In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by means of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem on approximation of absolutely continuous functions by Stancu Beta operators.Отримано точну оцiнку для абсолютного моменту бета-операторiв Станку першого порядку iз використанням формули Стiрлiнга та iнтегральних операцiй. Цю оцiнку використано для встановлення теореми про наближення абсолютно неперервних функцiй бета-операторами Станку

Xiao-Ming Zeng

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.5Xiao-Ming Zeng (Xiamen Univ., China)APPROXIMATION FOR ABSOLUTELY CONTINUOUSFUNCTIONS BY STANCU BETA OPERATORS *НАБЛИЖЕННЯ АБСОЛЮТНО НЕПЕРЕРВНИХ ФУНКЦIЙБЕТА-ОПЕРАТОРАМИ СТАНКУIn this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators bymeans of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem onapproximation of absolutely continuous functions by Stancu Beta operators.Отримано точну оцiнку для абсолютного моменту бета-операторiв Станку першого порядку iз вико-ристанням формули Стiрлiнга та iнтегральних операцiй. Цю оцiнку використано для встановленнятеореми про наближення абсолютно неперервних функцiй бета-операторами Станку.1. Introduction and definitions. For Lebesgue integrable functions f on the intervalI = (0,∞), Stancu Beta operators Ln are defined byLn(f ;x) =1B(nx, n+ 1)∞∫0tnx−1(1 + t)nx+n+1f(t)dt, (1)where B(p, q) is Beta function. The operators Ln were introduced first by Stancu [1]in 1995. Stancu [1] studied some approximation properties of these operators. Abel [2]derived the complete asymptotic expansion for the sequence of these operators. Abel,Gupta and others [3, 4] discussed the rates of convergence of the operators Ln forfunctions of bounded variation and functions with derivatives of bounded variation. Formore related work, one can refer to [5 – 7]. In present paper we first give an exactestimate for the first order absolute moment of the operators Ln, then by means ofthis estimate we establish an approximation theorem of operators Ln for the absolutelycontinuous functions f ∈ ΦDB . The class of functions ΦDB is defined as follows:ΦDB =f | f(x)− f(0) =x∫0φ(t)dt;φ is bounded on every finite subinterval of [0,∞);and f(t) = O(tr) as t→∞.For a bounded function f on every finite subinterval of [0,∞), we introduce the follow-ing metric form:Ωx(f, λ) = supt∈[x−λ, x+λ]∩[0,∞]|f(t)− f(x)|,where x ∈ [0,∞) is fixed, λ ≥ 0.For the properties of Ωx(f, λ), one can refer to reference [8].*This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), theNatural Science Foundation of Fujian Province of China (Grant No. 2010J01012) and the National DefenseBasic Scientific Research Program of China (Grant No. B1420110155).c© XIAO-MING ZENG, 20111570 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS . . . 15712. The first absolute moment of Stancu Beta operators. In this section we derivean exact estimate for the first order absolute moment of Stancu Beta operators. We needthe following lemma.Lemma 1 ([2], Proposition 2). For Stancu Beta operators Ln, we haveLn(1;x) = 1, Ln(t;x) = x, Ln((t− x)2;x) = x(1 + x)n− 1 . (2)Let x ∈ (0,∞) be fixed. For r = 0, 1, 2, . . . , there holdsLn((t− x)r;x) = O(n−b(r+1)/2c). (3)Theorem 1. For Stancu Beta operators Ln, we haveLn(|t− x|;x) = 2xnxn(1 + x)nx+nn!Γ(nx+ n+ 1)Γ(nx). (4)Proof. By direct computation and using Lemma 1, we haveLn(|t− x|;x) ==1B(nx, n+ 1) x∫0(x− t)tnx−1(1 + t)nx+n+1dt+∞∫x(t− x)tnx−1(1 + t)nx+n+1dt ==2B(nx, n+ 1)x∫0(x− t)tnx−1(1 + t)nx+n+1dt+ Ln((t− x);x) ==2B(nx, n+ 1) x∫0xtnx−1(1 + t)nx+n+1dt−x∫0tnx(1 + t)nx+n+1dt.Change of variable and integration by parts derivex∫0xtnx−1(1 + t)nx+n+1dt = xx/(1+x)∫0unx−1(1− u)ndu ==(x1 + x)nx(1(1 + x))nn+x/(1+x)∫0unx(1− u)n−1du, (5)andx∫0tnx(1 + t)nx+n+1dt =x/(1+x)∫0unx(1− u)n−1du. (6)From (5), (6) and simple computation, we obtainLn (|t− x|;x) = 2xnxn(1 + x)nx+nB(nx, n+ 1).ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 111572 XIAO-MING ZENGNote thatB(nx, n+ 1) =n!Γ(nx)Γ(nx+ n+ 1),we obtain (4).Theorem 1 is proved.From Theorem 1 we derive the following inequalities, which are suitable for actualapplications.Proposition 1. For Stancu Beta operators Ln, we have(2x(1 + x)pin)1/2(1− 1 + x12nx)< Ln(|t− x|;x) <(2x(1 + x)pin)1/2. (7)Proof. By Theorem 1 and using Stirling’s formula (cf. [9, 10])Γ(z + 1) =√2piz(z/e)zeθ, (12z + 1)−1 < θ < (12z)−1,we haveLn(|t− x|;x) = 2xnxn(1 + x)nx+nn!Γ(nx+ n+ 1)Γ(nx)=(2x(1 + x)pin)1/2eθ1−θ2−θ3 ,where112(nx+ n) + 1< θ1 <112(nx+ n),112n+ 1< θ2 <112n,112nx+ 1< θ3 <112nx.Set c(θ) = θ1 − θ2 − θ3, simple computation derives−1 + x12nx< c(θ) < 0.Thus we obtain(2x(1 + x)pin)1/2e−(1+x)/(12nx) < Ln(|t− x|;x) <(2x(1 + x)pin)1/2.It follows that(2x(1 + x)pin)1/2(1− 1 + x12nx)< Ln(|t− x|;x) <(2x(1 + x)pin)1/2.Proposition 1 is proved.Corollary 1. For Stancu Beta operators Ln and every x ∈ (0,∞), there holdsLn(|t− x|;x) =(2x(1 + x)pin)1/2+O(n−3/2).3. Approximation for absolutely continuous functions. In this section we studythe rate of convergence of Stancu Beta operators Ln for the functions f ∈ ΦDB . Weneed following two lemmas.ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS . . . 1573Lemma 2 ([3], Lemma 3). Let x ∈ (0,∞) be fixed, the functions Kn,x and Rn,xare defined byKn,x(t) =1B(nx, n+ 1)t∫0unx−1(1 + u)nx+n+1du, (8)Rn,x(t) = 1−Kn,x(t) = 1B(nx, n+ 1)∞∫tunx−1(1 + u)nx+n+1du. (9)Then for n ≥ 2, we haveKn,x(y) ≤ x(1 + x)(n− 1)(x− y)2 , 0 ≤ y < x, (10)Rn,x(z) = 1−Kn,x(z) ≤ x(1 + x)(n− 1)(z − x)2 , x < z <∞. (11)Lemma 3. Let r > 0 and m be a positive integer satisfying m > r/2, we have1B(nx, n+ 1)∞∫2xtrtnx−1(1 + t)nx+n+1dt =(2x)rx2mO(n−bm+1/2c). (12)Proof. For m > r/2 and t ∈ [2x,∞)ddttr(t− x)2m =tr−1(t− x)2m−1(rt− 2mt− rx)(t− x)4m < 0,which implies thattr(t− x)2m is monotone decreasing for t ∈ [2x,∞). Thus1B(nx, n+ 1)∞∫2xtrtnx−1(1 + t)nx+n+1dt ≤≤ 1B(nx, n+ 1)∞∫2x(2x)rx2m(t− x)2m tnx−1(1 + t)nx+n+1dt ≤≤ (2x)rx2mLn((t− x)2m;x).Equation (12) now follows from Lemma 1.Now we state the main result of this section.Theorem 2. Let f be a function in ΦDB . If φ(x+) and φ(x−) exist at a fixedpoint x ∈ (0,∞), write τ = φ(x+)− φ(x−)2, then for n ≥ 2 we have∣∣∣∣∣Ln(f ;x)− f(x)− τ(2x(1 + x)npi)1/2∣∣∣∣∣ ≤ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 111574 XIAO-MING ZENG≤ 3 + 7xn− 1[√n ]∑k=1Ωx(φx, x/k) +(1 + x)3/2√72xpin−3/2 +(2x)rx2mO(n−b(m+1)/2c), (13)where m is a positive integer satisfying m > r/2, andφx(t) =φ(t)− φ(x+), x < t ≤ 1,0, t = x,φ(t)− φ(x−), 0 ≤ t < x.(14)Proof. By direct computation we find thatLn(f ;x)− f(x) = φ(x+)− φ(x−)2Ln(|t− x|;x)−An,x(φx)++Bn,x(φx) +Dn,x(φx), (15)whereAn,x(φx) =x∫0 x∫tφx(u)du dtKn,x(t),Bn,x(φx) =2x∫x t∫xφx(u)du dtKn,x(t),Dn,x(φx) =∞∫2x t∫xφx(u)du dtKn(x, t),and Kn,x(t) is defined in (8).Integration by parts derivesAn,x(φx) =x∫0 x∫tφx(u)du dtKn,x(t) ==x∫tφx(u)duKn,x(t)∣∣∣∣∣∣x0+x∫0Kn,x(t)φx(t)dt =x∫0Kn,x(t)φx(t)dt == x−x/√n∫0+x∫x−x/√nKn,x(t)φx(t)dt.Note that Kn,x(t) ≤ 1 and φx(x) = 0, by monotonicity of Ωx (φx, λ) it follows that∣∣∣∣∣∣∣x∫x−x/√nKn,x(t)φx(t)dt∣∣∣∣∣∣∣ ≤x√nΩx(φx,x√n)≤ 2xn[√n ]∑k=1Ωx(φx, x/k).ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS . . . 1575On the other hand, by inequality (10) and using change of variable t = x − x/u, wehave ∣∣∣∣∣∣∣x−x/√n∫0Kn,x(t)φx(t)dt∣∣∣∣∣∣∣ ≤x(1 + x)n− 1x−x/√n∫0Ωx(φx, x− t)(x− t)2 dt ==1 + xn− 1√n∫1Ωx(φx, x/u)du ≤ 1 + xn− 1[√n ]∑k=1Ωx(φx, x/k).Thus, it follows that|An,x(φx)| ≤ 1 + 3xn− 1[√n ]∑k=1Ωx(φx, x/k). (16)Next we estimate |Bn,x(φx)|Bn,x(φx) =2x∫x t∫xφx(u)du dtKn,x(t) = − 2x∫x t∫xφx(u)du dtRn,x(t) == −t∫xφx(u)du ·Rn,x(t)∣∣∣∣∣∣2xx+2x∫xφx(t)Rn,x(t)dt == −2x∫xφx(u)du ·Rn,x(2x) +∫ 2xxφx(t)Rn,x(t)dt. (17)By Lemma 2∣∣∣∣∣∣−2x∫xφx(u)du ·Rn,x(2x)∣∣∣∣∣∣ ≤ xΩx(φx, x) x(1 + x)(n− 1)x2 = (1 + x)n− 1 Ωx(φx, x). (18)On the other hand, similar to the estimate of |An,x(φx)|, we have∣∣∣∣∣∣2x∫xφx(t)Rn(x, t)dt∣∣∣∣∣∣ ≤ 1 + 3xn− 1[√n ]∑k=1Ωx(φx, x/k).For estimate of |Dn,x(φx)|, note that f(t) = O(tr), thus there exists a constant M suchthat|Dn,x(φx)| ≤M∞∫2xtrtnx−1(1 + t)nx+n+1dt.Using Lemma 3 we obtain|Dn,x(φx)| = (2x)rx2mO(n−b(m+1)/2c). (19)Theorem 2 now follows from (15) – (19) combining with Proposition 1 and some simplecomputations.ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 111576 XIAO-MING ZENGRemark. If f is a function with derivative of bounded variation, then f ∈ ΦDB .Thus the approximation of functions with derivatives of bounded variation is a specialcase of Theorem 2. In this special case Theorem 2 is better than a result of Gupta, Abeland Ivan in [4].1. Stancu D. D. On the Beta approximation operators of second kind // Rev. Anal. Nume´r. The´or. Approxim.– 1995. – 2. – P. 231 – 239.2. Abel U. Asymptotic approximation with Stancu Beta operators // Rev. Anal. Nume´r. The´or. Approxim.– 1998. – 27. – P. 5 – 13.3. Abel U., Gupta V. Rate of convergence of Stancu Beta operators for functions of bounded variation //Rev. Anal. Nume´r. The´or. Approxim. – 2004. – 33. – P. 3 – 9.4. Gupta V., Abel U., Ivan M. Rate of convergence of Beta operators of second kind for functions withderivatives of bounded variation // Int. J. Math. and Math. Sci. – 2005. – 23. – P. 3827 – 3833.5. Bojanic R., Cheng F. Rate of convergence of Bernstein polynomials for functions with derivatives ofbounded variation // J. Math. Anal. and Appl. – 1989. – 141. – P. 136 – 151.6. Cheng F. On the rate of convergence of Bernstein polynomials of functions of bounded variation // J.Approxim. Theory. – 1983. – 39. – P. 259 – 274.7. Pych-Taberska P. Some properties of the Be´zier – Kantorovich type operators // J. Approxim. Theory. –2003. – 123. – P. 256 – 269.8. Zeng X. M. Approximation properties of Gamma operators // J. Math. Anal. and Appl. – 2005. – 311. –P. 389 – 401.9. Mermin N. D. Stiring’s formula // Amer. J. Phys. – 1984. – 52. – P. 362 – 365.10. Namias V. A simple derivation of Stiring’s asymptotic series // Amer. Math. Monthly. – 1986. – 93. –P. 25 – 29.Received 17.03.11ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11

Approximation for absolutely continuous functions by Stancu Beta operators

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Approximation for absolutely continuous functions by Stancu Beta operators

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)