# SCIENTIFIC ABSTRACT ZUKHOVITDSKIY, S.I. - ZUKOWSKI, R.

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CIA-RDP86-00513R002065610020-9

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RIF

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S

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100

Document Creation Date:

November 2, 2016

Document Release Date:

September 1, 2001

Sequence Number:

20

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Publication Date:

December 31, 1967

Content Type:

SCIENTIFIC ABSTRACT

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Some Remarks on the Beat Approximation~of Differential 807/20-1274-3/51
Equations by Polynomials
f-Ref 1,23 in-applioable,The uniform dionvergence of the
approximations for an increasing degree;'of the approximoting'
polynomial is discussed by an example of the Dirichlet problem
for the Laplace equation and hn other c4se. The authorb:give
proposals for the choice of the functions %fk.,They mention:
I.N.Vekua.
There are 3 Soviet references.,
PRESENTEDt May 8, 1959, by N.N.Bogolyubov, Academician
SUBMITTED: May 3, 1959
Card 2/2
AUTHORs an& Eakin$ G#I# ~20-119-6-4/56
TITLEs The Problem of Chebyshe-r Approximation 'Withinia Coymutative
Hilbert Ring (Zadacha chebyehovskogo priblizhoniya v kointantativ-
nom gillbertovom kolltse)
PERIODICALt Doklady Akademii nauk SSSRj'1qr8,Vol 119,0r 6ppp 1074-107,6(USSR)
ABSTRACTs On the compact Q let tfl(q) Tn(q) be continuous fill LCt,ons
with values in the infinite-dimensional.comm~ta:tive Hilbertl~
ring He It in (q)9-0(j where ?a is continuous on
%ok(q) T k k
Q and is the base of the orthogonal irredrcible idempotent.
Let the function f(j) continuous on Q with values in H be
n
approximated by the polynomials Y-a k 4tk C- ff. Let, T do-
keel
note the set of all a - (alto.dra d$ ak e H, for which~
n
Zak Tk(q) 9 G on Q. Let S be the orthogonal complement of T
ke, I n 2 1/2
Card 113 in the Hilbert space 9 with the norm II allHn aj
,The Problem of Chebyshev Approximation Within a Commitative 20-119-6-4/56
Hilbert Ring I:
Theorems In order that there exists a polynomial of bests
approximation for every f(q) it is neces6ary and sufficient
that 3 in finite-dimensional.
Theorems let dim 3 - t1f where t is an integer and I in the.
number of in&iaas 6L so that (q) f o~ oyj q for at least one
k_1,2,.*qna In order that there exists &'bingle polynomial of
beat approximation n (0) io) 0,14( 0))tE 5 for every
Y ak T k(q) f (s'
k-9
f(q) it is necessary and sufficient that every polynomial
n n
)r; S does not va'Ash'in
ak Tk(q) L 11 ak 11 > t n
kal k-1
more than t-1 points -of Q.,
Theorems Letfk(q) satisfy the condition that ovary !In k %ok(q)
n
(Z hakil >0) does not vanish in more th&'n n_1 points or qt :'Where
k-1
Q has more than n Pointes.Let the function f(q) possess a poly-
nomial of best approximation* In order'that
Card 2/3
The Problem of Chebyshev Approximation Within a Commutative 20-119-6-4/56
Hilbert Ring
n
a(0 (q) is this polynomial it is necessary that the
k 4 k
k-1
is 6ttained in at ledet n+1
deviation max (q) -f (q)
qeq k-i k k
points of q*
Theorem% In order that every function t,(q) possasaing,a poly-
nomial of best approximation admits only exactly one such
polynomial it is necessary and nuffidiant that every polynomial
n n
Y- ak Tk(q) ( Y- 11 ak 11 > 0) does not vanish in more than n-i
kol k-1
points of q&
There are 6 Soviet references.
ASSOCIATIONsLutskiy gosudarstvennyy pedagogicheskiyJnst1tut, imeni, Lesi
Ukrainki (Lutek State Pedagogical Inatitute imeni Losyaz, Uk-tainka)
PRESENTM December 4t 19570 by N.N&Bogolyubov, Leademician
SUB11ITTEDt December 21 1957
Card 3/3
ESKIN. G.I. (Eakin, H.1.3-, ZUKHOVITSKIY, S.t. (Zukhouytolkyl, 3.1.1
Some theorems on Tchabycheff's'approximatIon;of fuhatio s vit6
values belonging to a commutative complataly~~regul;D'r ring-.Cvit4
silm ry in Ingliah]. Dop. AM MR no. 4068-371 '58., (IIIRA.1;118)
1. lutelkly pedinstitut im. Lost Ukrainukt. fradstoviv akedemik
H.M.Boholinboy [N.1f.Bogolyubov) . I
Ofunctional analysis)
ZUKHOTITSKIT
Algor bm for constructing the Chabyehav aporiaximalion of a
continuous function by polynomIals. Dok1 .AS SSSR 120 no. 4:69)-
696 Ja 1~8. (MIRA 11:8)i
1. lutskiy pedagogichaskiy institut im. Lost 6waiiki. Predstivlemol
akedemikom N.N.Bogoljvubovym.
(Algorism)
(Functions. Continuous)
(Polygons)
AUTHORS: Eakin, G.I., Zukhovitski , S.T. SOV-21-58-4-3/29
TITLE- Some Theoremeon the Chebyshev Appi6ximation of Functions
with Values Belongingto a Commutative E+-Algebra (Neko:toryye
teoremy o Chebyshevskom priblizheriii funktsiy ao~zmaehe-
niyami v kommutativnom vpolne regui~arnom kol,tse);
PERIODICAL: Dopovidi Akademii nauk Ukrains1koi RSR, 1958, Nr 4,
PP 368-371 (USSR)
ABSTRACT: A continuous functionT(q) on some~compact Q is considered
with values in the Banach.commutatiye regular ring A with
unity ( R is a commutAtive of e -aigebra with unity) by
means of polynomials YK W Xn these po.lynomials
di 7 are complex numbers and
are some fixed continuous functions~,on Q;;to R. A polynomial
is sought which'satibfies the require-
Card 1/3 ment;
SOY-21-58-4-3/2~
Some Theorems on the Chebyshev Approximation of Functions with Values~
Belonging to a CommutativeC*-Algebra;
471 z'Xf'M'Q Ili: a,?. (q) (I)
4CR A K CEK
K=1
The author formulates:three theorems and otates that they
can be proven, starting from the corresponding theorems of
the Chebyshev approximation of num6rical functions~and'the
Gellfand-Naymark theorem ZRaf. g,' The'~,necessary Oondition
is given for the polynomial to be a Chebyshev polynomial as
well as the necessary condition fol, the uniqueness of such
a polynomial. A similar problem io then considered, r;elated
to the ring engendered by an Hermitian operator in,Hilbert
space, that is the uniqueness of'a~pdynomial d-1 'A
is asserted for which holds the following requirement.,
K
a'KA int (XxAk (2)
CLA
Card 2/3
SOV-21-58-4-3/29
Some Theorems on the Chebyshev Approximation of Functions with'ValueE
Belonging to a Commutative C4 -Algebra
wi.6re A is an arbitrary'Hermitiiri op4rator in Hilbert space
ov~ie t; ref er-
and B is any operator B E R(A)# There' are 3 5
ences.
ASSOCIATION: Lutskiy pedinstitut imeni Lesi Ukrainki (Lutsk'*Pedagogical
Institute i~eni Lesya Ukrainka);
PRESENTED: By Member of the AS USSR, N.N. togolyubov
SUBMITTED: September 5, 1957
17OTE- Russian title and Russian names of individuals~and insti-
tutions appearing in this articled haive been used in;;the
transliteration.
1. FiLnetions--Theory' 2. Polynomt4ls-40plicatione 3.' Complex
numbers--Applications 4, Operntolfs (Ma~h0matics)--Applicatlons
Card 3/3
ZMOTITSKIT, SIo; ININO Glo
Chebyshovl approximtlon in a coomtative flilbort r1fig- Dold~ AN
HER 119 ~o AP '58
,6 s 1074-1076 -TiMA ilt6Y
1. Latokly goffadarstveMy pedWgi6heshy iidtitut Im. Lea
Ukraluld. Predstavleno akadomikom X.So Bogowovrm.
(runotibnot continuoti)
AUTHORt ZUKROVITSFIY, S. 1. (Kiyev) 39-43-4-4/4
TITLE; Amendments to the Paper "Some Theorems of the Theory of
Chebyshey Approximations in the Hilbeif, *o#11'(Iepr&v--:
leniye k rabote "Nekotoryye teoremy teorii ebabyabovskikh
pribl:izheniy v proatranatya Gillbarta")"
FERIODIGAM Mathematicheskiy Sbornik, 1957, Vol 43,~~Rr 4, Pp 504 (USSR)
ABSTRACT: In a theorem published by the author in'1955 (Raf. I] tbo
assumptions mat be replaced by stronger onem, One Sovie,t
reference is quoted*
SUBMITTED: 6 ja=ar7 1958
AVAILABLEt Library of Congress
Card 1/1 1, Topology-Theory
AUTHORs Zukhovitakiy, SJ. (Lutsk) 20-1120-.4-2/67
TITLE& An Algorithm iii'e' Construction of the' Chebyshie~~ Approxima-
tion of a Continuous Function by a' Pol~nbmial Ulgori~rm dlya,
postroyeniya IChebyshevskogo pr1bl,izhen1y4 neprer yIvnoy funktaii
polinomom)
PERIODICALs Doklady Akademii nauk S.?SR,1958,Vol 120,Nr 4,PP:693-696(USSR)
ABSTRACTs On the compact Q let a real continuous function f(q) and
a.system of real continuous linearly independent fun*tions
kfj(q) ...... Pn(q) be given. The s,uthorproposes,~a ne* algorithm
for the determination or coefficients 5k for w1iich
n
max Tk(q) f (q)
qr k
zq kal
attainsIa minimum. Here only the,"funct ion valuesiin those
points of Q are essentially applied in ihich thwabsolute value
of the difference between the approximating polynoMU'Ll and
f(q) attains a maximum.
There are 5 Soviet references.
Card 1/2
An Algorlt2im for the Construction of the ChebysMv 20-120-4-216-1
Approximation of a Continuous Function by a Polynomial
ASSOCIATION's Lutskiy pedagogicheakiy institut im*ni Lasi Ukrainki
(Lutsk Pedagogical Institute imeni Zesya lVkrainka)
PRESENTEDt December 1), 1957, by N.V. Bogolyubov, Academician
SUBMITTEDs 'December 3, 1957
1. Mathematics
Card 2/2
Call Nr: APA108825
Transactions of the Third All-union Mathematical Congress (Cont~.)moscaw,
Jun-Jul '56, Trudy '56, V. 1, Sect. Rpts., Izdatellstvo:'AN SSSR, Hoscov, lq56, 237 PP.
There is I USSR reference. 82,83
Zukhovitskiy, S. I. (Kiyev). On a Minimum Problem of the
Probre-m of Momen-t-s. 81-84
There is 1 German reference.
KazImin, Yu. A. (Zernovoy). On Complete Systems in:,
Hilbert Spaces. 84:~85
There are 2 references, 1 of which,is USSR,, and the'
other German,
Kozmanova, A. A. (Sverdlovsk). The Theorem oflolya for N
Entire Functions of Two Complex Variable3
:85
-K&arev, P. P. (Tomsk). On the Method of Para:matric
Representation and 0. M, Goluzin Variational Method. ~85-786
Card 26/60
AUTHORs ZUKHOVITSKIY S.I., ESKIN G.I. 20~5-4/48
TITLEt On-the Lpproximati;n ofAbstract.continuouo ftnetione by;
Unbounde& Operator Functions'(0 priblishenli abstraktaftb
nopraryvnykh.funktaiy.naogrsAi.ohemWa,i oporiator-funktniyami)
JERIODICALs Boklady Akad.Rauk WSHO 195T,Vo1.I!6jMr-5ipp-73I-734 (USSR)
,LBSTRACTt Theorem It On the compactum Q lot be,defined~an operator function
A(q) with the following propertiest 1).For -every qE-q '' A(q) is a
closed.linear operator of the lilbert~spsoe".ffi iuto the 111,1bert
space ff Fo-r.all qF-Q" A~q) has -the S*Xe-regicn of definition
-2D denseAn R1. 2 For every fixed, zQD, A(q)x,is ,a
function continuous on q with
In -order. -that Xor every function f (q),' dantilMouo. on Q ~ with the
Values in R 2'there exists a vector z C--[D such that
0
inf max IIA(q)x-f(q) 112 - 2,
max IIA(q)~,o-f (q)
x 6D q SQ .qsq
it is necessary and sufficient that foreman m >0 and all ~xC-D
card 1/2 max IIA(q)x 112 >1 a 11Z 11
On the.-Approximation of Abstract Continuous Funoti'one bV 20-5-*4/40
Unbounded Operator Functions
Theorem 2s Lot A(q) satisfy-the oonditions-of-thearem. I and
let dim.K