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for j = 0 , . . . m , where 1$_{I}$$_{j}$( x ) = 1 if x ∈ I$_{j}$ and 1$_{I}$$_{j}$( x ) = 0 otherwise. Bases of order r > 1 are generated recursively according to 

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for j = − ( r − 1) , . . . , m where we adopt the convention 1 0 := 0. Finally, we rescale the basis by multiplying each N$_{j,r}$ by ( m + r ) 1 / 2 for j = − ( r − 1) , . . . , m . This results in a total of K = m + r splines of order r . Each spline is a polynomial of degree r − 1 on each interior interval I$_{1}$, . . . , I$_{m}$ and is ( r − 2)-times continuously differentiable on (0 , 1) whenever r > 2. The mesh ratio is defined as 

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We let the space BSpl( K, [0 , 1]) be the closed linear span of these K = m + r splines. The space BSpl( K, [0 , 1]) has uniformly bounded mesh ratio if mesh( K ) ≤ κ for all N ≥ 0 and some κ ∈ (0 , ∞ ). We let BSpl( K, [0 , 1] , γ ) denote the space BSpl( K, [0 , 1]) with degree γ and uniformly bounded mesh ratio. See De Boor (2001) and Schumaker (2007) for further details. 

Wavelets We construct a wavelet basis with support [0 , 1] following Cohen, Daubechies, and Vial (1993). Let ( ϕ, ψ ) be a Daubechies pair such that ϕ has support [ − N + 1 , N ]. Given j such that 2 j − 2 N > 0, the orthonormal (with respect to the L $^{2}$([0 , 1]) inner product) basis for the space V$_{j}$ consists of 2 j − 2 N interior scaling functions of the form ϕ$_{j,k}$ ( x ) = 2 j/ $^{2}$ϕ (2 $^{j}$x − k ), each of which has support [2 − $^{j}$( − N + 1 + k ) , 2 − $^{j}$( N + k )] for k = N, . . . , 2 j − N − 1. These are augmented with N left scaling functions of the form ϕ 0 $_{j,k}$( x ) = 2 j/ $^{2}$ϕ l $_{k}$(2$^{j}$x ) for k = 0 , . . . , N − 1 (where ϕ l $_{0}$, . . . , ϕ l N − 1 are fixed independent of j ), each of which has support [0 , 2 − $^{j}$( N + k )], and N right scaling func- tions of the form ϕ$_{j,}$$_{2}$ $^{j}$− $_{k}$( x ) = 2 j/ $^{2}$ϕ r − $_{k}$(2$^{j}$( x − 1)) for k = 1 , . . . , N (where ϕ r − $_{1}$, . . . , ϕ r − N are fixed independent of j ), each of which has support [1 − 2 − $^{j}$(1 − N − k ) , 1]. The resulting 2 j functions ϕ 0 j, $_{0}$, . . . , ϕ 0 j,N − $_{1}$, ϕ$_{j,N}$ , . . . , ϕ$_{j,}$$_{2}$ $^{j}$− N − $_{1}$, ϕ 1 j, 2 $^{j}$− $_{N}$, . . . , ϕ 1 j, 2 $^{j}$− 1 form an orthonormal basis (with respect to the L $^{2}$([0 , 1]) inner product) for the subspace they span, denoted V$_{j}$ . 

An orthonormal wavelet basis for the space W$_{j}$ , defined as the orthogonal complement of V$_{j}$ in V$_{j}$$_{+1}$ , is similarly constructed form the mother wavelet. This results in an orthonormal basis of 2 j functions ψ 0 j, $_{0}$, . . . , ψ 0 j,N − $_{1}$, ψ$_{j,N}$ , . . . , ψ$_{j,}$$_{2}$ $^{j}$− N − $_{1}$, ψ 1 j, 2 $^{j}$− $_{N}$, . . . , ψ 1 j, 2 $^{j}$− $_{1}$. To simplify notation we ignore the 

