We study the eigenvalue problem of perturbed Airy operator −4 + (x + V (x)) on positive real axis, with Dirichlet boundary condition. Given the asymptotic expansion of V (x) at +∞, we provide asymptotic expansion of the eigenvalues λn to certain order. Then we will provide an example to show that in general, merely from the asymptotic expansion of V (x), the asymptotic expansion of λn cannot be...

We study a quantum analogue of the iterative perturbation theory by Kolmogorov used in the proof of the Kolmogorov-Arnold-Moser (KAM) theorem. The method is based on sequent canonical transformations with a "running" coupling constant ; ; , etc. The proposed scheme, as its classical predecessor, is "superconvergent" in the sense that after the nth step, a theory is solved to the accuracy of ord...

We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of t...

This paper is concerned with the asymptotic behaviour of estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The results of the paper can be used to determine the form of the recursive procedure which is expected to have the same asymptotic properties as the corresponding nonrecursive one defined as a so...

Computing the optimal expansion of a signal in a redundant dictionary of waveforms is an NP-complete problem. We introduce a greedy algorithm called a matching pursuit which computes a sub-optimal expansion. The dictionary waveforms which best match a signal's structures are chosen iteratively. An orthogonalized version of the matching pursuit is also developed. Matching pursuits are general pr...