Multivariate spline functions. I. Construction, properties and computation

On Smooth Multivariate Spline Functions

In this paper the dimensions of bivariate spline spaces with simple cross-cut grid partitions are determined and expressions of their basis functions are given. Consequently, the closures of these spaces over all partitions of the same type can be determined. A somewhat more detailed study on bivariate splines with rectangular grid partitions is included. The results in this paper can be applie...

Construction of dual B-spline functions

Using new properties of dual functions, we show that the method of constructing the dual bases recently presented in [P. Woźny, J. Comput. Appl. Math. 245 (2013), 75–85] can be significantly simplified. Next, we propose a simple algorithm of constructing the dual B-spline functions, which can be useful in solving some approximation problems related to numerical analysis or computer graphics.

Developing a Computation-Friendly Mathematical Foundation for Spline Functions

Construction of Lyapunov functions by validated computation

We provide a systematic method to validate the domain of Lyapunov functions around hyperbolic fixed points both for continuous and for discrete dynamical systems with computer assistance. We also give the construction of Lyapunov functions around periodic orbits as fixed points of Poincaré maps as well as validation examples in our talk.

Construction of spline functions in spreadsheets to smooth experimental data

Experimental data are often in need of smoothing without a pre-determined trend line equation. This manuscript demonstrates how spreadsheet software can be programmed to smooth experimental data via cubic splines. Once the trend lines (splines) have been constructed, it is also simple to interpolate values and calculate derivatives and integrals. The formulas to carry out the calculation are li...