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- W3100018246 abstract "Consider an arbitrary complex-valued, twice continuously differentiable, nonvanishing function $phi$ defined on a finite segment $[a,b]subset mathbb{R}$. Let us introduce an infinite system of functions constructed in the following way. Each subsequent function is a primitive of the preceding one multiplied or divided by $phi$ alternately. The obtained system of functions is a generalization of the system of powers ${(x-x_{0}%)^{k}}_{k=0}^{infty}$. We study its completeness as well as the completeness of its subsets in different functional spaces. This system of recursive integrals results to be closely related to so-called $L$-bases arising in the theory of transmutation operators for linear ordinary differential equations. Besides the results on the completeness of the system of recursive integrals we show a deep analogy between the expansions in terms of the recursive integrals and Taylor expansions. We prove a generalization of the Taylor theorem with the Lagrange form of the remainder term and find an explicit formula for transforming a generalized Taylor expansion of a function in terms of the recursive integrals into a usual Taylor expansion. As a direct corollary of the formula we obtain the following new result concerning solutions of the Sturm-Liouville equation. Given a regular nonvanishing complex valued solution $y_{0}$ of the equation $y^{primeprime}+q(x)y=0$, $xin(a,b)$, assume that it is $n$ times differentiable at a point $x_{0}% inlbrack a,b]$. We present explicit formulas for calculating the first $n$ derivatives at $x_{0}$ for any solution of the equation $u^{primeprime}+q(x)u=lambda u$. That is, an explicit map transforming the Taylor expansion of $y_{0}$ into the Taylor expansion of $u$ is constructed." @default.
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- W3100018246 date "2012-02-01" @default.
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- W3100018246 title "Complete systems of recursive integrals and Taylor series for solutions of Sturm-Liouville equations" @default.
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- W3100018246 doi "https://doi.org/10.1002/mma.1596" @default.
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