Wavelet packets: Uniform approximation and numerical integration

In this paper, it is proved that under some conditions, wavelet packet basis of [Formula: see text] can be used as a tool for the uniform approximation of an [Formula: see text]-times ([Formula: see text]) continuously differentiable and square integrable function [Formula: see text]. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function [Formula: see text] at lower levels are uniformly reliable and they uniformly approach zero as [Formula: see text] are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in [Formula: see text] is given and its error analysis has been discussed.

LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.

THE BEST WEIGHTED GRADIENT APPROXIMATION TO AN OBSERVED FUNCTION

We find the potential function whose gradient best approximates an observed square integrable function on a bounded open set subject to prescribed weight factors. With an appropriate choice of topology, we show that the gradient operator is a bounded linear operator and that the desired potential function is obtained by solving a second-order, self-adjoint, linear, elliptic partial differential equation. The main result makes a precise analogy with a standard procedure for the best approximate solution of a system of linear algebraic equations. The use of bounded operators means that the definitive equation is expressed in terms of well-defined functions and that the error in a numerical solution can be calculated by direct substitution into this equation. The proposed method is illustrated with a hypothetical example.

Nonlinear System Identification Using Laguerre Wavelet Models

A compact and efficient model that is capable of approximating both the linear and nonlinear components of the process is in high demand. In this paper, a novel black-box modeling technique viz. Wiener type Laguerre-Wavelet model is proposed. The Laguerre-Wavelet model has the capability to approximate a function with moderate/reasonable number of data with appreciable approximation accuracy. The Laguerre filter is used to approximate the linear dynamic components of the process, whereas wavelet structure is used for the static nonlinear components. The ability of wavelets to approximate any square-integrable function to any arbitrary precision by input-output mappings is utilised for the nonlinear approximation following a modified single scaling method. The performance efficiency of the proposed Wiener type model structure, Laguerre-Wavelet model, is demonstrated using simulation case study on a continuous bioreactor.