The paper is concerned with the approximation and interpolation employing polyharmonic splines in multivariate problems. The properties of approximants and interpolants based on these radial basis functions are shown. The methods of such data fitting are applied in practice to treat the problems of, e.g., geographic information systems, signal processing, etc. A simple 1D computational example is presented.

The paper contains short description of $\Sigma \Pi$-algorithm for the approximation of the function with two independent variables by the sum of products of one-dimensional functions. Some realizations of this algorithm based on the continuous and discrete splines are presented here. Few examples concerning with compression in the solving of approximation problems and colour image processing are described and discussed.

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative (${𝐃}_{\phi }$) and integration matrix ($𝐏$) are introduced. These matrices are utilized to reduce the solution of nonlinear constrained quadratic optimal control to the solution of nonlinear programming one to which existing well-developed...

Two universally applicable smoothing operations adjustable to meet the specific properties of the given smoothing problem are widely used: 1. Smoothing splines and 2. Smoothing digital convolution filters. The first operation is related to the data vector $r={(r_0,...,r_{n-1})}^T$ with respect to the operations $𝒜$, $ℒ$ and to the smoothing parameter $\alpha$. The resulting function is denoted by ${\sigma }_{\alpha }\left(t\right)$. The measured sample $r$ is defined on an equally spaced mesh $\Delta ={\{t_i=ih\}}_{i=0}^{n-1}$$...$