SOURCE. Daubechies, I. (1988), "Orthonormal bases of compactly supported wavelets", Communications on Pure and Applied Mathematics 41(7):909-996, doi:10.1002/cpa.3160410705.
Value
List with basis, coefficients, h,
g, orthonormality, normalisation,
double_shift, vanishing_moments, gram_error,
energy_in, energy_out, n, level,
filter_length.
Details
The scaling filter h of length L = 2N is what the paper constructs:
the compactly supported solution of the two-scale relation whose
defining conditions are sum h_k = sqrt(2), sum_k h_k h_{k+2m} =
delta_{m0}, and sum_k (-1)^k k^m h_k = 0 for m = 0 ... N-1. The
wavelet filter is the quadrature mirror g_k = (-1)^k h_{L-1-k}. Those
conditions determine h up to reflection and are checked numerically
here rather than taken on trust: orthonormality,
double_shift and vanishing_moments are the residuals.
FILTERS. db1 (Haar), db2 and db3, each from its exact radical closed form rather than a decimal table: db1 h = (1,1)/sqrt(2); db2 h = (1+r3, 3+r3, 3-r3, 1-r3)/(4 sqrt(2)) with r3 = sqrt(3); db3 h = c/sqrt(2) with 16c = (1+a+b, 5+a+3b, 10-2a+2b, 10-2a-2b, 5+a-3b, 1+a-b), a = sqrt(10), b = sqrt(5+2a). Longer filters have no compact radical form and are NOT provided; that is this implementation's scope choice, stated rather than attributed.
BASIS. For length N = 2^J the periodic pyramid transform is orthonormal; its matrix W is built by transforming the standard basis vectors, so W W' = I is a real check on the pyramid code.
References
Daubechies, I. (1988). Communications on Pure and Applied Mathematics 41(7):909-996. doi:10.1002/cpa.3160410705.
Examples
Wave(c(1, 2, 3, 4), "db1")$coefficients
#> [1] 5.0000000 -2.0000000 -0.7071068 -0.7071068