Alexandru Kristály — Page 4

arXiv2013

A constructive approach to triangular trigonometric patches

We construct a constrained trivariate extension of the univariate normalized B-basis of the vector space of trigonometric polynomials of arbitrary (finite) order n defined on any compact interval [0,α], where αis a fixed (shape) parameter in (0,π). Our triangular extension is a normalized linearly independent constrained trivariate trigonometric function system of dimension 3n(n+1)+1 that spans the same vector space of functions as the constrained trivariate extension of the canonical basis of truncated Fourier series of order n over [0,α]. Although the explicit general basis transformation is yet unknown, the coincidence of these vector spaces is proved by means of an appropriate equivalence relation. As a possible application of our triangular extension, we introduce the notion of (rational) triangular trigonometric patches of order n and of singularity free parametrization that could be used as control point based modeling tools in CAGD.

math.NA↗

arXiv2013

Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.

math.DG↗