Power series of several variables with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

This paper is devoted to the numerical scheme for a class of fractional order integrodifferential equations by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. Reproducing kernel function in the form of Jacobi polynomials is established for the first time. It is implemented as a reproducing kernel method. The numerical solutions obtained by taking the different values of parameter are compared; Schmidt orthogonalization process is avoided. It is proved that this method is feasible and accurate through some numerical examples.

The Combined Reproducing Kernel Method and Taylor Series for Solving Weakly Singular Fredholm Integral Equations

<p>In this paper, a numerical method is proposed for solving weakly singular Fredholm integral equations in Hilbert reproducing kernel space (RKHS). The Taylor series is used to remove singularity and reproducing kernel function are used as a basis. The effectiveness and stability of the numerical scheme is illustrated through two numerical examples.</p>

APPROXIMATE SOLUTION OF NONLINEAR MULTI-POINT BOUNDARY VALUE PROBLEM ON THE HALF-LINE

In this work, we construct a novel weighted reproducing kernel space and give the expression of reproducing kernel function skillfully. Based on the orthogonal basis established in the reproducing kernel space, an efficient algorithm is provided to solve the nonlinear multi-point boundary value problem on the half-line. Uniformly convergence of the approximate solution and convergence estimation of our algorithm are studied. Numerical results show our method has high accuracy and efficiency.