Let（S）L<sup>2</sup>（S’（IR）,μ）（S）<sup>*</sup> be the Gel’fand triple over the white noise space （S’（IR）,μ）.Let（e<sub>n</sub>,n≥0）be the ONB of L<sup>2</s...Let（S）L<sup>2</sup>（S’（IR）,μ）（S）<sup>*</sup> be the Gel’fand triple over the white noise space （S’（IR）,μ）.Let（e<sub>n</sub>,n≥0）be the ONB of L<sup>2</sup>（IR）consisting of the eigenfunctions of the s.a. operator-（d/（dt））<sup>2</sup>+1+t<sup>2</sup>.In this paper the Euler operator △<sub>E</sub> is defined as the sum ∑<sub>i</sub>【,e<sub>i</sub>)<sub>i</sub>, where <sub>i</sub> stands for the differential operator D<sub>ei</sub>.It is shown that △<sub>E</sub> is the infinitesimal gen- erator of the semigroup（T<sub>t</sub>）,where(T<sub>t</sub>)（x）=（e<sup>t</sup>x）for ∈（S）.Similarly to the finite dimensional case,the λ-order homogeneous test functionals are characterized by the Euler equa- tion:△<sub>E</sub>=λ.Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out.展开更多
基金Supported by the National Natural Science Foundation of China
文摘Let（S）L<sup>2</sup>（S’（IR）,μ）（S）<sup>*</sup> be the Gel’fand triple over the white noise space （S’（IR）,μ）.Let（e<sub>n</sub>,n≥0）be the ONB of L<sup>2</sup>（IR）consisting of the eigenfunctions of the s.a. operator-（d/（dt））<sup>2</sup>+1+t<sup>2</sup>.In this paper the Euler operator △<sub>E</sub> is defined as the sum ∑<sub>i</sub>【,e<sub>i</sub>)<sub>i</sub>, where <sub>i</sub> stands for the differential operator D<sub>ei</sub>.It is shown that △<sub>E</sub> is the infinitesimal gen- erator of the semigroup（T<sub>t</sub>）,where(T<sub>t</sub>)（x）=（e<sup>t</sup>x）for ∈（S）.Similarly to the finite dimensional case,the λ-order homogeneous test functionals are characterized by the Euler equa- tion:△<sub>E</sub>=λ.Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out.
