空间L_2(R^2;e^(-x^2-y^2))中的函数逼近

Approximation of Functions in the space L_2(R^2;e^(-x^2-y^2))

利用L2(R2;e-x2-y2)的一个平移算子Fh定义了差分Δhk(f)和广义连续模Ωk(f;δ),根据Hermite多项式的性质引入了一个二阶微分算子D,由此来定义函数类Wφ(r,k)(D)和KH(α).借助于参考文献中的一些结论及研究方法可以得到f∈Wt(r,kv)(D)的充分必要条件,同时得到关于f∈KH(α),α>2的Fourier-Hermite系数cij(f)的级数∑i=0 to ∞∑j=0 to ∞cij(f)一定绝对收敛的结论.

In this paper,using the shift operator Fh in L2(R2;e-x2-y2),the difference operator Δkh(f) and the generalized modulus of continuity Ωk(f;δ) are defined.According to the property of Hermite polynomial,we introduce the Second-order differential operator D.And then we can define classes of functions Wr,kφ(D) and KH(α).By means of some conclusions and research methods from bibliography,[3],we can obtain the necessary and sufficient condition of f∈Wr,ktv(D) At the same time,we can also obtain the conclusion that the series ∑i=0 to ∞ ∑i=0 to ∞cij(f) about the Fourier-Hermite coefficient cij(f) of f∈KH(α),α>2 is absolutely convergence.