International Journal of Pure and Applied Mathematics Volume 60 No ,

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1 International Journal of Pure and Applied Mathematics Volume 60 No , ON CERTAIN CLASS OF SZÁSZ-MIRAKYAN OPERATORS IN EXPONENTIAL WEIGHT SPACES Lucyna Rempulska, Szymon Graczyk 2,2 Institute of Mathematics Poznan University of Technology 3a, Ul. Piotrowo, Poznan, , POLAND lucyna.rempulska@put.poznan.pl 2 szymon.graczyk@euronauka.com.pl Abstract: Applying the Borel methods of summability of sequences, we introduce the class of Szász-Mirakyan operators acting from the exponential weight space C q to C q and we give approximation theorems for them. AMS Subject Classification: 4A36 Key Words: Szász-Mirakyan operator, exponential weight space, degree of approximation, Voronovskaya type theorem. Introduction.. The approximation properties of the Szász-Mirakyan operators S n f;x) := e nx nx) k ) k f, x R 0,n N, ) k! n R 0 = [0, ), N = {,2,...}) and their various modifications were examined in many papers and monographs e.g. []-[5], [8]-[0], [2], [3]). In the paper [2] the operators S n were investigated in the exponential weight space C q, q = const. > 0, with the weight function v q x) = e qx, x R 0. There C q is the set of all functions f : R 0 R for which v q f is uniformly continuous and bounded on R 0 and the norm is defined by Received: February 20, 200 c 200 Academic Publications Correspondence author

2 260 L. Rempulska, S. Graczyk f q f ) q := sup x R 0 v q x) fx). 2) In [2] it was proved that S n is a positive linear operator acting from the space C q to C r provided that r > q > 0 and n > n 0, where n 0 > q/lnr/q) is a fixed integer..2. In the paper [8] the following modified Szász-Mirakyan operators: S n f;q;x) := e nx nx) k ) k f, x R 0,n N, 3) k! n + q were introduced for f C q. In [8] was proved that S n q) is a positive linear operator acting from the space C q to C q and S n f;q) q f q for f C q,n N, 4) and approximation theorems were given for them..3. In this paper Section 2) we generalize the formulas ) and 3) for S n, applying the Borel methods of summability of sequences. We introduce the class of Szász-Mirakyan operators in the exponential weight space C q and we examine approximation properties these operators. The sequence a n ) 0 of numbers a n R is summable to g by the Borel method B r, r N, [6] and [7]) if the series xrk rk)! a k is convergent on R 0 and x rk lim x re x rk)! a k = g. In the paper [0] was defined the class of Szász-Mirakyan operators S n;r. nx) rk ) rk S n;r f;x) := A r nx) rk)! f, x R 0,n N,r N, 5) n A r t) := t rk rk)!, t R 0, 6) for functions f belonging to the polynomial weight spaces. By ), 5) and 6) it is obvious that S n; f) S n f) for n N. Now we shall consider certain modification of the operators S n;r in exponential weight spaces C q.

3 ON CERTAIN CLASS OF SZÁSZ-MIRAKYAN Definition and Auxiliary Results 2.. Let r N and q > 0 be fixed numbers. For f C q we define the opertaors: nx) rk ) rk S n;r,q f;x) := A r nx) rk)! f, x R 0,n N, 7) n + q where A r is the function given by 6). We shall denote by Ω q, q > 0, the class of all operators S n;r,q with n,r N. By 7) and 3) it is obvious that S n;,q f;x) S n f;q;x) for f C q, x R 0 and n N. Let f k x) = x k for k = 0,,2. We immediately obtain from 7) and 6): S n;r,q f 0 ;x) =, S n;r,q f ;x) = x A rnx) n + q A r nx), 8) x 2 A S n;r,q f 2 ;x) = r nx) n + q) 2 A r nx) + x A r nx) n + q) 2 A r nx), 9) for x R 0 and n N. Moreover, we have S n;r,q f;0) = f0) for f C q,n N. 0) 2.2. In the paper [0] the following lemma was proved. Lemma. The function A r, 3 r N, defined by 6) can be rewritten in the following form: [ A 2m t) = m cosh t + exp t cos kπ ) cos t sin kπ ) ] ) m m m k= for 2 m N, and [ A 2m+ t) = e t 2) 2m + m ) 2kπ +2 exp t cos cos t sin 2kπ ) ] 2m + 2m + for m N and t R 0. k= Moreover, A t) = e t and A 2 t) = cosh t = 2 et + e t ) for t R 0. Applying ) and 2) or 6)), we easily derive the following Lemma 2. Let r N be fixed. Then lim t A r t) e t = r.

4 262 L. Rempulska, S. Graczyk Moreover, there exists a positive constant M r depending only on r such that et A r t) M r for t R 0. 3) 2.3. Now we give some elementary properties of the operators S n;r,q. By 7), 6), 3) and 3) is obvious the following Corollary. For q > 0, r N and every non-negative function f C q there holds the inequality where M r S n;r,q f;x) M r S n f;q;x) for x R 0,n N, = const. > 0 is given in 3). Applying Corollary and the formulas for S n t x) k ;q;x ), k =,2,4, and S n t x) 2 /v q t);q;x ) given in the paper [8], we immediately obtain Corollary 2. Let q > 0 and r N be fixed numbers. Then there exists M q,r = const. > 0 depending only on q and r such that S n;r,q t x;x) = enx A r nx) S x nt x;q;x) M q,r n + q, 4) and analogously S n;r,q t x) 2 ;x ) x 2 + x M q,r n + q, 5) S n;r,q t x) 4 ;x ) M q,r x 2 + x) 2 n + q) 2, 6) v q x)s n;r,q t x) 2 /v q t);x ) M q,r x 2 + x n + q, 7) for x R 0 and n N. Applying 8), 9) and Lemma 2, we can easily obtain the following Lemma 3. For fixed q > 0 and r N there holds: lim ns n;r,qt x;x) = qx and lim ns n;r,q t x) 2 ;x ) = x, n n at every x R 0. Lemma 4. Let q > 0 and r N be fixed and let Mr = const. > 0 be given by 3). Then for S n;r,q Ω q we have S n;r,q f) q M r f q for f C q and n N. 8) The formulas 7) and 6) and the inequality 8) show that S n;r,q, n N, is a positive linear operator acting from the space C q to C q.

5 ON CERTAIN CLASS OF SZÁSZ-MIRAKYAN Proof. From 7), 6), 2)-4) and Corollary we get v q x) S n;r,q f;x) f q v q x)s n;r,q /v q t);x) Mr f q S n /v q ;q) q Mr f q for x R 0,n N, which by 2) yields the inequality 8). 3. Theorems 3.. First we shall give two theorems on the degree of approximation of f C q by S n;r,q f). We shall use the modulus of continuity ω f;c q ; ) and the modulus of smoothness ω 2 f;c q ; ) of a function f C q, q > 0, i.e. ω k f;c q ;t) := sup k h f ) q for t 0,k =,2, 9) 0 h t where h fx) = fx + h) fx) and 2 hfx) = fx) 2fx + h) + fx + 2h) for x,h R 0 [4], []). Moreover, let ϕx) := x 2 + ) for x R 0, 20) δ n,q := n + q) /2 for n N,q > 0, 2) and let C 2 q, q > 0, be the set of all functions f C q which the derivatives f and f belong to C q also. Theorem. Let q > 0 and r N be fixed numbers. Then there exists M q,r = const. > 0 depending only on q and r such that for every f Cq 2 the following inequality holds: S n;r,q f) f)ϕ q M q,r f q + f ) q n N. 22) n + q Proof. Choosing f C 2 q and x R 0, we have ft) = fx) + f x)t x) + t x t u)f u)du and by 2) t t u)f u)du f q x v q t) + ) t x) 2, v q x) for t R 0. Using now the operator S n;r,q and 8), we get S n;r,q ft);x) = fx) + f x)s n;r,q t x;x) t ) +S n;r,q t u)f u)du;x, n N. x

6 264 L. Rempulska, S. Graczyk From the above and 4), 5) and 7) we deduce that v q x) S n;r,q f;x) fx) f q S n;r,q t x;x) t ) + v q x)s n;r,q t u)f u)du ;x f q S n;r,q t x;x) x + f { q vq x)s n;r,q t x) 2 /v q t);x ) + S n;r,q t x) 2 ;x )} x 2 + x M q,r f q + f ) q, n N. n + q Now using 20) and 2), we obtain 22). Theorem 2. Suppose that q > 0 and r N are fixed and ϕ and δ n,q are defined by 20) and 2). Then there exists M q,r = const. > 0 depending only on q and r such that: S n;r,q f) f)ϕ q M q,r {δ n,q ω f;c q ;δ n,q ) + ω 2 f;c q ;δ n,q )}, 23) for every f C q and n N, where ω k f;c q ; ) is defined by 9). and Proof. Analogously to [2] we use the Steklov function f h for f C q : f h x) := 4 h 2 h/2 0 h/2 0 [2fx + s + t) fx + 2s + t))] ds dt, x 0,h > 0. It is known [2]) that f h C 2 q if f C q and f h f q ω 2 f;c q ;h), 24) f h q 5e hq h ω f;c q ;h), 25) f h q 9h 2 ω 2 f;c q ;h), for h > 0. 26) By the linearity of the operator S n;r,q : C q C q and 20) we can write [S n;r,q f) f]ϕ q [S n;r,q f f h )]ϕ q + [S n;r,q f h ) f h ]ϕ q + f h f q := Z + Z 2 + Z 3. Next, by 20), 8) and 24), we have Z S n;r,q f f h ) q M r f f h q M r ω f;c q ;h). Applying Theorem for f h Cq 2 Z 2 M q,r n + q M q,r n + q and next 25) and 26), we get ) f h q + f h q e hq h ω f;c q ;h) + h 2 ω 2 f;c q ;h) ).

7 ON CERTAIN CLASS OF SZÁSZ-MIRAKYAN Consequently, [S n;r,q f) f]ϕ q M q;r {e hq h n + q) ω f;c q ;h) + n + q) h 2 + ) ω 2 f;c q ;h) } for n N,h > 0. Choosing now h = n + q) /2 δ n,q for fixed n N and q > 0, we obtain the desired estimation 23). The property lim t 0 + ω kf;c q ;t) = 0 for f C q and k =,2, and Theorem 2 imply the following Corollary 3. For fixed q > 0, r N and every f C q we have lim S n;r,qf;x) = fx) at every x R 0. n This convergence is uniform on every interval [a,b], a Applying Corollary 3 and Lemma 3, we shall prove the Voronovskaya type theorem for the operators S n;r,q. Theorem 3. Suppose that f C 2 q with a fixed q > 0 and r N. Then lim n S n;r,qf;x) fx)) = qxf x) + x n 2 f x) at every x R 0. 27) Proof. The statement 27) is obvious for x = 0 by 0). Let now x > 0 be fixed. By the Taylor formula for f C 2 q we have ft) = fx) + f x)t x) + 2 f x)t x) 2 + ψt,x)t x) 2, for t R 0, where ψt) ψt,x) is a function belonging to C q and ψx) = 0. From the above 7) and 8) we deduce that S n;r,q f;x) = fx) + f x)s n;r,q t x;x) + 2 f x)s n;r,q t x) 2 ;x ) Using now Lemma 3, we get +S n;r,q ψt)t x) 2 ;x ), for n N. lim n S n;r,qf;x) fx)) = qxf x) + x n 2 f x) 28) + lim ns n;r,q ψt)t x) 2 ;x ). n Next by the Hölder inequality, we have n Sn;r,q ψt)t x) 2 ;x ) 29) S n;r,q ψ 2 t);x )) /2 n 2 S n;r,q t x) 4 ;x )) /2

8 266 L. Rempulska, S. Graczyk and, by Corollary 3, lim S n;r,q ψ 2 t);x ) = ψ 2 x) = 0. 30) n From 29), 30) and 6) it follows that lim ns n;r,q ψx)t x) 2 ;x ) = 0, n which used to 28) yields the statement 27). Remark. Theorems -3 show that the approximation properties of the Szász-Mirakyan operators S n;r,q Ω q are independent on r N but these are dependent only on n N and q > 0. References [] M. Becker, Global approximation theorems for Szász-Mirakyan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27, No. 978), [2] M. Becker, D. Kucharski, R.J. Nessel, Global approximation theorems for the Szász-Mirakyan operators in exponential weight spaces, In: Linear Spaces and Approximation, Proc. Conf. Oberwolfach, 977, Birkhäuser Verlag, Basel, ISNM ), [3] A. Ciupa, Approximation by a generalized Szász type operator, J. Comput. Anal. and Applic., 5, No ), [4] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer-Verlag, New-York 987). [5] P. Gupta, V. Gupta, Rate of convergence on Baskakov-Szász type operators, Fasc. Math., 3 200), [6] G.H. Hardy, Divergent Series, Oxford Univ. Press, Oxford 949). [7] K. Knopp, Infinite Series, PWN, Warsaw 956). [8] L. Rempulska, Z. Walczak, Approximation properties of certain modified Szász-Mirakyan operators, Le Matematiche, 55, No. 2000), [9] L. Rempulska, Z. Walczak, Modified Szász-Mirakyan operators, Math. Balcanica, ), [0] L. Rempulska, Sz. Graczyk, Approximation by modified Szász-Mirakyan operators, J. of Inequal. in Pure and Appl. Math., 0, No ), -8.

9 ON CERTAIN CLASS OF SZÁSZ-MIRAKYAN [] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow 960), In Russian. [2] V. Totik, Uniform approximation by Szász-Mirakyan type operators, Acta Math. Hung., 4, No-s: ), [3] R.A. De Vore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, New York 993).

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