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December 31, 1967
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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