L²[0, 1]中拉盖尔多项式及欧拉微分方程相关的Hardy算子的结构

The Structure of the Hardy Operator Related to Laguerre Polynomials and Euler Differential Equations on L²[0, 1]

摘要: 本文探讨了拉盖尔多项式与Hardy算子之间的关系, 以及求解欧拉微分方程的方法. 首先介绍了拉盖尔多项式的基本性质, 并证明Hardy算子的伴随算子可表示为恒等算子与移位算子之差. 随后, 研究了一阶欧拉微分方程在L²[0, 1]上的解, 表明该解可视为L²(0,∞)上解的截断形式, 并在加权Lebesgue空间中推导了该方程解的表达式. 最后, 本文将Hardy算子推广至更一般的加权空间, 并给出了相应的表示形式, 为后续研究提供了方向. 这些结果不仅加深了对Hardy 算子的理解, 也为相关微分方程的求解提供了新方法。

Abstract: This paper explores the relationship between Laguerre polynomials and the Hardy operator, as well as methods for solving Euler differential equations. First, the fundamental properties of Laguerre polynomials are introduced, and it is shown that the adjoint of the Hardy operator can be expressed as the difference between the identity operator and a shift operator. Then, the solution of a first-order Euler differential equation on L²[0, 1] is studied, demonstrating that it can be viewed as a truncation of the solution on L²(0,∞). An explicit expression for the solution is derived within weighted Lebesgue spaces. Finally, the Hardy operator is extended to more general weighted spaces, and corresponding representations are provided, offering potential directions for future research. These results not only deepen the understanding of the Hardy operator but also provide new methods for solving related differential equations.