfrak{T}_j^B } rightrangle = delta _{i,j}$ (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.