Atakut, Çiğdem

İspir, Nurhayat

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaÇiğdem Atakut; Nurhayat İspirOn Bernstein type rational functions of two variablesMathematica Slovaca, Vol. 54 (2004), No. 3, 291--301Persistent URL: http://dml.cz/dmlcz/136908Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2004Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©2004 . /-. WA / , - .™,\ ... o ,-.«., .-.«., Mathematical Insti tute Math. SlOVaCa, 5 4 ( 2 0 0 4 ) , NO. 3 , 2 9 1 - 3 0 1 Slovák Academy of Sciences ON BERNSTEIN TYPE RATIONAL FUNCTIONS OF TWO VARIABLES QlGDEM ATAKUT* — NURHAYAT ÍSPÍR** (Communicated by Pavel Kostyrko ) ABSTRACT. In this paper the authors define Bernstein type rational functions of two variables and prove the approximation theorems for them. Moreover, asym-ptotic approximation theorem is proved for Bernstein type rational functions of two variables. 1. Introduction K. B a l az s [1] introduced and considered some approximation properties of the Bernstein type rational functions ^'•*>-<r+W|.G)(v)'/(£)' (1) and proved that if / is continuous in [0, oo), f(x) = 0(eax) (x —> oo) with some real numbers a , then in any interval [0, A] (A > 0) the estimate \f(x) - Rn(f,x)\ < c o ^ n -1 ! 3 ) (0 < x < A) (2) holds for sufficiently large n ' s provided an = n~1/ /3 , bn = n2/ /3 . Here c0 depends only on A and a , and CJ2J4(-) is the modulus of continuity of / on the interval [0, 2A]. As it was noted in [1], the convergence of Rn(f-> x) holds under the more general conditions an = ^- —> 0, bn —J> oo (n —> oo) as well. The positive linear operator Rn was investigated by J. S z a b a d o s [2], K. B a l a z s [3] and V. T o t i k [4]. K. B a l a z s and J. S z a b a d o s [2] improved the estimate (2) by an ap-propriate choice of an and bn whenever / is uniformly continuous on [0, oo). 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i on : Primary 41A20. Keywords : Bernstein type rational function, order of approximation, modulus of continuity, asymptotic approximation. Supported by Research Foundation of Ankara University 99-05-02-01. 291 CIGDEM ATAKUT — NURHAYAT ISPIR Furthermore they showed that these results could been applied to approximate certain improper integrals by quadrature sums of positive coefficients based on finite number of equidistant nodes. In [4], they settled the saturation proper-ties of Rn(f,x) and proved a general convergence theorem for itn-like rational functions. K. B a l a z s [3] investigated approximation by modified Bernstein type ra-tional functions Rn on the real axis. In the present paper we define Bernstein type rational functions of twro vari-ables and give an estimate, in therm of the first usual moduli of continuity, for the approximation of functions by means of the operators Rn m defined at (3). Moreover, we prove an asymptotic approximation theorem. Let f(x,y) be a function of two variables, defined in [0, oo) x [0,oo). By Bernstein type rational functions of two variables corresponding to f(x,y) we mean the following: 1 1 n m / 7, • \ "•.-"•'•rt = (i t , .) .(Hy)-^^>'") ' '-W ' U ' t ) (n , ra = 1,2,. . .) (3) where a n ,O n , c m , ( i m are suitably chosen real numbers, independent of x and y and Pntk(x) = (t)(anx)k and therefore PmJ(y) = {™)(cmy)i. 2. Convergence theorem Let Rn)in(f,x,y) be the functions defined by (3) with an = ^- , 6n = n2 / 3 , cm = in., dm = ra2//3 (n , ra = 1,2, . . . ) and let w2A(5) be the modulus of continuity of the function f(x,y) in [0, 2A] x [0, 2A]. We shall prove the following theorem: THEOREM 1. Let f(x,y) be a continuous function defined in [0, oo) x [0, oo) such that f(x,y) = 0(ea(x+y>) (x -> oo, y —r oo) for some real number a. Then in any square 0 < x < A, 0 <y < A, (A>0) the inequality | / (Z, V) - Rn,mV>X>V)\ < C^2A ( V ^ 7 ¥ + ^ 7 3 j (4) is valid if n, ra are sufficiently large, where c0 is a constant depending on A and a only. The inequality (4) shows that -Rn>m(/5-C,2/) -r f(x,y) when x > 0, y > 0 if n,ra —> oo, and this convergence is uniform in every finite square. 292 ON BERNSTEIN TYPE RATIONAL FUNCTIONS OF TWO VARIABLES To prove the theorem some lemmas are needed. LEMMA 1. If x > 0. y > 0. then the following identities hold: 1 1 n m (l + V)«(l + C.,)-££^'«J->>-1- (5) 1 1 n m h 2 (1+v).(1 + .ri.E£^w^(»)(* - w = - f ^ • («) fc=0 j=0 n m U T ^ O T ^ F ŽŽ" .^ 'W»)W " O ) - ^ . (7) (1 + a„x)» (1 + cmy)«» h U m'3 ' ~ ( 1 + a " x ) 2 '(8) £ + *n*)n£ + cmv)mhh n'ki) m'j[y){J mV) ~ (i + cmy)2 ( 9 ) 1 1 n m (1 + n T W l + r ?AmS^EPn,/c(a;)I3mJ(y)(fc - W O ' - O ) U + a„x; ^- t -c my; fc=oi=0 = _aA£2/L^V_ (1 + a„x)(l + cmy) ' t«/»ereon = ^ , cm = ^ . LEMMA 2. If x >0, y > 0, then the inequality 1 1 m a2 x4 + — (TT^roT^F,.E E ^ « . » « ^ s c j f ^ (ID tWds /Or sufficiently large n,m, where S > 0 and 7 are arbitrary fixed real numbers, &n = ^7 ~> 0; kn ~̂  °°; c™ ~ 1^ ~* ^ J dm -» 00 as n, m -> 00. The proofs of Lemma 1 and Lemma 2 are evident from [1]. It is well known that if A and 5 are arbitrary positive numbers, then co2A(X5)<u2A(5)(X + l). (12) Now we prove the convergence theorem. 293 CIGDEM ATAKUT — NURHAYAT ISPIR P r o o f o f T h e o r e m 1 . By (3) and (5) K,mU,x,y) = •\f(x,y)-Rntm(f,x,y)\ 1 1 < (1 + anx)n (1 + cmyY £ £ p „ , . W p m j ( y ) k=0 j=0 f{x,y)-f( — ,j-un am < (l + anx)n(l + cmy)™\ " E E + E E (13) x\<2A M y\<2A \j*--x\>2A U- y\>2A + E E + E E \£-x\<2A | a Ł - - y | > 2 A |çŁ--x|>2Л | a І - _ J , | < 2 A ^ + S ^ + sз + ^ . We obtain by (12) f{x,y)~f k_ J_ Ьn dm < Lü, 2A X~VÌ +{y~Z - U2A ( Sn,m. ' g 1 æ - т - ì + (y- J < x~ъ~) + { y - f ~ ) + 1 ^(U-(14) By (13), (14) and (5) S - - W 2 л ( Ä " ' r a ) t 7 ( l + -яx)»(l + L ) ? S . £ ^t^-tø-| £ - * | < 2 A | ; £ - , i | < 2 A I _ £ ) +(v~i) +^('--) = 5i+W2A(5n,m)-Using the Schwarz inequality, then considering (5), (8) and (9) we obtain 294 (15) ON BERNSTEIN TYPE RATIONAL FUNCTIONS OF TWO VARIABLES Sí < "2A (*„,,») őnm { b2n ( 1 + ӣnX)n ( 1 + Cmy)r h=0j=0 1 ^ 1 n m x 1/2 + ^ (1 + «„*)« (1 + cmy)m E £ - ^M-W-W - <U)2} ?n v n / v vn^ f /c=0 7=0 ř* ^J_ í±<Š?l±Kň , 1 < t o 4 + <L^1/2 V n . m i r 1 L2 t i • „ „ \ 2 "^ W 2 M V m ; ^ , m \ ^ (i + v )2 "r<*m (i + cmy)2 (16) Since b = n 2/ 3 and cL, = m 2 / / 3, in this case a„ = —- = n 1//3 and c = It Tib »£ 7T 776 -^ = m - 1 / 3 . So by (15) and (16) we have s , < ^ ( u { ^ ( ^ + ^ ) 1 / 2 + i}- (") Since 0 < x < A and 0 < y < A, then by (17) ( 1 / 1 1 ^ / 2 N n,m S,<c2ш2A{5nJ — ( — + — ) + l } . (18) where c2 = \A44 + A. 1 \ i / 2 Assuming 5nm = (--^- + --£--) , by (18) we have 5 i -- 4-2A ( \ / ^ 7 3 + diTs ) • ( 1 9 ) where c2 = 2c2. Since f{x,y) = 0(ea^x+2/ ;) (x,y —> oo, a fixed), the estimation of S^ is a trivial consequence of Lemma 2: ^ ( l + U-d+c!;,)- E .E'V.W.wO'^*'**"-| ^ - - c r | > 2 A | ^ - - - 7 / | > 2 A K ^ ^ / /^4+^\ < C,CAC. ~ ^ ^ \ { l + anx)2) \{l + cmyY < clCic5 (anx* + ^) (cmy* + J- ) (20) ^ ^ 7 ^ ( - 4 + ̂ 4 + y) 2 < C 6 C 0 _ C 7 n2/3m2/3 n2/3m.2/3 295 CIGDEM ATAKUT — NURHAYAT ISPÍR where c6 = c1c4c5 and c7 = C6CQ, CQ = A4 + A. Since "->- ' (£•£ ) < f(x,y) ~ -AA + ' (£•0 Чb„'dÜ we have c . 1 1 x^ ү^ P t~Л P l*Л . 3 - (1 + anxГ (1 + cmУy --x\<2A ď---y > 2 A f(x,y)-f [ -ç,У (1 + апх)п(1 + стУу £ £ V-)^(»)' -*—x|<2A U ^ - - y | > 2 A (21) "b*)Ahi < Lü, '2Л Щl^ + ̂  + iì + ^ ŕ + y) < CQ ІU), 9 1 - 2 Л l „ 1 / 3 + m 2/3 The estimate of 5 4 is similar to S3. Since f(x,y)-f k j b 'd n m < f(x,y)-f[x, d_ '"•£)-'(£•£ then we obtain ^4 < C10 | W 2 Л ( - - - -1/3 J ' n2/3j * (2 2 ) Now, on the basis of (19), (20), (21) and (22), the inequality (13) may be written in the following way ABim(/,.r>y) < c'2u2A (^-1- + - ^ J + -^-^ { ^ ( ^ ) + ^ } + C I O { - 2 A ( ^ / + CQ < cn < üü0 /З / n 2 / 3 0 ì "2A V ^ ^ m2/3 + + -o-77 + 2/3 m 2 / 3 n^i° m + Üün + í^9 2 A l n l / 3 J - Î - - 2 Л І - Л 7 296 ON BERNSTEIN TYPE RATIONAL FUNCTIONS OF TWO VARIABLES Since 5 < c[0u(5), then l i l i i , i i + ™2/3 - V ^2/3 + ̂ 273 - ClO^A \ 3Ž3 + n2/3 ^ m 2 3  Y „ / ^ m /3 - "10<"2A \V rfß "" m 2/3 Also, 1 / 1 / 1 1 „1/3 V n2/3 ^ ү n2/3 m2/3 <\/-Ti-- + and we obtain U i/зj W2A l riTI ) < U2A \ \l ^273 + m2/3 I ' 1 1 An,J/,x,y)<cbo;aл(У--L +J_j ( 0 < z < A, 0 < y < A), which gives the proof. • 3. Asymptot ic approximation In this part of the paper we prove an asymptotic approximation theorem similar to [1]. THEOREM 2. Let /(£,r) be a function defined in [0,oo) x [0,oo) such that /(£, r) = 0(e a ( t + r ' ) (£, r -> 00. a zs a fixed real number), then at each point (x, y) m ivtac/i /£-., f'Jy and f£y exist finitely Rn,rnU^y) = f^y) + «J'^y) (rp^) + cmfy^y) (T^J) ( abx4 + ̂ - \ . 26„(l + anx)- / \ n v. П ' I + a«CmҐ:«{X'У)L + af)t+cmy)) ( c d y* + JL \ + cmf'^(xty) [2dm(l + cmy)Ч + ( a " + C ™ ) p ">™ ' V . V2 ( 2 3 ) «Лere p П i m -> 0, a n = àj- -» 0, cm = ̂  -> 0, -££-- -> 0 and ^ -> 0 as n,m -> oo. 297 CIGDEM ATAKUT — NURHAYAT JSPJR P r o o f . By the conditions of the theorem, f'Jx, f'J and fvv are finite, thus 'УУ we may write f(t,T) = f(x,y) + (t-x)f'x(x,y) + (T-y)f'y(x,y) + \(t-x)2f':x(x,y) + (t-x)(T -y)f':y(x,y) + \(T -y)2f'^(x,y) (24) + [(t-x)2 + (T-y)2]X(t,T,x,y), where X(t,T,x,y) ->0 as i - > x , r ^ y . Hence f(hi)"^+(t*)>*^+(i-«)<>'* +i{£-*)'u*.,» (£-*)(£-,) CM +\(i-y1 r>>{x'y) + ź-xî+(í-;' \ l k з K dm (25) Substituting this expression in Rnm(f,x,y) and taking into account the iden­tities (5)-(10), we get Rn,mU,X,У) = ľÅxìУ) k=0j=0 n m + ) n (l + cmyУ Ьn (1 + anx)n (1 + cmyГ ЪoЪPn'k{X)Pm'iШk ~ KX) f y t y ) ( i + L - ( i + l y ) - ШPn'k(x)Pm'jШ ~dmy) fll ( \ л 1 n m + g a + •.«>-(. + ...•)- £ £ ' Л » - ^ 2 + + + îLfay) i fc=0j=0 n m i xy i+o x)"(i+c ^ S E ^ w ^ ^ - w - o ) M™ (1 + onx)» (1 + crop) /y„(g.y) i fc = 0 j = 0 n m i+Lx)« (i+L)« ? £ ^ ( ^ J Ы Ü - 0) 2 2d^ (1 + anx)" (1  cro2/) k=0 j = 0 298 ON BERNSTEIN TYPE RATIONAL FUNCTIONS OF TWO VARIABLES - - n m + (i+„„*)» (i+cm!l)» S S P"'*(I)P,"''(!') č-Mi-' /c=0 j = 0 2 A , £ ' £ ' x , y (26) = f(x,y) + f'(x,y) «A^_ í+/ ; (X)2/) C m d m í / 2 + f" (r v) ( anbnx4 + KX \ , f,i , _ V) ( anbnCmdmx2y2 + f"(xv)(d™C™y4+dmy)+r + ;yy[x>y)\2d?n(i + cmyy)+rn>™> i! + Cmž/) where (1 + anxY (1 + cmyY . 2 EE^«/i/)' A; fc=0 j = 0 .2 X ' + ' d V n m (27) For given an e > 0, let us choose á > 0 so small that \\(t,r,x,y)\ < e when |č - x\ < 6 and \r — y\ < S. For this 5, decompose the sum (27) into four parts: 1 r = n,m í E E + E E + E E + E E | ^ - - x | < ( 5 | / I bn I I a?7 |>(5 | f c - x | > ( 5 / I — | O f l I — I d 7 7 -</<<* (28) - ^ + A2 + A3 + A4 . By (8) and (9) and considering the property of X(t,r,x,y) we obtain r a 2 x 4 , ^ . 2 4 • _JL^| t)2 (l + c™!/)2 Now we give upper estimation for \A2\. Henceforth c{, i = 11,12,... , are positive numbers depending only on x,y and a. By f(t,r) = 0(e a ( t + r ) ) ( í , r -> oo, a, fixed), it follows from (25) for some c u that }-x) +&-" x[hhx'v <cuea^ea"" (30) 299 CIGDEM ATAKUT — NURHAYAT ÍSPÍR Using (27), (28), (30) and (11) we get | A 2 | < c : 12 Lmy ^ drn il + anx)*) l ( l + cmH)2 a 2 4 , x_ a n x ^ bn \A\<c (C^l±±) 1 3 ' 1 3 d + c m y )2 ' | A 4 | < C 14 <^ + ţ (l + a„x)2 Let now Pn.r. an + C m By (34), (28), (29), (31), (32) and (33), the relation lim lim \p \ < m—»oo n—>oo < lim m—»oo + lim < e n—>oo I 2 4 af X 4 + CmУ4 Ҝ + c m )(l + anxY bn(an + c J ( l + anx) У + + c K + 0 ( 1 + cmV)2 dm{an + c m )(l + cm2/)2j a n * 4 + 1 2 l Ч + cJ(i + v ) 2 ' ь„Ҝ + 0 ( i + -ҷ-)2 У 2 4 C H m& + (1 + CmУ)2 ^ ( 1 + C m ^ + C 13 c 2 H4 m& + 2/ + c 14 >n + C J ( 1 + Cm2/)2 <U«„ + C J ( 1 + СтУ)2 X aíx4 + Ҝ + C J ( 1 + V ) 2 Ь„(on + c m )( l + anxf = 0 (31) (32) (33) (34) (35) holds, because a = ^ -> 0, ^ ^ 0, cm = ^ -> 0 and ^ > 0 as n,m -> co. (26), (27), (34) and (35) give the proof of Theorem 2. • A c k n o w l e d g m e n t The authors would like to thank Professor A. D. Gadjiev for his valuable suggestions. 300 ON BERNSTEIN TYPE RATIONAL FUNCTIONS OF TWO VARIABLES REFERENCES [1] BALAZS, K.: Approximation by Bernstein type rational functions, Acta Ma th. Acad. Sci. Hungar. 26 (1975), 123-134. [2] BALAZS, K.—SZABADOS, J. : Approximation Bernstein type rational functions II, Ac ta Ma th. Acad. Sci. Hungar. 4 0 (1982), 331-337. [3] BALAZS, K.: Approximation by Bernstein type rational functions on the real axis, Ac ta Ma th. Hungar. 46 (1985), 195-204. [4] TOTIK, V . : Saturation for Bernstein type rational functions, Ac ta Ma th. Hungar. 4 3 (1984), 219-250. Received June 4, 2002 * Department of Mathematics Faculty of Science Ankara University 06100, Tandogan Ankara TURKEY E-mail: atakut@science.ankara.edu.tr Department of Mathematics Sciences and Arts Faculty Gazi University 06500, Teknikokullar Ankara TURKEY E-mail: nispir@gazi.edu.tr 301 

English

On Bernstein type rational functions of two variables

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/136908/MathSlov_54-2004-3_9.pdf

On Bernstein type rational functions of two variables

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library