Locally tensor product functions
A class of tensor product functions in subdomains (usually with different ``steps'' of the grid for different subdomains) is constructed. The global function is not a tensor product function. Some fine error bounds for these functions are unknown. The property of smoothness and some discontinuities of global functions are not investigated. These functions cumulate main advantages of tensor product functions and the possibility of numerous (dense grid) degrees of freedom in some parts of the space, while less (coarse grid) degrees of freedom in other parts. This allows the user to have a fine grid in some regions, a coarse grid in other regions without any interaction between these two grids. The cubic B-splines are used in the space of two dimensions, because a cubic space spline has more knots than the dimension of the space (``knot''- a junction between two different polynomials). The choice of the knots of the B-splines is something strategic. The form and the properties of the constructed functions depend on it. If the underlying process presents some irregularity in an isolated point, centres of the B-splines can be concentrated around this point, and the levels of the B-splines will be greater there. Approximation of functions, or data, with such functions gives satisfactory results, as shown by numerical experimentation.
scientific article; zbMATH DE number 2170146
scientific article