Structure of the Hardy Operator Related to Laguerre Polynomials and the Euler Differential Equation
Structure of the Hardy Operator Related to Laguerre Polynomials and the Euler Differential Equation
复制标题
与拉盖尔多项式和欧拉微分方程相关的Hardy算子的结构
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
L. Persson
中科院分区：
文献类型：
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作者：
N. Kruglyak;L. Maligranda;L. Persson
本文直接证明了空间L 2=L 2(0，1)中的Hardy算子Hf(X)=1xRx0f(T)dt可记为H=IU，其中U是某一正交基{En}的移位算子(Uen=En+1，n2Z).利用经典拉盖尔多项式构造了基{En}。我们还解释了与一阶欧拉微分方程y 0 1 y=g的关系，并指出了加权L 2(a，b)情形的一些推广
We present a direct proof of a known result that the Hardy operator Hf(x) = 1 x R x 0 f(t)dt in the space L 2 = L 2 (0,1) can be written as H = I U, where U is a shift operator (Uen = en+1, n 2 Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler dierential equation of the first order y 0 1 y = g and point out some generalizations to the case with weighted L 2(a,b)