# Dual Discrete Canonical Systems and Dual Orthogonal Polynomials

• L. Sakhnovich
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

## Abstract

The string equation
$$\frac{{d^2 \phi \left( {x,\lambda } \right)}}{{dx^2 }} = \lambda \rho ^2 \left( x \right)\phi \left( {x,\lambda } \right),\rho \left( x \right) > 0,0 \leqslant x \leqslant l$$
can be written in the form
$$\frac{{d^2 \phi \left( {x,\lambda } \right)}}{{dx^2 }} = \lambda \frac{{dM}}{{dx}}\phi \left( {x,\lambda } \right),$$
(0.1)
where
$$M\left( x \right) = \int\limits_0^x {\rho ^2 \left( t \right)dt.}$$
The equation
$$\frac{{d^2 \tilde \phi \left( {M,\lambda } \right)}}{{dM^2 }} = \lambda \frac{{dx}}{{dM}}\tilde \phi \left( {M,\lambda } \right)$$
(0.2)
is said to be dual to equation (0.1). The notion of a dual string was investigated by I.S. Kac and M.G. Krein . Kac and Krein obtained the dual string equation from the original by interchanging the variables x and M(x). Let us add conditions
$$\phi \left( {0,\lambda } \right) = 1,\phi '\left( {0,\lambda } \right) = 0,$$
(0.3)
$$\tilde \phi \left( {0,\lambda } \right) = 0,\tilde \phi '\left( {0,\lambda } \right) = 1$$
(0.4)
to equations (0.1) and (0.2).

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