The principal investigator studies cubature rules, which are numerical integration formulas for higher dimensional integrals, and approximation of functions on regular domains such as cubes, balls, spheres and simplexes. The project combines several topics: numerical analysis, discrete Fourier analysis, orthogonal polynomials, and approximation theory. Two approaches are emphasized. The first approach is based on a connection between cubature rules and discrete Fourier analysis with translation tiling. The approach allows one to study, in several stages, cubature rules and interpolation by exponential functions on the fundamental domain of the translation tiling, by trigonometric functions on the fundamental simplex of the domain, and by algebraic polynomials on corresponding domains, and it yields results on cubature rules, interpolation, orthogonal polynomials and approximation. The second approach starts with a characterization of best approximation by polynomials on the sphere and on the ball in terms of the smoothness of the functions being approximated, while the smoothness is measured by the differences of the function values in Euler angles. This line of work is based on recent results of the investigator and his collaborators that for some problems it is necessary to work with these angles, even though their number is much larger than the dimension.
Cubature rules, which are multidimensional numerical integration formulas, and approximation on regular domains in higher dimensional spaces are fundamental tools in a variety of applications, because most integrals can only be evaluated numerically and very few problems can be evaluated exactly. At the current stage, in contrast to the situation in one dimension, many fundamental problems in these two areas have not been resolved, despite increasing need for them in applications. The project aims at finding new methods to construct numerically efficient algorithms, such as accurate cubature rules with fewer nodes, fast discrete Fourier transforms, and approximation operators on regular domains such as cubes, balls, spheres and simplexes. The algorithms have applications in scientific computing, imaging, statistics, and geosciences.
Approximation is essential for understanding many natural phenomenon. Broadly speaking, it considers a simplified model that is close to the reality, or a finite process that is getting progressively close to an infinite process that describes reality. It is increasingly important for problems in high dimensions, which are characteristics of complex problems that depend on many parameters. The project aims at understanding numerical integration formulas for higher dimensional integrals and approximation of functions on regular domains such as cubes, balls, spheres and simplexes. These provide crucial tools for problems that appear in many applied fields, such as scientific computing, imaging, statistics and geo-sciences. The outcome of the project includes several important findings. For example, in numerical integration formulas, several new families of high precision formulas are found, including two families on the rectangular domains that have the smallest number of points among formulas with the same precision. In approximation side, a substantial progress was made in understanding the orthogonality in the Sobolev space and its connection to approximation on the regular domains, which has a close connection to the spectral theory in numerical solution of partial differential equations. The product of the project includes two published books, ``Orthogonal Polynomials of SeveralVariables" and "Approximation Theory and Harmonic Analysis on Spheres and Balls," and close to twenty published, or accepted, research papers.
