MULTIWAVELETS OF ODD DEGREE, ORTHOGONAL TO POLYNOMIALS

For the space of Hermitian splines of 2r + 1-st degree of a kind S-L(x) = Sigma(r)(k=0) h(k) Sigma(i=0)2(L) C-i(L,k) N-i,k(L)(x), a <= x <= b, where the coefficients C-i(L,k), k = 0, ..., r, are values and corresponding derivatives of the approximated function in the knots of uniform net Delta(L): u(i) = a + (b - a) i / 2(L), i = 0, 1, ..., 2(L), L >= 0, and basic functions are N-i,k(L)(u(j)) = delta(j)(i) delta(l)(k), l = 0, 1, ..., r, with the centers in integers, it is offered to use as wavelets the functions M(L)i,k(x), k = 0, 1, ..., r for all i, with the centers in odd integers, the linear combinations of basic Hermitian splines with the grid Delta(L+1), that are orthogonal to polynomials of 2r + 1-st order, integral(b)(a) M-i,k(L)(x)x(m) dx = 0, k = 0, 1, ..., r for all i (m = 0, 1, ..., 2r+1). If the corresponding spline-coefficients are collected in the vector, C-L = [C-0(L,0), C-0(L,1), ..., C-0(L,r), C-1(L,0), ..., C-2L(L,r)](T), and the corresponding wavelet-coefficients - in the vector, D-L = D-1(L,0), D-1(L,1), ..., D-1(L,r), ..., D-2L(L,r)](T), then with use of designations for block matrices the formulas for evaluation of spline-coefficients CL-1 on the thinned grid Delta(L-1) and wavelet-coefficients DL-1 in the form of the solution of sparse system of linear algebraic equations are proved: [P-L vertical bar Q(L)][CL-1/DL-1] = C-L. Here blocks of the matrix P-L are composed from coefficients of the scale relations for basic splines and blocks of the matrix Q(L) are composed from coefficients of the decomposition for basic wavelets M-L(i,k) (x). For the purpose of using of the rarefied structure of a matrix [P-L vertical bar Q(L)] there is offered to make it block tri-diagonal, having changed an order of unknowns so that blocks of matrixes P-L and Q(L) been alternated, to be able to apply an algorithm of a block matrix sweeping to the solution of the received system. The problem of stability of algorithm of multiwavelets-transformation on big grids by means of observation of behavior of condition numbers in Euclidean norm of sweeping matrixes is investigated. The numerical example of approximation and compression of data for a case of Hermitian splines of the 7th degree is given.