L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in ... L. A. Aizenberg, "Integral representations of holomorphic functions of several ...
If the system (5) has no roots for w = 0, then a,lao is equal to the con iV r s -- 1 efficient of the monomial z~... zn in the polynomial ~ (-- I)~+~-I (zl + ... + z~) v,I,det A, PROPOSITION.
$=I
and 2a2/a o - (az/ao)2= ~ dk
.
N
where d k i s the coefficient of the monomial ZXl... z~N+* .. zn
in
k=1
the polynomial
~
(-- l)n+s-l%;s%~l(zl-~-... nU Zn)
ue~AJsahl.
8,/=1
LITERATURE CITED
2. 3. 4. 5. 6. .
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1978). A. K. Tsikh and A. P. Yuzhakov, "Properties of a total sum of residues with respect to a polynomial mapping," Sib. Mat. Zh., 25, No. 4, 208-213 (1984). L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis [in Russian], Nauka, Novosibirsk (1979). L. A. Aizenberg, "Integral representations of holomorphic functions of several complex variables," Dokl. Akad. Nauk SSSR, 155, No. i, 9-12 (1964). M. Z. Lazman, G. S. Yablonskii, and V. I. Bykov, "A stationary kinetic equation. Nonlinear unidirectional mechanism," Khim. Fiz., No. 2, 239-248 (1983). A. M. Kytmanov, "On a system of algebraic equations arising in chemical kinetics," in: Multidimensional Complex Analysis [in Russian], Inst. Fiz., Sib. Otd., Akad. Nauk SSSR, Krasnoyarsk (1985), pp. 97-108. V. I. Bykov, A. M. Kytmanov, M. Z. Lazman, and G. S. Yablonskii, "A kinetic polynomial for a unidirectional n-stage catalytic reaction," in: Chemical Kinetics in Catalysis [in Russian], Inst. Khim. Fiz., Akad. Nauk SSSR, Chernogolovka (1985), pp. 69-74.
INTEGRAL REPRESENTATION OF COMPLETELY CONTINUOUS OPERATORS IN L 2 UDC 517.983
I. M. Novitskii
A linear operator T: L2(O, ~) + L2(O, ~) is said to be integral if there exists an almost-everywhere finite measurable function K(s, t), defined on (0, ~)2, such that
T/(s) = .! K (s, t) / (t) dt 0
almost everywhere for all f e L2(O , ~). integral operator T.
The function K(s, t) is called the kernel of the
An integral operator T: L2(O, =) + L2(O, ~) is called a Carleman operator if its kernel satisfies the Carleman condition [i] oo
~lK(s, t)[2dt 2, f o r
increasing
i = 1, 2, 3,
sequence in ....
i, l= i,
2, 3,
[ 0 , ~) s u c h t h a t
..~ ~ (a~+x
a0-'/20
= ~ lim l gv (s + e) - - gp (~ Ill rgp ~
O,
=
s ~ (0, co).
~=18~0
The theorem is proved. LITERATURE CITED i~ 2. 3o 4o 5.
T. Carleman, Sur les ~quations Integrales SinguliSres a Noyau Reel et Symetrique, A.-B. Lundequistska Bokhandeln, Uppsala (1923). V . B . Korotkov, Integral Operators [in Russian], Nauka, Novosibirsk (1983). i . M . Novitskii, "On the representation of the kernels of integral operators by bilinear series," Sib. Mat. Zh., 23, No. 5, 114-118 (1984). A. Pietsch, Operator Ideals, North-Holland, Amsterdam (1980). G . M . Fikhtengol'ts, A Course of Differential and Integral Calculus [in Russian], Vol. 2, 7th ed., Nauka, Moscow (1969).
ON AN EXAMPLE OF DIVERGENCE OF RIEMANN SUMS UDC 517.5
B. V. Pannikov
In the present article, we assume all functions to be measurable and 1-periodic. following theorem has been proved in [i].
The
THEOREM A. If an increasing sequence of natural numbers n m satisfies the condition nmlnm+ l and f(x) ~ L(0, i), then for almost all x "nm
lim nTn1 rn-~co
1
] x qh=l
nmj
=
/ (x) dx.
(1)
o
It is interesting to compare the following theorem, proved in the present article, with Theorem A. THEOREM i. For each increasing sequence of natural numbers n m (m = I,...) and each e > 0 there exists a 1-periodic open set E such that mes [E N (0, i)] 5 s and n m
lira n ~ l Z
/ {yq-k~ =
i
Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 3, pp. 208210, May-June, 1988. Original article submitted April 7, 1986.
0037-4466/88/2903-0503
$12.50
9 1989 Plenum Publishing Corporation
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