Lower and upper bounds for the error of the Jth resolution via optimal wavelet choice for a signal

Selection of a wavelet for a given signal such that the error of the discrete wavelet representation up to a given scale is minimized is investigated. Lower and upper bounds are derived for the error of the Jth resolution f_j of f with respect to f itself when the Fourier spectrum of f is mostly concentrated in -2^jπ, 2^jπ. The lower and upper bounds only differ from each other by a positive constant multiple. Based on the error bounds, the cost function is chosen as the upper bound with quadratic-form of unknown coefficients. The optimal coefficients of the Daubechies wavelets are formulated as solutions of certain quadratic equations which depend on the signal and J. Optimal wavelets for the signal-independent case are considered