In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.
The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classi...The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.展开更多
Stimulating renewable energy consumption is a major focus of the Sustainable Development Goals in combating climate change and global warming.The International Energy Agency estimates that renewable energy consumption...Stimulating renewable energy consumption is a major focus of the Sustainable Development Goals in combating climate change and global warming.The International Energy Agency estimates that renewable energy consumption should be doubled to achieve the COP21 targets.In this context,the question is whether renewable energy types promote the improvement of ecological quality and economic growth.Most studies have investigated the influence of renewable energy on ecological pollution using carbon dioxide emissions or ecological footprint indicators,which only represent the pollution caused by human consumption patterns,and these indicators neglect the supply side.Motivated by this point,this study uses the LCF(Load Capacity Factor)as an environmental indicator and examines the causality relationship among different types of renewable energy,income,and environmental quality in the USA,while also incorporating employment and capital stock into the analysis.Through using the Fourier causality test with the wavelet-decomposed series,the study explores for the validity of the renewable energy-based growth hypothesis and answers to the question of whether there is a causal effect of renewable energy types on environmental quality.The results demonstrate that there is a bidirectional causality between total renewable energy,wood,biomass,and economic growth as well as between these renewable energy types and the LCF.展开更多
A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical schem...A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.展开更多
文摘In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.
文摘The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.
文摘Stimulating renewable energy consumption is a major focus of the Sustainable Development Goals in combating climate change and global warming.The International Energy Agency estimates that renewable energy consumption should be doubled to achieve the COP21 targets.In this context,the question is whether renewable energy types promote the improvement of ecological quality and economic growth.Most studies have investigated the influence of renewable energy on ecological pollution using carbon dioxide emissions or ecological footprint indicators,which only represent the pollution caused by human consumption patterns,and these indicators neglect the supply side.Motivated by this point,this study uses the LCF(Load Capacity Factor)as an environmental indicator and examines the causality relationship among different types of renewable energy,income,and environmental quality in the USA,while also incorporating employment and capital stock into the analysis.Through using the Fourier causality test with the wavelet-decomposed series,the study explores for the validity of the renewable energy-based growth hypothesis and answers to the question of whether there is a causal effect of renewable energy types on environmental quality.The results demonstrate that there is a bidirectional causality between total renewable energy,wood,biomass,and economic growth as well as between these renewable energy types and the LCF.
文摘A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.
