Regularization of Certain Divergent Series of Polynomials
نویسنده
چکیده
We investigate the generalized convergence and sums of series of the form P n≥0 anT P (x), where P ∈ R[x], an ∈ R, ∀n ≥ 0, and T : R[x] → R[x] is a linear operator that commutes with the differentiation d dx : R[x]→ R[x]. CONTENTS 1. The main result 1 2. Some applications 3 References 9 1. THE MAIN RESULT We consider series of the form ∑ n≥0 anT P (x), (†) where P ∈ R[x], and T : R[x]→ R[x] is a linear operator such that TD = DT , (∗) where D is the differentiation operator D = d dx . The condition (∗) is equivalent with the translation invariance of T , i.e., TU = UT , ∀h ∈ R, (I) where U : R[x]→ R[x] is the translation operator R[x] 3 p(x) 7→ p(x+ h) ∈ R[x]. For simplicity we set U := U. Clearly U ∈ O so a special case of the series (†) is the series ∑
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تاریخ انتشار 2009