Spline-wavelets, orthogonal to polynomials, and optimization of calculations of wavelet-transformation

R 2 For the space of the Hermitian splines of any odd degree 2r+1 of a kind S (x) = ^ ^ C N k (x), a < x < b, with a uni- k=0 i=0 form grid of nodes A : щ = a + (b - a) i / 2 , i = 0, 1,..., 2 , L > 0, and the basic functions NL _k (u _} ) = 5j • b k, l = 0,1,..., r, with the centers in integers, it is proposed to use as wavelets the functions M ik(x), satisfying the conditions of orthogonality to all polynomials of the 2r+2-nd order, i.e. f MW(x)x dx = 0, k = 0,1,.,r Vi (m = 0,1,...,2r +1). For wavelets with the centers in even integers Ja and the supports, which are equal to the supports of basic splines on a grid A , the formulas for calculating the coefficients in the thinned grid A from the spline coefficients C°L'°, С'О'' ,..., С°"' , С D -i,°, Dj ' , ,L-1,r L-1, r -rL,r -rL,0 ..., D ..., D ..., С L in a ' 2 dense grid A in the form of the solution of the linear algebraic equations system with a block three-diagonal matrix of a kind K u = C , /~*L,r P)L,0 pjL,\ ' 0 , 1 , 1 , " Hi д center O ' K = H 2 I H 2 O center 2 H; · CLL ] _uL = ГсL,° СL1 |_ 0 , 0 , , Dj , , CL,°, D Г h 0 ^ 0 0 0 I left 0 H2 a;- h left 2 0 0 A H; H 2 K = L >1. 0 right A°° H ₂ A H° Hi 0 I are received. Here H ₂ = U AU, H ₁ = diag(1,2 ,...,2 ), H ₀ = SH ₂S , and the matrix Uof dimensions (r +1) x (r +1) is set by the elements U, , =(-1) + + - (Г + + , k, j = 0,1,...,r, and Л = diag(2 ,...,2 ), j \ ! ( +1 + k - j)! )' S = diag(1, -1,...,(-1) ) , O designates a matrix of the r+1-st order with zero coefficients whereas I is a single matrix of the r+1-st , g t reputing t K . T [AO™ /A2 ] = -[R |R ] R are set by the elements, respectively j+1 RjJ = J Ф;(21 - j)t dt, j = 0,1,2,l = 0,1,...,r, m = 0,1,.,2r +1, j-1 where _Фк ( _t ) = ^ - ) к (- )пР - - - [ ra _k (t) при 0 - t - 1, and ® _k (t) = (1 - t) + У t +P, k = 0,1,., r. y> y ' _pt0 k!P!r! Results of numerical experiments for different degree of a spline at the same number of basic functions in comparison with a classical method of the least squares (MLS) are presented. The improvement of adjustment on condition of coincidence of a spline knot to a break of the approximated function is revealed. It is shown that for lack of spline knots the MLS-solution and the wavelet-decision are close to each other. In the presence of spline knot the MLS -solution and the wavelet-decision differ from each other more considerably.

Keywords

Authors

References