三次弦边值问题与Cauchy双正交多项式
Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials
摘要：
柯西双正交多项式出现在被称为Degasperis-Procesi（DP）方程的色散非线性偏微分方程的特解的研究中，以及某些双矩阵随机矩阵模型中。此类双正交多项式发挥作用的另一个背景是三次弦;三阶ODE边值问题-f '=zg f，这是MG研究的非齐次弦问题的推广克莱恩一类一般的这样的边值问题超出了原来的立方弦问题与DP方程的假设下，讨论的不均匀性g的源是一个离散的措施。结果表明，通过适当地选择与这些边值问题相关联的广义傅里叶变换，可以建立一个Parseval型恒等式，该恒等式将Cauchy双正交多项式与L^2_g上的某些自然正交系对齐。
英文摘要：
Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem -f'''=zg f which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L^2_g.
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