Structure of the Hardy Operator Related to Laguerre Polynomials and the Euler Differential Equation
نویسندگان
چکیده
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 = L2(0,∞) can be written as H = I − U , where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y′− 1 x y = g and point out some generalizations to the case with weighted Lw(a, b) spaces.
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