Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree $n$ via hyperinterpolation. Hyperinterpolation of degree $n$ is a discrete approximation of the $L^2$-orthogonal projection of degree $n$ with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most $2n$. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.Comment: 22 pages, 7 figure

Theoretical Studies on Dehydrogenation Reactions in Mg-2(BH4)(2)(NH2)(2) Compounds

National Nature Science Foundation of China [21133004]Borohydrides have been recently hightlighted as prospective new materials due to their high gravimetric capacities for hydrogen storage. It is, therefore, important to understand the underlying dehydrogenation mechanisms for further development of these materials. We present a systematic theoretical investigation on the dehydrogenation mechanisms of the Mg-2(BH4)(2)(NH2)(2) compounds. We found that dehydrogenation takes place most likely via the intermolecular process, which is favorable both kinetically and thermodynamically in comparison with that of the intramolecular process. The dehydrogenation of Mg-2(BH4)(2)(NH2)(2) initially takes place via the direct combination of the hydridic H in BH4- and the protic H in NH2-, followed by the formation of Mg-H and subsequent ionic recombination of Mg-H delta-center dot center dot center dot H delta+-N