Abstract
In this paper we compare the approximation properties of degree p spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, $$\mathcal {C}^{p-1}$$ splines provide better a priori error bounds for the approximation of functions in $$H^{p+1}(0,1)$$ . Our result holds for all practically interesting cases when comparing $$\mathcal {C}^{p-1}$$ splines with $$\mathcal {C}^{-1}$$ (discontinuous) splines. When comparing $$\mathcal {C}^{p-1}$$ splines with $$\mathcal {C}^{0}$$ splines our proof covers almost all cases for $$p\ge 3$$ , but we can not conclude anything for $$p=2$$ . The results are generalized to the approximation of functions in $$H^{q+1}(0,1)$$ for $$q<p$$ , to broken Sobolev spaces and to tensor product spaces.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
