n this paper, the relative widths of some sets in are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers gre...n this paper, the relative widths of some sets in are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid)展开更多
It is found that in free space, the curves of the mean-squared beam width may each have a cross point at a certain propagation distance Zc. For Gaussian array beams, the analytical expressions of zc are derived. For t...It is found that in free space, the curves of the mean-squared beam width may each have a cross point at a certain propagation distance Zc. For Gaussian array beams, the analytical expressions of zc are derived. For the coherent com- bination, Zc is larger than that for the incoherent combination. However, in non-Kolmogorov turbulence, the cross point disappears, and the Gaussian array beams will have the same directionality in terms of the angular spread. Furthermore, a short propagation distance is needed to reach the same directionality when the generalized exponent is equal to 3.108. In particular, it is shown that the condition obtained in previous studies is not necessary for laser beams to have the same directionality in turbulence, which is explained physically. On the other hand, the relative average intensity distributions at the position where the Gaussian array beams have the same mean-squared beam width are also examined.展开更多
For the approximation in L_(p)-norm,we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots.For p=1,∞,we ...For the approximation in L_(p)-norm,we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots.For p=1,∞,we obtain its values.By these results we know that for the Sobolev classes,the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.At the same time,the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.展开更多
文摘n this paper, the relative widths of some sets in are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid)
基金supported by the National Natural Science Foundation of China(Grant No.61178070)the Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan Province,China(Grant No.12TD008)
文摘It is found that in free space, the curves of the mean-squared beam width may each have a cross point at a certain propagation distance Zc. For Gaussian array beams, the analytical expressions of zc are derived. For the coherent com- bination, Zc is larger than that for the incoherent combination. However, in non-Kolmogorov turbulence, the cross point disappears, and the Gaussian array beams will have the same directionality in terms of the angular spread. Furthermore, a short propagation distance is needed to reach the same directionality when the generalized exponent is equal to 3.108. In particular, it is shown that the condition obtained in previous studies is not necessary for laser beams to have the same directionality in turbulence, which is explained physically. On the other hand, the relative average intensity distributions at the position where the Gaussian array beams have the same mean-squared beam width are also examined.
基金supported by the National Natural Science Foundations of China(Grant No.11271263).
文摘For the approximation in L_(p)-norm,we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots.For p=1,∞,we obtain its values.By these results we know that for the Sobolev classes,the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.At the same time,the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.
