The Generalized Haar Spaces and Their Adaptive Decomposition
International Journal of Circuits Systems and Signal Processing · 2020
This paper is devoted to the numericalinformation flows and adaptive decompositions of thegeneral Haar functions connected with them. The aim ofthis paper is to propose an adaptive wavelet decompositionusing an adaptive compression algorithm for a flow ofnumerical information of length M with complexity𝑶(𝑴) and with a given precision of 𝜀> 0. The numericalflows are associated with irregular spline grids. This paperdiscusses the calibration relations, the embedding of thegeneral Haar spaces and their wavelet decompositions.The structure of the decomposition/ reconstructionalgorithms are done. The cases of the finite and the infiniteflows are considered. The paper discusses various methodsof adaptive Haar approximations for the flow of functionvalues. Assuming that the values of the first derivative ofthe approximated function is known (exactly orapproximately), the complexity of using an adaptive grid isestimated for a priori specified approximation accuracy.The number of K knots in the adaptive grid determine therequired amount of memory for storage of thecompression results. The number of M knots of the initialgrid characterizes the number of operations required toobtain the adaptive compression. In the case of access tothe derivative values (or their approximations) thenumber of digital operations is proportional to the numberM. If it does not have access to the last ones then thenumber of required operations has the order of M2 (in thegeneral case). If additionally, the approximated flow isconvex, then the number of required operations has theorder of M log2M. In all cases the result requires thecomputer memory amount to be of the order of K