Rate of approximation for Bézier variant of Bleimann-Butzer and Hahn operators

Transcription

1 General Mathematics Vol 13, No 1 25), Rate of approximation for Bézier variant of Bleimann-Butzer and Hahn operators Vijay Gupta and Alexandru Lupaş Dedicated to Professor Emil C Popa on his 6th anniversary Abstract In the present paper we obtain the rate of convergence for the Bézier variant of the Bleimann-Butzer and Hahn operators L n,α, for functions of bounded variation We consider the case when < α < 1 2 Mathematics Subject Classification: 65B99 Key words and phrases: Bleimann-Butzer and Hahn operators, Bézier variant, rate of approximation 1 Introduction In the year 198, Bleimann, Butzer and Hahn [5] introduced an interesting sequence of positive linear operators defined on the space of real functions on the infinite interval [, ) by 1) L n f, x) = p n, x)f = n 1 41 ), x [, ), n N

2 42 Vijay Gupta and Alexandru Lupaş where p n, x) = ) n x 1 + x) n The approximation properties of the Bleimann-Butzer and Hahn operators BBH operators) were studied by many researchers see eg [1], [2], [3], [5] and [6] etc Very recently Srivastava and Gupta [8], introduced the Bézier variant of the BBH operators, which is defined as ) L n,α f, x) = Q 2) n,x)f, x [, ), n N n + 1 = where Q α) n, x) = J α n, x) J α n,+1 x) and J n,x) = α j= p n,j x) Some basic properties of J n, x) can be found in [8] It is easily verified that L n,α f, x) are positive linear operators The authors [8] have obtained the rate of convergence for functions of bounded variation for the case whenever α 1 As a special case α = 1, L n,α f, x) reduce to the operators L n,1 f, x) L n f, x) defined by 1) The other case < α < 1 of the operators defined by 2) is also equally important In the present paper we extend the study in this direction and obtain teh rate of convergence for functions of bounded variation, for the operators L n,α, < α < 1 Our main theorem is stated as: Theorem 1 Let f be a function of bounded variation on every finite subinterval of [, ) Let ft) = Ot r ) for some r N as t Then for x, ), < α < 1 and n sufficiently large, we have L n,αf, x) 1 2 α fx+) 1 1 ) 2 α fx ) 71 + x)2 n + 2)x n =1 V x+x/ n x/ f x) x 2enx ε n x) fx) fx ) + 1 x 6 fx+) fx ) + 2πn + 1)x Cα, f, x, r) n m, m > r

3 Rate of approximation for Bézier variant 43 where { 1, ifn + 1)px N ε n x) =, otherwise ft) fx ), t < x f x t) =, t = x ft) fx+), x < t < and V b a f x ) is the total variation of f x on [a, b] We recall the Lebesque-Stieltjes integral representation L n,α f, x) = ft)d t K n,α x, t)),, where Q α) n, x), < t < K n,α x, t) = n +1)t, t = Also we define H n,α x, t) = { 1 Kn,α x, t), < t <, t = 2 Auxiliary Results In this section we give certain results, which are necessary to prove the main results Lemma 1 [3] For all x [, ) and n 1 we have the following inequality: L n t x) 2, x) 3x1 + x)2 n + 2

4 44 Vijay Gupta and Alexandru Lupaş Lemma 2 For x, ), we have p n, x) x 6 2πn + 1)x /n +1)>x Proof Apart from at most one term p n, x) when /n +1) = x, using the transform p n, x) = ) n x 1 + x) n = = with y = x/1 + x), we have ) ) ) n n x 1 = 1 + x 1 + x ) n y 1 y) n b n, y), /n +1)>x this is approximately equal to with the incomplete Beta function We have 1 2 B y a, b) = p n, x) = >n+1)y b n, y) B y n + 1)y, n n + 1)y + 1) Bn + 1)y, n n + 1)y + 1) /n +1)>x t a 1 1 t) b 1 dt, a >, b > p n, x) = 1 2 = Bn + 1)y, n n + 1)y + 1) n+1)>y b n, y) = t n+1)y 1 1 t) n n+1)y dt

5 Rate of approximation for Bézier variant 45 where Now we estimate the right hand side as follows: Let y, 1), we have Bn + 1)y, n + 1)1 y)) Bn + 1)y, n + 1)1 y)) = 1 I y n + 1) Bn + 1)y, n + 1)1 y)) I y n + 1) = 1 t n+1)y 1 1 t) n+1)1 y) 1 dt = 1 gt)e n+1)hyt) dt y y with gt) = t1 t)) 1 Since h y = h y is strictly decreasing on y, 1) Furthermore h yt) = h y t) = y log t + 1 y) log1 t) t + y) t1 t) < y < t < 1) and h yy) = t y) 2 + y1 y) t 2 1 t) 2 < and h yy) = y1 y)) 1 Thus it is well nown eg I y n) meets the assumptions of [7, Th 1, Kap 3]) that there holds the complete asymptotic expansion I y n + 1) 1 2 en+1)h y = Γ + 1)/2) n + 1) +1)/2 [ ] y y 1 y) 1 y a π ) 2n + 1) + a 1 1/2 2n + 1) + a 2 4n + 1) + 3/2 On 2 ), for n, with the coefficients a = 1 ) +1 d! dt gt) t y h y y) h y t) t=1/y

6 46 Vijay Gupta and Alexandru Lupaş By direct calculation we obtain the explicit expressions y) a =, a 1 = y1 y) 3y1 y), a 1 y + y 2 2 = 3 2y1 y)) 3/2 Thus we have [ π 2nπ1 y) I y n + 1) = y y 1 y) 1 y ) n 1 2y π 3ny1 y) Also by Stirling s formula, we have 1 Bn + 1)y, n + 1)1 y)) [ n 1 y + y 2 + 1)y1 y) 1 y + y 2 y1 y)) 3/2 + On 2 ) = Γn + 1) Γn + 1)x) = 1 2π y y 1 y) 1 y ) n 12 ny1 y) + 1 y + y 2 ) 2 299n 3/2 y1 y)) 3/2 + On 5/2 ) Combining the both asymptotic expansions, we have I y n + 1) Bn + 1)y, n + 1)1 y)) = y 3 2πn + 1)y1 y) + On 3/2 ) Thus /n +1)>x p n, x) x 6 2πn + 1)x + On 3/2 ) Lemma 3 For all x, ), α 1 and N, there holds Proof Q α) n, x) αp n,x) < 1 + x) 2enx For the Bernstein functions the optimum bound was obtained by Zeng [8], and Bastien and Rogalsi [4] in a problem posed by the author, which is as follows: n Substituting y = ) y 1 y) n 1 2eny1 y), < y < x x, we get the required result ] ]

7 Rate of approximation for Bézier variant 47 Lemma 4 For < α < 1 and < x < t <, we have H n,α x, t) C n m t x) m, where C is a positive constant that depends on x but independent of n n + 1 x Proof Since < x < t <, so 1, for nt Thus t x for m N, we have H n,α x, t) = 1 K n,α x, t) = 1 1 t x) 2m n +1)t n +1)t Q α) n, x) 2m/α n + 1 p t x) 2m/α n, x) n + 1 x = 2m/α n +1)t α p n, x)) α For all conjugate p, q 1, ie 1/p + 1/q = 1, we have n + 1 x = = n + 1 x = 2m/α n + 1 x = α/q since p n, x)) = 1 = 2m/α p n, x)) α = p 1/p n, x)p1/q n, x) ) α 2mp/α p n, x) ) α/p Q α) n, x)

8 48 Vijay Gupta and Alexandru Lupaş Choosing p = m q [ mα + 1 ] we have that 2mp/α is an even positive integer By the well nown result B n,1 ψx 2r, x) = On r ), as n r = 1, 2, 3, ) we obtain n + 1 x = as n 2mp/α This completes the proof of Lemma 4 ) α/p p n, x) = L n,1ψx 2mp/α, x)) α/p = On m ) 3 Proof of main Theorem Our main theorem is stated as: Proof We have ft) = 2 α fx+)+1 2 α )fx )+g x t)+2 α ff+) fx ))sign α) t)+ +fx) 2 α fx+) 1 2 α )fx ))δ x t), where 2 α 1, if t > x sign α) t x) :=, if t = x 1, if t < x Therefore and δ x t) { 1, if x = t, if x t 3) L n,αf, x) 1 2 α fx+) 1 1 ) 2 α fx ) L n,αf x, x) + + fx+) fx ) 2 α L n,α sign α) t x), x)+ [ + fx) 1 2 α fx+) 1 1 ) ] 2 α fx ) L n,α δ x, x)

9 Rate of approximation for Bézier variant 49 and We first estimate L n,α sign α) t x), x) = 2 α = 2 α >n +1)x >n +1)x p n, x) L n,α δ x, x) = ε n x)q α) n, x) α Q α) n, x) 1 + e nx)q α) n, x) = 1 + ε n x)q α) n, x) Hence, we have fx+) fx ) 4) 2 α L n,α sign α t x), x)+ = [ + fx) 1 2 α fx+) fx+) fx ) 2 α 2 α >n +1)x α α ) ] fx ) By mean value theorem, we have α p n,j x) 1 2 α = αξ n,jx)) α 1 j>n j+1)x where ξ n,j x) lies between 1 2 and L n,α δ x, x) = p n, x) 1 + [fx) fx )]ε n Q α) j>n j+1)x j>n j+1)x n, x) p n,j x) 1 2 p n,j x) In view of Lemma 2, it is observed that for n sufficiently large, the intermediate point ζ n,j is arbitrary close to 1/2 ie ζ n,j = ε with an arbitrary small ε Then we have αξ n,j x)) α 1 α2 + ε) 1 α

10 5 Vijay Gupta and Alexandru Lupaş The later expression is positive and strictly increasing for α, 1), since α α2 + ε)1 α = 2 + ε) 1 α [1 α log2 + ε)] >, for sufficiently small ε This it taes maximum value at α = 1 This implies αξ n,j x)) α 1 1 5) Hence j>n j+1)x p n,j x) α 1 2 α 1 x 6 2πn + 1)x Also we have Q α) n, x) = J α n, x) J α n, +1x) = αξ n, ) α 1 p n, x), where J n, +1x) < ξ n, x) < J n, x) Thus by Lemma 3, we have 6) Q α) n, x) 1 + x 2enx Combining the estimates of 4), 5) and 6), we have fx+) fx ) 7) 2 α L n,α signt x), x)+ + [ fx) 1 2 α fx+) 1 x 6 2πn + 1)x 1 1 ) ] 2 α fx ) L n,α δ x, x) fx+) fx ) x 2enx ε n x) fx) fx ) Next we estimate L n,α f x, x) We decompose the integral into four parts as follows: 8) L n,α f x, x)= f x t)d t K n,α x, t))=

11 Rate of approximation for Bézier variant 51 = I 1 + I 2 + I 3 + I 4 f x t)d t K n,α x, t)) = E 1 + E 2 + E 3 + E 4 say, where I 1 = [, x x/ n], I 2 = [x x/ n, x + x/ n], I 3 = [x + x/ n, 2x] and I 4 = [2x, ] We first estimate E 2 For t [x x/ n, x + x/ n], we have and therefore Since b a f x t) = f x t) f x x) V x+x/ n x x/ n f x) E 2 V x+x/ n x x/ n f x) x+x/ n x x/ n d t K n,α x, t)) d t K n,α x, t)) 1 for a, b) [, ), therefore 9) E 2 V x+x/ n x x/ n f x) 1 n n =1 V x+x/ n x x/ n f x) Next, we estimate E 1, writing y = x x/ n and using Lebesque-Stieltjes integration by parts, we have E 1 = f x t)d t K n,α x, t)) = f x y)k n,α x, y) K n,α x, t)d t f x t)) Since f x y) V x y f x ), it follows that E 1 Vy x f x )K n,α x, y) + Applying Lemma 1, we have K n,α x, t)d t V x t f x )) 1) K n,α x, t) 3x1 + x)2, t < x n + 2)

12 52 Vijay Gupta and Alexandru Lupaş Using the inequality 1), we get E 1 Vy x 3x1 + x) 2 3x1 + x)2 f x ) 2 + n + 2)x y) n + 2) Integrating by parts the last term, we have Hence 1 x t) 2 d t Vt x f x )) = V y x f x ) x y) 2 + V x f x ) x E 1 3x1 + x)2 n + 2) V x f x ) x x t) 2 d t V x t f x )) dt t f x ) x t) 3 V x Vt x f x ) x t) 3 Now replacing the variable y in the last integral by x x/ u, we get 11) E x)2 n + 2)x n =1 V x x x/ f x) Using the similar method to estimate E 3, we get 12) E x)2 n + 2)x n =1 V x+x/ x f x ) Finally, by the assumption that f x t) = t r ), r N, t, we have ) r t x f x t) Mt r M, for t 2x x Now E 4 = 2x f x t)k n,α x, t) 2x f x t) d t K n,α x, t) Mx r t x) r d t K n,α x, t) Mx r t x) r d t 1 K n,α x, t)) = 2x

13 Rate of approximation for Bézier variant 53 = Mx r 2x = Mx r lim = Mx r lim t x) r d t H n,α x, t)) Mx r R R = Mx r lim R t x) r H n,α x, t) R 2x + t x) r H n,α x, t) R 2x + R 2x R 2x t x) r C n m t x) m R 2x + rc n m t x) r d t 1 K n,α x, t)) = H n,α x, t)d t t x) r = H n,α x, t)d t t x) r 1 dt = R = M C n m x m + rc n m m r, m > r m r)x 2x t x) r m 1 dt = By combining the estimates given by 3), 7) - 9) and 11) to 13), we obtain the desired result This completes the proof of Theorem References [1] U Abel, M Ivan, Some identities for the operator Bleimann-Butzer and Hahn involving divided differences, Calcolo 36 3), 1999, [2] U Abel, On the asymptotic approximation with Bivariate operators of Bleimann-Butzer and Hahn, J Approx Theory 97, 1999, [3] U Abel, M Ivan, Best constant for a Bleimann-Butzer-Hahn moment estimation, East J Approx 6 3), 2, 1-7 [4] G Bastien, M Rogalsi, Convexite, complete monotonie et inegalities sur les functions zeta et Gamma, sur le functions des operateurs de Basaov et sur des functions arthemetiques, Canadian J Math 54 5), 22,

14 54 Vijay Gupta and Alexandru Lupaş [5] G Bleimann, P L Butzer, L Hahn, A Bernstein-type operator approximating continuous functions on the semi axis, Indag Math 42, 198, [6] B Della Vechia, Some properties of a rational operator of Bernstein type, pp , Progress in Approximation Theory P nevai, A Pinus, Eds), pp , Academic Press Boston, 1991 [7] G Doetsch, Handbuch del Laplace Transformation, Vol II, Birhuserm, Basel and Stuttgart, 1972 [8] H M Srivastava, V Gupta, Rate of convergence for the Bézier variant of the Bleimann-Butzer-Hahn operators, Applied Math Latters to appear [9] X M Zeng, Bounds for Bernstein basis functions and Meyer-König and Zeller basis functions, J Math Anal Appl 219, 1998, School of Applied Sciences Netaji Subhas Institute of Technology Sector 3 Dwara 1145 New Delhi, India Lucian Blaga University of Sibiu Department of Mathematics Str Dr I Raṭiu, No Sibiu, Romania address: alexandrulupas@ulbsibiuro