In the present paper, we characterize the Eigenfunctions of a weighted composition operator on space of holomorphic function on the unit disk.

Functional data analysis is a relatively new and rapidly growing area of statistics. This is partly due to technological advancements which have made it possible to generate new types of data that are in the form of curves. Because the data are functions, they lie in function spaces, which are of infinite dimension. To analyse functional data, one way, which is widely used, is to employ principal component analysis, allowing finite dimensional analysis of the problem. The authors gave stochastic expansions of estimators eigenvalues and Eigenfunctions, providing not only a new understanding of the effects of truncating to a finite number of principal components, but also pointing to new methodology, such as simultaneous and individual bootstrap confidence statements for eigenvalues and Eigenfunctions. The expansions explicitly include terms of sizes n-1/2, n-1, and a remainder of order n-3/2, where n denotes sample size. The terms of size n-1/2 are related to limit theory. Because for many situations, the exact statistical properties of the eigenvalues and Eigenfunctions estimators are not directly obtainable, the way by which we can approximate their distributions is of interest in practice. In this paper, we discuss asymptotic results for eigenvalues and Eigenfunctions. The work shows that eigenvalue spacings have only a second-order effect on properties of eigenvalue estimators, but a first-order effect on properties of eigenfunction estimators.

We establish a local gradient estimate for positive Finsler p-eigenfunctions on a complete non-compact Finsler measure space M with its weighted Ricci curvature Ric∞ bounded from below by a non-positive constant. As an application, we obtain the corresponding Harnack inequality.

In this paper we study some properties of the Eigenfunctions corresponding to the eigenvalues of Sturm-Liouville Problem in two turning points case. There is connection between the eigenvalues of Sturm-Liouville Problem and infinite product representation. Before representing the solution in the form of an infinite product, we make a slight digression into an asymptotic solution of these problems.First, in this case we discuss the asymptotic form of the eigenvalues of Sturm-Liouville Problems. Then by making use of the eigenvalues and by substituting them in solution and infinite product representation of solution we get the asymptotic form of eigenfunction and the asymptotic form of the derivative of eigenfunction.

determination of eigenvalues and Eigenfunctions of a High-order Sturm-Liouville problem (HSLP) is considered. To this end, the Differential Transformation Method (DTM) is applied which is an efficient technique for solving differential equations. The results of the proposed approach are compared with those of some well-known methods reported in the literature. Four illustrative real life examples of mechanical engineering are provided to show the ability and the cost-effectiveness of this numerical approach.

We combine sinc method and self-consistent method to obtain eigenvalues andEigenfunctions of a class of nonlinear differential equations. Some propertiesof sinc method and self-consistent method are given. Numerical examples aregiven to demonstrate the validity of the present method. We compare theseresults with the existing results and show that the sinc-self-consistent methodis suitable method to obtain eigenvalue and eigenfunction of equations on aninfinite domain and produces the smallest eigenvalue with the most accuracy.

In this paper we consider the regular Sturm-Liouville problem -rg”+(q-rl)g=0On the interval [0,p] with initial conditionsg’1(0)= p  وg’1(0)=0, و g’2(0) وg’(0)= p,where  and are positive functions, we will determine the asymptotics form of the Eigenfunctions of Sturm-Liouville problem in a particular case.

In this paper, we prove the completeness in the elliptic part of the domain of the system of Eigenfunctions of the Gellerstedt problem for a degenerating equation of mixed type. Simultaneously, we consider the completeness of the system of Eigenfunctions of the Tricomi and Neuman – Tricomi problems, which arise in studing the Gellerstedt problem.

In this paper, we give the spectral theory for eigenvalues and Eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding Eigenfunctions are orthogonal.