M. Derevyagin, L. Vinet and A. Zhedanov introduced in Derevyagin et al. (2012) a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real parameter $\lambda$. In Derevyagin et al. (2012) the authors prove that this map yields a natural link between the Jacobi polynomials on the unit circle and the little and big $-1$ Jacobi polynomials on the real line. They also provide explicit expressions for the measure and orthogonal polynomials associated with the Jacobi matrix in terms of those related to the CMV matrix, but only for the value $\lambda =1$ which simplifies the connection –basic DVZ connection–. However, similar explicit expressions for an arbitrary value of $\lambda$ –(general) DVZ connection– are missing in Derevyagin et al. (2012). This is the main problem overcome in this paper.

This work introduces a new approach to the DVZ connection which formulates it as a two-dimensional eigenproblem by using known properties of CMV matrices. This allows us to go further than Derevyagin et al. (2012), providing explicit relations between the measures and orthogonal polynomials for the general DVZ connection. It turns out that this connection maps a measure on the unit circle into a rational perturbation of an even measure supported on two symmetric intervals of the real line, which reduce to a single interval for the basic DVZ connection, while the perturbation becomes a degree one polynomial. Some instances of the DVZ connection are shown to give new one-parameter families of orthogonal polynomials on the real line.

Derevyagin等人介绍了M. Derevyagin，L。Vinet和A. Zhedanov。（2012年）单位圆和实线上的正交多项式之间的新连接。它根据实际参数将任何实际CMV矩阵映射到Jacobi矩阵$\lambda$。在Derevyagin等人中。（2012年）作者证明，该图在单位圆上的Jacobi多项式与小点和大点之间产生了自然联系$-\mathrm{1个}$实线上的Jacobi多项式。它们还提供了与Jacobi矩阵相关的度量和正交多项式的显式表达式，仅涉及与CMV矩阵相关的度量和正交多项式，但仅针对值$\lambda =\mathrm{1个}$这简化了连接–基本的DVZ连接–。但是，对于的任意值，类似的显式表达式$\lambda$– （通用）DVZ连接– Derevyagin等人缺少。（2012）。这是本文要克服的主要问题。

这项工作为DVZ连接引入了一种新方法，该方法通过使用CMV矩阵的已知属性将其表示为二维本征问题。这使我们比Derevyagin等人走得更远。（2012年），为一般DVZ连接提供了量度和正交多项式之间的明确关系。事实证明，此连接将单位圆上的一个测度映射为在实线的两个对称间隔上支持的偶数测度的合理扰动，对于基本DVZ连接，该偶数测度减小为单个间隔，而扰动变为一阶多项式。DVZ连接的某些实例显示为在实线上给出正交多项式的新一参数系列。