Bounds for characteristic functions of polynomials in asymptotically normal random variables

Abstract

In the three-dimensional holography of phase objects, scatter screens have been used to provide m u s e illumination. In the reconstruction process a diaphragm is used to select a particular angle or position of view. The paper shows that there are upper and lower limits to the size of the diaphragm, determined by the optical path variations within the object. These limits are not necessarily compatible with'one another. Thus the scatter screen technique is shown to have limitations. An alternative is simply to use an array of parallel beams separated by discrete angular intervals. The advantage of continuous variation of viewing angle is lost but the resistrictions on the type of object are less severe. In addition the reconstructions may take the form of schlieren or shadow photographs as well as interferograms.

On Properties of Polynomials in Random Elements

“…VIII, section 4, Statement 4.2]). Our improvements of estimate (1) obtained in [6] are similar in form to the improvements which gave the Vinogradov method for trigonometric sums comparing the Weyl results (see the introduction in [21]). 2,Lemma 4]) and its generalizations to the multidimensional case (see [22, Chap.…”

“…As noted in the introduction, (1) could be improved so that D becomes larger (see [6], [7]). The improvement is obtained in the following cases:…”

“…The expressions for c5, c6 (see [6,Theorem 3]). For components with differentiable densities, Theorem 2 could be refined (see [6,Theorem 4]).…”

“…The expressions for c7, c8 (see [6,Theorem 5]). Notice that until now we gave the upper bounds for characteristic functionals.…”

“…In fact, one cannot only get a better upper bound but also prove a lower bound of the same order in t (see [6,Theorem 6]). …”