论文标题
与扰动的$ R_I $类型复发关系相关的光谱转换
Spectral transformation associated with a perturbed $R_I$ type recurrence relation
论文作者
论文摘要
在这项工作中，正交多项式满足$ r_i $ type recurrence关系％$ \ MATHCAL {p} _ {n+1}（n+1}（z）=（z-c_n）\ Mathcal {p} _n（z-a_n（z-a_n）（z-a_n）\ Mathcal {p} _n（z-a_n） $ \ MATHCAL {p} _ { - 1}（z）= 0 $和$ \ Mathcal {p} _0（z）= 1 $，当修改复发系数时。研究了扰动和不受干扰的多项式以及光谱特性以及持续分数的光谱转化之间的结构关系。已经证明，转移矩阵方法比获得扰动$ r_i $多项式的经典方法在计算上的计算效率更高。此外，提出了对Carathéodary功能共同污染的有趣结果。最后，借助插图进行了与单位圆相关的共恢复和共同污染的研究。证明了L-雅各比多项式及其扰动形式之间零的隔离和单调性。
In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-λ_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are modified. The structural relationship between the perturbed and the unperturbed polynomials along with the spectral properties and spectral transformation of continued fraction are investigated. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of co-dilation on the Carathéodary function is presented. Finally, the study of co-recursion and co-dilation in connection to the unit circle is carried out with the help of an illustration. The interlacing and monotonicity of zeros between L-Jacobi polynomials and their perturbed forms are demonstrated.
