An algorithm with splitting of the wavelet transforms offirst degree splines

For the space of first degree splines of a kind ( ) ( )2 10S N ,LL L Li iiu C u−== ƒ a ≤ u ≤ b, with a uniformnet of knots ƒL : ui = a + (b - a) i / 2L, i = 0, 1,…, 2L, L ≥ 0, and basic functionsNL( ) ƒji uj = i , with the centers in integers, it is proposed to use as wavelets functions NLiwith the centers in even integers («lazy» wavelets). The formulas for splines-coefficients on athinned net ƒL-1 and wavelet-coefficientsC0L-1=2 C L1, CiL-1=2 CL2i-1+ 2 CL2i+1, i = 1, 2,…, 2L-1,D0L-1= C L0 - 2 C L1, DiL-1=C L2i - 2CL2i-1- 2CL2i+1, i = 0, 1,…, 2L-1,are obtained. Here values of spline-coefficients CiL in odd knots are counted from the solutionof three-diagonal system of the linear equations with matrix2 11 2 1 .1 1⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢⎣ ⎥⎦

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