RIESZ FRAMES AND APPROXIMATION OF THE FRAME COEFFICIENTS

A frame is a fmaily {f_i}_(i=1)~∞ of elements in a Hilbert space with the property that every element in can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coef- ficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on The projection method is introduced in order to avoid this problem. The basic idea is to con- sider finite subfamilies {f_i}_(i=1)~n of the frame and the orthogonal projection P_n onto its span. For f∈P_n f has a representation as a linear combination of f_i,i=1,2,…,n and the correspond- ing coefficients, can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case we have avoided the inversion problem. In the same spirit we approximate the solution to a moment prob- lem. It turns out, that the class of 'well-behaving frames' are identical for the two problems we consider.

Photoelectron momentum distributions of triatomic CO_(2)molecules by circularly polarized attosecond pulses

Molecular-frame photoelectron momentum distributions(MF-PMDs)have been studied for imaging molecular structures.We investigate the MF-PMDs of CO_(2)molecules exposed to circularly polarized(CP)attosecond laser pulses bysolving the time-dependent Schrodinger equations based on the single-active-electron approximation frames.Results showthat high-frequency photons lead to photoelectron diffraction patterns,indicating molecular orbitals.These diffractionpatterns can be illustrated by the ultrafast photoionization models.However,for the driving pulses with 30 nm,a deviationbetween MF-PMDs and theoretically predicted results of the ultrafast photoionization models is produced because theCoulomb effect strongly influences the molecular photoionization.Meanwhile,the MF-PMDs rotate in the same directionas the helicity of driving laser pulses.Our results also demonstrate that the MF-PMDs in a CP laser pulse are the superpositionof those in the parallel and perpendicular linearly polarized cases.The simulations efficiently visualize molecularorbital geometries and structures by ultrafast photoelectron imaging.Furthermore,we determine the contribution of HOMOand HOMO-1 orbitals to ionization by varying the relative phase and the ratio of these two orbitals.