论文标题
正交多项式的渐近公式中的普遍关系
Universal relations in asymptotic formulas for orthogonal polynomials
论文作者
论文摘要
正交多项式$ p_ {n}（λ）$是$ n $的振荡函数，为$ n \ to \ to \ infty $，在相应的jacobi操作员$ j $的绝对连续频谱中，$λ$。我们表明，无论对操作员$ j $系数的任何特定假设如何，对于$ p_ {n}（λ）$的渐近公式中的幅度和相位因素，都与本文中的某些普遍关系联系在一起。 我们的方法依赖于对雅各比操作员对角线的研究。对角运算符是根据正交多项式$ p_ {n}（λ）$构建的。他们从space $ \ ell^2（{\ bbb z} _ {+}）$的空间$ l^2（\ bbb r）$的函数作用。我们在相当普遍的环境中考虑此类操作员，并找到其界限的必要条件。
Orthogonal polynomials $P_{n}(λ)$ are oscillating functions of $n$ as $n\to\infty$ for $λ$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on coefficients of the operator $J$, amplitude and phase factors in asymptotic formulas for $P_{n}(λ)$ are linked by certain universal relations found in the paper. Our approach relies on a study of operators diagonalizing Jacobi operators. Diagonalizing operators are constructed in terms of orthogonal polynomials $P_{n}(λ)$. They act from the space $L^2 (\Bbb R)$ of functions into the space $\ell^2 ({\Bbb Z}_{+})$ of sequences. We consider such operators in a rather general setting and find necessary and sufficient conditions of their boundedness.
