Abstract
A new analytical method for the computation of reproducing kernel is proposed and tested on some examples. The expression of reproducing kernel on infinite interval is obtained concisely in polynomial form for the first time. Furthermore, as a particular effective application of this method, we give an explicit representation formula for calculation of reproducing kernel in reproducing kernel space with boundary value conditions.
Highlights
It is well known that reproducing kernel theory has been used in many research fields such as complex analysis, dilation of linear operators, stochastic processes [1,2,3,4,5], and solution of various differential and integral equations [6,7,8,9,10]
Compared with the procedure for computation of the reproducing kernel in [22], we can see that our method is easier to implement, and it avoids the complexity of δ function
In this paper, a new method for the calculation of reproducing kernel on infinite interval was introduced, and the representation in polynomial form was obtained for the first time
Summary
It is well known that reproducing kernel theory has been used in many research fields such as complex analysis, dilation of linear operators, stochastic processes [1,2,3,4,5], and solution of various differential and integral equations [6,7,8,9,10]. One is using Green’s function for a differential operator to construct a reproducing kernel [11, 12] Another very standard method involves boundary value conditions depending on the property of δ function [13, 14]. In order to apply the new approach to solving differential equations with multiform boundary value problems, the explicit formula for calculation of reproducing kernel in the appropriate reproducing kernel space is provided successfully by using the orthogonal decomposition property.
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