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)^{n}y^{(n)}(x) +
p_{2}(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| >
c_{0})

$|G($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 $L$.

Location: http://mech.math.msu.su/~fpm/eng/k06/k066/k06614h.htm

Last modified: February 26, 2007