FPM 2006, vol. 12, no. 6, pp. 231-239

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 6, PAGES 231-239

Birkhoff regularity in terms of the growth of the norm for the Green function

E. A. Shiryaev

Abstract

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We consider the ordinary differential operator $L$ generated on $[0,1]$ by the differential expression

l(y) = (

-i)ⁿy⁽ⁿ⁾(x) + p₂(x)y^(n-2) + ... + p_n-1(x)y' + p_n(x)y

and $n$ linearly independent homogeneous boundary conditions at the endpoints. We assume that the coefficients $p$ _k(x) are Lebesgue integrable complex functions. If the boundary conditions are Birkhoff regular, then the Green function $G($ l), being the kernel of the operator $(L$ - l)^-1, admits the asymptotic estimate (for sufficiently large $|$ l| > c₀)

|G(

l)| £ M|l|^(-n+1)/(n),

where $M = M(c$ ₀) is a certain constant. In the present paper, we prove the converse assertion: the fulfillment of this estimate on some rays implies the regularity of the operator $L$ .

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