Approximation by the 𝑞-Szász-Mirakjan Operators

Abstract

This paper deals with approximating properties of the q-generalization of the Szász-Mirakjan operators in the case 𝑞>1. Quantitative estimates of the convergence in the polynomial-weighted spaces and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by the q-Szász-Mirakjan operators (𝑞>1 ) is of order 𝑞^−𝑛 versus 1/n for the classical Szász-Mirakjan operators.

1. Introduction

The approximation of functions by using linear positive operators introduced via 𝑞-Calculus is currently under intensive research. The pioneer work has been made by Lupaş [1] and Phillips [2] who proposed generalizations of Bernstein polynomials based on the 𝑞-integers. The 𝑞-Bernstein polynomials quickly gained the popularity, see [3–11]. Other important classes of discrete operators have been investigated by using 𝑞-Calculus in the case 0<𝑞<1, for example, 𝑞-Meyer-König operators [12–14], 𝑞-Bleimann, Butzer and Hahn operators [15–17], 𝑞-Szász-Mirakjan operators [18–21], and 𝑞-Baskakov operators [22, 23].

In the present paper, we introduce a 𝑞-generalization of the Szász operators in the case 𝑞>1. Notice that different 𝑞-generalizations of Szász-Mirakjan operators were introduced and studied by Aral and Gupta [18, 19], by Radu [20], and by Mahmudov [21] in the case 0<𝑞<1. Since we define 𝑞-Szász-Mirakjan operators for 𝑞>1, the rate of approximation by the 𝑞-Szász-Mirakjan operators (𝑞>1) is of order 𝑞^−𝑛, which is essentially better than 1/𝑛 (rate of approximation for the classical Szász-Mirakjan operators). Thus our 𝑞-Szász-Mirakjan operators have better approximation properties than the classical Szász-Mirakjan operators and the other 𝑞-Szász-Mirakjan operators.

The paper is organized as follows. In Section 2, we give standard notations that will be used throughout the paper, introduce 𝑞-Szász-Mirakjan operators, and evaluate the moments of 𝑀_𝑛,𝑞. In Section 3 we study convergence properties of the 𝑞-Szász-Mirakjan operators in the polynomial-weighted spaces. In Section 4, we give the quantitative Voronovskaja-type asymptotic formula.

2. Construction of 𝑀_𝑛,𝑞 and Estimation of Moments

Throughout the paper we employ the standard notations of 𝑞-calculus, see [24, 25].

𝑞-integer and 𝑞-factorial are defined by [𝑛]_𝑞∶=1−𝑞^𝑛,1−𝑞if𝑞∈𝑅⁺⧵{1},𝑛,if𝑞=1,for[0][𝑛]𝑛∈ℕ,=0,_𝑞!∶=[1]_𝑞[2]_𝑞…[𝑛]_𝑞,for𝑛∈ℕ,[0]!=1. (2.1) For integers 0≤𝑘≤𝑛  𝑞-binomial is defined by 𝑛𝑘_𝑞[𝑛]∶=^𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!. (2.2) The 𝑞-derivative of a function 𝑓(𝑥), denoted by 𝐷_𝑞𝑓, is defined by 𝐷_𝑞𝑓(𝑥)𝑓(𝑥)∶=𝑞𝑥−𝑓𝑥𝐷𝑞−1,𝑥≠0,_𝑞𝑓(0)∶=lim_𝑥→0𝐷_𝑞𝑓(𝑥). (2.3) The formula for the 𝑞-derivative of a product and quotient are 𝐷_𝑞(𝑢(𝑥)𝑣(𝑥))=𝐷_𝑞(𝑢(𝑥))𝑣(𝑥)+𝑢𝑞𝑥𝐷_𝑞(𝑣(𝑥)). (2.4) Also, it is known that 𝐷_𝑞𝑥^𝑛=[𝑛]_𝑞𝑥^𝑛−1,𝐷_𝑞𝐸(𝑎𝑥)=𝑎𝐸𝑞𝑎𝑥. (2.5) If |𝑞|>1, or 0<|𝑞|<1 and |𝑧|<1/(1−𝑞), the 𝑞-exponential function 𝑒_𝑞(𝑥) was defined by Jackson 𝑒_𝑞(𝑧)∶=^∞_𝑘=0𝑧^𝑘[𝑘]_𝑞!. (2.6) If |𝑞|>1, 𝑒_𝑞(𝑧) is an entire function and 𝑒_𝑞(𝑧)=^∞_𝑗=0𝑧1+𝑞−1𝑞^𝑗+1,𝑞>1. (2.7) There is another 𝑞-exponential function which is entire when 0<|𝑞|<1 and which converges when |𝑧|<1/|1−𝑞| if |𝑞|>1. To obtain it we must invert the base in (2.6), that is, 𝑞→1/𝑞: 𝐸_𝑞(𝑧)∶=𝑒_1/𝑞(𝑧)=^∞_𝑘=0𝑞^{𝑘(𝑘−1)/2}𝑧^𝑘[𝑘]_𝑞!. (2.8) We immediately obtain from (2.7) that 𝐸_𝑞(𝑧)=^∞_𝑗=01+1−𝑞𝑧𝑞^𝑗,0<𝑞<1. (2.9) The 𝑞-difference equations corresponding to 𝑒_𝑞(𝑧) and 𝐸_𝑞(𝑧) are 𝐷_𝑞𝑒_𝑞(𝑎𝑧)=𝑎𝑒_𝑞𝑞𝑧,𝐷_𝑞𝐸_𝑞(𝑎𝑧)=𝑎𝐸_𝑞,𝐷𝑞𝑎𝑧_1/𝑞𝑒_𝑞(𝑧)=𝐷_1/𝑞𝐸_1/𝑞(𝑧)=𝐸_1/𝑞𝑞⁻¹𝑧=𝑒_𝑞𝑞⁻¹𝑧,𝑞≠0. (2.10) Let 𝐶_𝑝 be the set of all real valued functions 𝑓, continuous on [0,∞), such that 𝑤_𝑝𝑓 is uniformly continuous and bounded on [0,∞) endowed with the norm 𝑓_𝑝∶=sup_{[)𝑥∈0,∞}𝑤_𝑝(𝑥)𝑓(𝑥). (2.11) Here 𝑤₀(𝑥)∶=1,𝑤_𝑝(𝑥)∶=1+𝑥^𝑝−1,if𝑝∈ℕ. (2.12) The corresponding Lipschitz classes are given for 0<𝛼≤2 by Δ²_ℎ𝑓(𝑥)()()(𝑥),𝜔∶=𝑓𝑥+2ℎ−2𝑓𝑥+ℎ+𝑓_2𝑝𝑓;𝛿∶=sup_0<ℎ≤𝛿Δ²_ℎ𝑓_𝑝,Lip²_𝑝𝛼∶=𝑓∈𝐶_𝑝∶𝜔²_𝑝𝑓;𝛿=0𝛿^𝛼,𝛿→0⁺. (2.13)

Now we introduce the 𝑞-parametric Szász-Mirakjan operator.

Definition 2.1. Let 𝑞>1 and 𝑛∈ℕ. For 𝑓∶[0,∞)→𝑅 one defines the Szász-Mirakjan operator based on the 𝑞-integers 𝑀_𝑛,𝑞𝑓;𝑥∶=^∞_𝑘=0𝑓[𝑘]_𝑞[𝑛]_𝑞1𝑞^{𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑥. (2.14)

Similarly as a classical Szász-Mirakjan operator 𝑆_𝑛, the operator 𝑀_𝑛,𝑞 is linear and positive. Furthermore, in the case of 𝑞→1⁺ we obtain classical Szász-Mirakjan operators.

Moments 𝑀_𝑛,𝑞(𝑡^𝑚;𝑥) are of particular importance in the theory of approximation by positive operators. From (2.14) one easily derives the following recurrence formula and explicit formulas for moments 𝑀_𝑛,𝑞(𝑡^𝑚;𝑥),  𝑚=0,1,2,3,4.

Lemma 2.2. Let 𝑞>1. The following recurrence formula holds 𝑀_𝑛,𝑞𝑡^𝑚+1;𝑥=^𝑚_𝑗=0𝑚𝑗𝑥𝑞^𝑗[𝑛]_𝑞^𝑚−𝑗𝑀_𝑛,𝑞𝑡^𝑗;𝑞⁻¹𝑥. (2.15)

Proof. The recurrence formula (2.15) easily follows from the definition of 𝑀_𝑛,𝑞 and 𝑞[𝑘]_𝑞+1=[𝑘+1]_𝑞 as show below: 𝑀_𝑛,𝑞𝑡^𝑚+1=;𝑥_∞𝑘=0[𝑘]_𝑞^𝑚+1[𝑛]_𝑞^𝑚+11𝑞^{𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑥=^∞_𝑘=1[𝑘]^𝑚_𝑞[𝑛]^𝑚_𝑞1𝑞^{𝑘(𝑘−1)/2}[𝑛]_𝑞^𝑘−1𝑥^𝑘[𝑘−1]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑥=^∞_𝑘=0𝑞[𝑘]_𝑞+1^𝑚[𝑛]^𝑚_𝑞1𝑞^{𝑘(𝑘+1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘+1[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑞⁻¹𝑥=^∞_𝑘=01[𝑛]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑘]^𝑗_𝑞1𝑞^{𝑘(𝑘+1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘+1[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑞⁻¹𝑥=^𝑚_𝑗=0𝑚𝑗𝑥𝑞^𝑗[𝑛]_𝑞^{∞𝑚−𝑗}_𝑘=0[𝑘]^𝑗_𝑞[𝑛]^𝑗_𝑞1𝑞^{𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘[𝑘]_𝑞!𝑞^𝑘𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑞⁻¹𝑥=^𝑚_𝑗=0𝑚𝑗𝑥𝑞^𝑗[𝑛]_𝑞^𝑚−𝑗𝑀_𝑛,𝑞𝑡^𝑗;𝑞⁻¹𝑥. (2.16)

Lemma 2.3. The following identities hold for all 𝑞>1,  𝑥∈[0,∞), 𝑛∈ℕ, and 𝑘≥0: 𝑥𝐷_𝑞𝑠_𝑛𝑘𝑞;𝑥=[𝑛]_𝑞[𝑘]_𝑞[𝑛]_𝑞−𝑥𝑠_𝑛𝑘,𝑀𝑞;𝑥_𝑛,𝑞𝑡^𝑚+1=𝑥;𝑥[𝑛]_𝑞𝐷_𝑞𝑀_𝑛,𝑞𝑡^𝑚;𝑥+𝑥𝑀_𝑛,𝑞𝑡^𝑚;𝑥, (2.17) where 𝑠_𝑛𝑘(𝑞;𝑥)∶=(1/𝑞^{𝑘(𝑘−1)/2})([𝑛]^𝑘_𝑞𝑥^𝑘/[𝑘]_𝑞!)𝑒_𝑞(−[𝑛]_𝑞𝑞^−𝑘𝑥).

Proof. The first identitiy follows from the following simple calculations 𝑥𝐷_𝑞𝑠_𝑛𝑘=[𝑘]𝑞;𝑥_𝑞1𝑞^{𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]𝑞^−𝑘𝑥−𝑥𝑞^−𝑘[𝑛]_𝑞1𝑞^{𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑞^𝑘𝑥^𝑘[𝑘]_𝑞!𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑥=[𝑘]_𝑞𝑠_𝑛𝑘𝑞;𝑥−𝑥[𝑛]_𝑞𝑠_𝑛𝑘𝑞;𝑥=[𝑛]_𝑞[𝑘]_𝑞[𝑛]_𝑞−𝑥𝑠_𝑛𝑘𝑞;𝑥. (2.18) The second one follows from the first: 𝑥𝐷_𝑞𝑀_𝑛,𝑞𝑡^𝑚;𝑥=[𝑛]_𝑞^∞_𝑘=0[𝑘]_𝑞[𝑛]_𝑞^𝑚[𝑘]_𝑞[𝑛]_𝑞−𝑥𝑠_𝑛𝑘=[𝑛]𝑞;𝑥_{𝑞∞𝑘=0}[𝑘]_𝑞[𝑛]_𝑞^𝑚+1𝑠_𝑛𝑘𝑞;𝑥−[𝑛]_𝑞𝑥^∞_𝑘=0[𝑘]_𝑞[𝑛]_𝑞^𝑚𝑠_𝑛𝑘=[𝑛]𝑞;𝑥_𝑞𝑀_𝑛,𝑞𝑡^𝑚+1;𝑥−[𝑛]_𝑞𝑥𝑀_𝑛,𝑞𝑡^𝑚;𝑥. (2.19)

Lemma 2.4. Let 𝑞>1. One has 𝑀_𝑛,𝑞(1;𝑥)=1,𝑀_𝑛,𝑞(𝑡;𝑥)=𝑥,𝑀_𝑛,𝑞𝑡²;𝑥=𝑥²+1[𝑛]_𝑞𝑀𝑥,_𝑛,𝑞𝑡³;𝑥=𝑥³+2+𝑞[𝑛]_𝑞𝑥²+1[𝑛]²_𝑞𝑀𝑥,_𝑛,𝑞𝑡⁴;𝑥=𝑥⁴+3+2𝑞+𝑞²𝑥³[𝑛]_𝑞+3+3𝑞+𝑞²𝑥²[𝑛]²_𝑞+1[𝑛]³_𝑞𝑥. (2.20)

Proof. For a fixed 𝑥∈𝑅₊, by the 𝑞-Taylor theorem [24], we obtain 𝜑_𝑛(𝑡)=^∞_𝑘=0(𝑡−𝑥)^𝑘_1/𝑞[𝑘]_1/𝑞!𝐷^𝑘_1/𝑞𝜑_𝑛(𝑥). (2.21) Choosing 𝑡=0 and taking into account (−𝑥)^𝑘_1/𝑞=(−1)^𝑘𝑥^𝑘𝑞^{−𝑘(𝑘−1)/2},𝐷^𝑘_1/𝑞𝑒_𝑞−[𝑛]_𝑞𝑥=(−1)^𝑘𝑞^{−𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑥 (2.22) we get for 𝜑_𝑛(𝑥)=𝑒_𝑞(−[𝑛]_𝑞𝑥) that 1=𝜑_𝑛(0)=^∞_𝑘=0(−1)^𝑘𝑥^𝑘𝑞^{𝑘(𝑘−1)/2}[𝑘]_1/𝑞!𝐷^𝑘_1/𝑞𝜑_𝑛(𝑥)=^∞_𝑘=0(−1)^𝑘𝑥^𝑘[𝑘]_𝑞!(−1)^𝑘𝑞^{−𝑘(𝑘−1)/2}[𝑛]^𝑘_𝑞𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑥=^∞_𝑘=0[𝑛]^𝑘_𝑞𝑥^𝑘[𝑘]_𝑞!𝑞^{𝑘(𝑘−1)/2}𝑒_𝑞−[𝑛]_𝑞𝑞^−𝑘𝑧. (2.23) In other words 𝑀_𝑛,𝑞(1;𝑥)=1.
Calculation of 𝑀_𝑛,𝑞(𝑡^𝑖;𝑥), 𝑖=1,2,3,4, based on the recurrence formula (2.17) (or (2.15)). We only calculate 𝑀_𝑛,𝑞(𝑡³;𝑥) and 𝑀_𝑛,𝑞(𝑡⁴;𝑥): 𝑀_𝑛,𝑞𝑡³=𝑥;𝑥[𝑛]_𝑞𝐷_𝑞𝑀_𝑛,𝑞𝑡²;𝑥+𝑥𝑀_𝑛,𝑞𝑡²=𝑥;𝑥[𝑛]_𝑞[2]_𝑞1𝑥+[𝑛]_𝑞+𝑥𝑥²+1[𝑛]_𝑞𝑥=1[𝑛]²_𝑞𝑥+2+𝑞[𝑛]_𝑞𝑥²+𝑥³,𝑀_𝑛,𝑞𝑡⁴=𝑥;𝑥[𝑛]_𝑞𝐷_𝑞𝑀_𝑛,𝑞𝑡³;𝑥+𝑥𝑀_𝑛,𝑞𝑡³=𝑥;𝑥[𝑛]_𝑞1[𝑛]²_𝑞+2+𝑞[𝑛]_𝑞[2]_𝑞𝑥+[3]_𝑞𝑥²1+𝑥[𝑛]²_𝑞𝑥+2+𝑞[𝑛]_𝑞𝑥²+𝑥³=1[𝑛]³_𝑞𝑥+3+3𝑞+𝑞²𝑥²[𝑛]²_𝑞+3+2𝑞+𝑞²𝑥³[𝑛]_𝑞+𝑥⁴. (2.24)

Lemma 2.5. Assume that 𝑞>1. For every 𝑥∈[0,∞) there hold 𝑀_𝑛,𝑞(𝑡−𝑥)²=𝑥;𝑥[𝑛]_𝑞,𝑀_𝑛,𝑞(𝑡−𝑥)³=1;𝑥[𝑛]²_𝑞𝑥𝑥+𝑞−1²[𝑛]_𝑞,𝑀_𝑛,𝑞(𝑡−𝑥)⁴=1;𝑥[𝑛]³_𝑞𝑥+𝑞²𝑥+3𝑞−1²[𝑛]²_𝑞+𝑞−1²𝑥³[𝑛]_𝑞. (2.25)(2.26)(2.27)

Proof. First of all we give an explicit formula for 𝑀_𝑛,𝑞((𝑡−𝑥)⁴;𝑥). 𝑀_𝑛,𝑞(𝑡−𝑥)³;𝑥=𝑀_𝑛,𝑞𝑡³;𝑥−3𝑥𝑀_𝑛,𝑞𝑡²;𝑥+3𝑥²𝑀_𝑛,𝑞(𝑡;𝑥)−𝑥³=𝑥³+2+𝑞[𝑛]_𝑞𝑥²+1[𝑛]²_𝑞𝑥−3𝑥𝑥²+𝑥[𝑛]_𝑞+3𝑥³−𝑥³=1[𝑛]²_𝑞𝑥𝑥+𝑞−1²[𝑛]_𝑞,𝑀_𝑛,𝑞(𝑡−𝑥)⁴;𝑥=𝑀_𝑛,𝑞𝑡⁴;𝑥−4𝑥𝑀_𝑛,𝑞𝑡³;𝑥+6𝑥²𝑀_𝑛,𝑞𝑡²;𝑥−4𝑥³𝑀_𝑛,𝑞(𝑡;𝑥)+𝑥⁴=1[𝑛]³_𝑞𝑥+3+3𝑞+𝑞²𝑥²[𝑛]²_𝑞+3+2𝑞+𝑞²𝑥³[𝑛]_𝑞+𝑥⁴1−4𝑥[𝑛]²_𝑞𝑥+2+𝑞[𝑛]_𝑞𝑥²+𝑥³+6𝑥²𝑥²+𝑥[𝑛]_𝑞−4𝑥⁴+𝑥⁴=1[𝑛]³_𝑞𝑥+−1+3𝑞+𝑞²𝑥²[𝑛]²_𝑞+𝑞−1²𝑥³[𝑛]_𝑞. (2.28)

Now we prove explicit formula for the moments 𝑀_𝑛,𝑞(𝑡^𝑚;𝑥), which is a 𝑞-analogue of a result of Becker, see [26, Lemma 3].

Lemma 2.6. For 𝑞>1,  𝑚∈ℕ there holds 𝑀_𝑛,𝑞𝑡^𝑚;𝑥=^𝑚_𝑗=1𝕊_𝑞𝑥𝑚,𝑗^𝑗[𝑛]_𝑞^𝑚−𝑗, (2.29) where 𝕊_𝑞𝑚+1,𝑗=𝑗𝕊_𝑞𝑚,𝑗+𝕊_𝑞𝕊𝑚,𝑗−1,𝑚≥0,𝑗≥1,_𝑞(0,0)=1,𝕊_𝑞(𝑚,0)=0,𝑚>0,𝕊_𝑞𝑚,𝑗=0,𝑚<𝑗. (2.30) In particular 𝑀_𝑛,𝑞(𝑡^𝑚;𝑥) is a polynomial of degree 𝑚 without a constant term.

Proof. Because of 𝑀_𝑛,𝑞(𝑡;𝑥)=𝑥, 𝑀_𝑛,𝑞(𝑡²;𝑥)=𝑥²+𝑥/[𝑛]_𝑞, the representation (2.29) holds true for 𝑚=1,2 with 𝕊_𝑞(2,1)=1,  𝕊_𝑞(1,1)=1.
Now assume (2.29) to be valued for 𝑚 then by Lemma 2.3 we have 𝑀_𝑛,𝑞𝑡^𝑚+1=𝑥;𝑥[𝑛]_𝑞𝐷_𝑞𝑀_𝑛,𝑞𝑡^𝑚;𝑥+𝑥𝑀_𝑛,𝑞𝑡^𝑚=𝑥;𝑥[𝑛]_𝑞^𝑚_𝑗=1𝑗_𝑞𝕊_𝑞𝑥𝑚,𝑗^𝑗−1[𝑛]_𝑞^𝑚−𝑗+𝑥^𝑚_𝑗=1𝕊_𝑞𝑥𝑚,𝑗^𝑗[𝑛]_𝑞^𝑚−𝑗=^𝑚_𝑗=1𝑗_𝑞𝕊_𝑞𝑥𝑚,𝑗^𝑗[𝑛]_𝑞^{𝑚−𝑗+1}+^𝑚_𝑗=1𝕊_𝑞𝑥𝑚,𝑗^𝑗+1[𝑛]_𝑞^𝑚−𝑗=𝑥[𝑛]^𝑚𝕊_𝑞(𝑚,1)+𝑥^𝑚+1𝕊_𝑞()+𝑚,𝑚_𝑚𝑗=2𝑗_𝑞𝕊_𝑞𝑚,𝑗+𝕊_𝑞𝑥𝑚,𝑗−1^𝑗[𝑛]_𝑞^{𝑚−𝑗+1}. (2.31)

Remark 2.7. Notice that 𝕊_𝑞(𝑚,𝑗) are Stirling numbers of the second kind introduced by Goodman et al. in [8]. For 𝑞=1 the formulae (2.30) become recurrence formulas satisfied by Stirling numbers of the second type.

3. 𝑀_𝑛,𝑞 in Polynomial-Weighted Spaces

Lemma 3.1. Let 𝑝∈ℕ∪{0} and 𝑞∈(1,∞) be fixed. Then there exists a positive constant 𝐾₁(𝑞,𝑝) such that 𝑀_𝑛,𝑞1/𝑤_𝑝;𝑥_𝑝≤𝐾₁𝑞,𝑝,𝑛∈ℕ. (3.1) Moreover for every 𝑓∈𝐶_𝑝 one has 𝑀_𝑛,𝑞𝑓_𝑝≤𝐾₁𝑞,𝑝𝑓_𝑝,𝑛∈ℕ. (3.2) Thus 𝑀_𝑛,𝑞 is a linear positive operator from 𝐶_𝑝 into 𝐶_𝑝 for any 𝑝∈ℕ∪{0}.

Proof. The inequality (3.1) is obvious for 𝑝=0. Let 𝑝≥1. Then by (2.29) we have 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞1/𝑤_𝑝;𝑥=𝑤_𝑝(𝑥)+𝑤_𝑝(𝑥)^𝑝_𝑗=1𝕊_𝑞𝑥𝑝,𝑗^𝑗[𝑛]_𝑞^𝑝−𝑗≤𝐾₁𝑞,𝑝, (3.3)𝐾₁(𝑞,𝑝) is a positive constant depending on 𝑝 and 𝑞. From this follows (3.1). On the other hand 𝑀_𝑛,𝑞𝑓_𝑝≤𝑓_𝑝𝑀_𝑛,𝑞1𝑤_𝑝𝑝, (3.4) for every 𝑓∈𝐶_𝑝. By applying (3.1), we obtain (3.2).

Lemma 3.2. Let 𝑝∈ℕ∪{0} and 𝑞∈(1,∞) be fixed. Then there exists a positive constant 𝐾₂(𝑞,𝑝) such that 𝑀_𝑛,𝑞(𝑡−⋅)²𝑤_𝑝(𝑡);⋅_𝑝≤𝐾₂𝑞,𝑝[𝑛]_𝑞,𝑛∈ℕ. (3.5)

Proof. The formula (2.25) imply (3.5) for 𝑝=0. We have 𝑀_𝑛,𝑞(𝑡−𝑥)²𝑤_𝑝(𝑡);𝑥=𝑀_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝑀_𝑛,𝑞(𝑡−𝑥)²𝑡^𝑝;𝑥, (3.6) for 𝑝,𝑛∈ℕ. If 𝑝=1 then we get 𝑀_𝑛,𝑞(𝑡−𝑥)²(1+𝑡);𝑥=𝑀_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝑀_𝑛,𝑞(𝑡−𝑥)²𝑡;𝑥=𝑀_𝑛,𝑞(𝑡−𝑥)³;𝑥+(1+𝑥)𝑀_𝑛,𝑞(𝑡−𝑥)²;𝑥, (3.7) which by Lemma 2.5 yields (3.5) for 𝑝=1.
Let 𝑝≥2. By applying (2.29), we get 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞(𝑡−𝑥)²𝑡^𝑝;𝑥=𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑡^𝑝+2;𝑥−2𝑥𝑀_𝑛,𝑞𝑡^𝑝+1;𝑥+𝑥²𝑀_𝑛,𝑞𝑡^𝑝;𝑥=𝑤_𝑝(𝑥)𝑥^𝑝+2+^𝑝+1_𝑗=1𝕊_𝑞𝑥𝑝+2,𝑗^𝑗[𝑛]_𝑞^{𝑝+2−𝑗}−2𝑥^𝑝+2−2^𝑝_𝑗=1𝕊_𝑞𝑥𝑝+1,𝑗^𝑗+1[𝑛]_𝑞^{𝑝+1−𝑗}+𝑥^𝑝+2+^𝑝−1_𝑗=1𝕊_𝑞𝑥𝑝,𝑗^𝑗+2[𝑛]_𝑞^𝑝−𝑗=𝑤_𝑝(𝑥)^𝑝_𝑗=2𝕊_𝑞𝑝+2,𝑗−2𝕊_𝑞𝑝+1,𝑗+𝕊_𝑞𝑥𝑝,𝑗^𝑗+1[𝑛]_𝑞^{𝑝+1−𝑗}+𝕊_𝑞𝑥𝑝+2,1[𝑛]_𝑞^𝑝+1+𝕊_𝑞𝑝+2,2−2𝕊_𝑞𝑥𝑝+2,1²[𝑛]^𝑝_𝑞=𝑤_𝑝(𝑥)𝑥[𝑛]_𝑞𝒫_𝑝𝑞;𝑥, (3.8) where 𝒫_𝑝(𝑞;𝑥) is a polynomial of degree 𝑝. Therefore one has 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞(𝑡−𝑥)²𝑡^𝑝;𝑥≤𝐾₂𝑥𝑞,𝑝[𝑛]_𝑞. (3.9)

Our first main result in this section is a local approximation property of 𝑀_𝑛,𝑞 stated below.

Theorem 3.3. There exists an absolute constant 𝐶>0 such that 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑔;𝑥−𝑔(𝑥)≤𝐾₃𝑞,𝑝𝑔^′′𝑥[𝑛]_𝑞, (3.10) where 𝑔∈𝐶²_𝑝, 𝑞>1 and 𝑥∈[0,∞).

Proof. Using the Taylor formula 𝑔(𝑡)=𝑔(𝑥)+𝑔^′(𝑥)(𝑡−𝑥)+^𝑡_𝑥^𝑠_𝑥𝑔^′′(𝑢)𝑑𝑢𝑑𝑠,𝑔∈𝐶²_𝑝, (3.11) we obtain that 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑔;𝑥−𝑔(𝑥)=𝑤_𝑝(𝑥)𝑀_𝑛,𝑞^𝑡_𝑥^𝑠_𝑥𝑔^′′(𝑢)𝑑𝑢𝑑𝑠;𝑥≤𝑤_𝑝(𝑥)𝑀_𝑛,𝑞^𝑡_𝑥^𝑠_𝑥𝑔^′′(𝑢)𝑑𝑢𝑑𝑠;𝑥≤𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑔^′′_𝑝^𝑡_𝑥^𝑠_𝑥1+𝑢^𝑚𝑑𝑢𝑑𝑠;𝑥≤𝑤_𝑝(𝑥)12𝑔^′′_𝑝𝑀_𝑛,𝑞(𝑡−𝑥)²1/𝑤_𝑝(𝑥)+1/𝑤_𝑝(𝑡)≤1;𝑥2𝑔^′′_𝑝𝑀_𝑛,𝑞(𝑡−𝑥)²;𝑥+𝑤_𝑝(𝑥)𝑀_𝑛,𝑞(𝑡−𝑥)²𝑤_𝑝(𝑡);𝑥≤𝐾₃𝑞,𝑥𝑔^′′_𝑝𝑥[𝑛]_𝑞. (3.12)

Now we consider the modified Steklov means 𝑓_ℎ(𝑥)4∶=ℎ^20ℎ/22𝑓(𝑥+𝑠+𝑡)−𝑓(𝑥+2(𝑠+𝑡))𝑑𝑠𝑑𝑡. (3.13)𝑓_ℎ(𝑥) has the following properties: 𝑓(𝑥)−𝑓_ℎ(𝑥)=4ℎ^20ℎ/2Δ²_𝑠+𝑡𝑓(𝑥)𝑑𝑠𝑑𝑡,𝑓_ℎ^′′(𝑥)=ℎ⁻²8Δ²_ℎ/2𝑓(𝑥)−Δ²_ℎ𝑓(𝑥) (3.14) and therefore 𝑓−𝑓_ℎ𝑝≤𝜔²_𝑝𝑓;ℎ,𝑓_ℎ^′′_𝑝≤19ℎ²𝜔²_𝑝𝑓;ℎ. (3.15) We have the following direct approximation theorem.

Theorem 3.4. For every 𝑝∈ℕ∪{0},𝑓∈𝐶_𝑝 and 𝑥∈[0,∞),  𝑞>1, one has 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑀_𝑝𝜔²_𝑝𝑓;𝑥[𝑛]_𝑞=𝑀_𝑝𝜔²_𝑝𝑓;𝑞−1𝑥𝑞^𝑛.−1 (3.16) Particularly, if Lip²_𝑝𝛼 for some 𝛼∈(0,2], then 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑀_𝑝𝑥[𝑛]_𝑞^𝛼/2. (3.17)

Proof. For 𝑓∈𝐶_𝑝 and ℎ>0𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑀_𝑛,𝑞𝑓−𝑓_ℎ;𝑥−𝑓−𝑓_ℎ(𝑥)+𝑀_𝑛,𝑞𝑓_ℎ;𝑥−𝑓_ℎ(𝑥) (3.18) and therefore 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑓−𝑓_ℎ𝑝𝑤_𝑝(𝑥)𝑀_𝑛,𝑞1𝑤_𝑝(𝑡);𝑥+1+𝐾₃𝑞,𝑝𝑓_ℎ^′′_𝑝𝑥[𝑛]_𝑞. (3.19) Since 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞(1/𝑤_𝑝(𝑡);𝑥)≤𝐾₁(𝑞,𝑝), we get that 𝑤_𝑝(𝑥)𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝑀𝑞,𝑝𝜔²_𝑝𝑥𝑓;ℎ1+[𝑛]_𝑞ℎ². (3.20) Thus, choosing ℎ=𝑥/[𝑛]_𝑞, the proof is completed.

Corollary 3.5. If 𝑝∈ℕ∪{0},𝑓∈𝐶_𝑝,  𝑞>1 and 𝑥∈[0,∞),  then lim_𝑛→∞𝑀_𝑛,𝑞𝑓;𝑥=𝑓(𝑥). (3.21) This converegnce is uniform on every [𝑎,𝑏], 0≤𝑎<𝑏.

Remark 3.6. Theorem 3.4 shows the rate of approximation by the 𝑞-Szász-Mirakjan operators (𝑞>1) is of order 𝑞^−𝑛 versus 1/𝑛 for the classical Szász-Mirakjan operators.

4. Convergence of 𝑞-Szász-Mirakjan Operators

An interesting problem is to determine the class of all continuous functions 𝑓 such that 𝑀_𝑛,𝑞(𝑓) converges to 𝑓 uniformly on the whole interval [0,∞) as 𝑛→∞. This problem was investigated by Totik [27, Theorem 1] and de la Cal and Cárcamo [28, Theorem 1]. The following result is a 𝑞-analogue of Theorem 1 [28].

Theorem 4.1. Assume that 𝑓∶[0,∞)→𝑅 is bounded or uniformly continuous. Let 𝑓^∗(𝑧)=𝑓𝑧²,𝑧∈[0,∞). (4.1) One has, for all 𝑡>0 and 𝑥≥0, 𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤2𝜔𝑓^∗;1[𝑛]_𝑞. (4.2) Therefore, 𝑀_𝑛,𝑞(𝑓;𝑥) converges to 𝑓 uniformly on [0,∞) as 𝑛→∞, whenever 𝑓^∗ is uniformly continuous.

Proof. By the definition of 𝑓^∗ we have 𝑀_𝑛,𝑞𝑓;𝑥=𝑀_𝑛,𝑞𝑓^∗√⋅;𝑥. (4.3) Thus we can write 𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)=𝑀_𝑛,𝑞𝑓^∗√⋅;𝑥−𝑓^∗√𝑥=^∞_𝑘=0𝑓^∗[𝑘]_𝑞[𝑛]_𝑞−𝑓^∗√𝑥𝑠_𝑛,𝑘≤𝑞;𝑥_∞𝑘=0𝑓^∗[𝑘]_𝑞[𝑛]_𝑞−𝑓^∗√𝑥𝑠_𝑛,𝑘≤𝑞;𝑥_∞𝑘=0𝜔𝑓^∗;[𝑘]_𝑞[𝑛]_𝑞−√𝑥𝑠_𝑛,𝑘≤𝑞;𝑥_∞𝑘=0𝜔𝑓^∗;[𝑘]_𝑞/[𝑛]_𝑞−√𝑥𝑀_𝑛,𝑞√⋅−√𝑥𝑀;𝑥^𝑛,𝑞√⋅−√𝑥;𝑥𝑠_𝑛,𝑘𝑞;𝑥. (4.4) Finally, from the inequality 𝜔𝑓^∗;𝛼𝛿≤(1+𝛼)𝜔𝑓^∗;𝛿,𝛼,𝛿≥0, (4.5) we obtain 𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)≤𝜔𝑓^∗;𝑀_𝑛,𝑞√⋅−√𝑥;𝑥^∞_𝑘=01+[𝑘]_𝑞/[𝑛]_𝑞−√𝑥𝑀_𝑛,𝑞√⋅−√𝑥𝑠;𝑥^𝑛,𝑘𝑓𝑞;𝑥=2𝜔^∗;𝑀_𝑛,𝑞√⋅−√𝑥;𝑥. (4.6) In order to complete the proof we need to show that we have for all 𝑡>0 and 𝑥>0, 𝑀_𝑛,𝑞√⋅−√𝑥;𝑥≤1[𝑛]_𝑞. (4.7) Indeed we obtain from the Cauchy-Schwarz inequality: 𝑀_𝑛,𝑞√⋅−√𝑥;𝑥=^∞_𝑘=0[𝑘]_𝑞[𝑛]_𝑞−√𝑥𝑠_𝑛,𝑘=𝑞;𝑥_∞𝑘=0[𝑘]_𝑞/[𝑛]_𝑞−𝑥[𝑘]_𝑞/[𝑛]_𝑞+√𝑥𝑠_𝑛,𝑘≤1𝑞;𝑥√𝑥^∞_𝑘=0[𝑘]_𝑞[𝑛]_𝑞−𝑥𝑠_𝑛,𝑘≤1𝑞;𝑥√𝑥^∞_𝑘=0[𝑘]_𝑞[𝑛]_𝑞−𝑥²𝑠_𝑛,𝑘=1𝑞;𝑥√𝑥𝑀_𝑛,𝑞(⋅−𝑥)²=1;𝑥√𝑥1[𝑛]_𝑞𝑥=1[𝑛]_𝑞 (4.8) showing (4.2), and completing the proof.

Next we prove Voronovskaja type result for 𝑞-Szász-Mirakjan operators.

Theorem 4.2. Assume that 𝑞∈(1,∞). For any 𝑓∈𝐶²_𝑝 the following equality holds lim_𝑛→∞[𝑛]_𝑞𝑀_𝑛,𝑞(𝑥=1𝑓;𝑥−𝑓)2𝑓^′′(𝑥)𝑥, (4.9) for every 𝑥∈[0,∞).

Proof. Let 𝑥∈[0,∞) be fixed. By the Taylor formula we may write 𝑓(𝑡)=𝑓(𝑥)+𝑓^′(𝑥)+1)(𝑡−𝑥2𝑓^′′(𝑥)(𝑡−𝑥)²+𝑟(𝑡;𝑥)(𝑡−𝑥)², (4.10) where 𝑟(𝑡;𝑥) is the Peano form of the remainder, 𝑟(⋅;𝑥)∈𝐶_𝑝, and lim_𝑡→𝑥𝑟(𝑡;𝑥)=0. Applying 𝑀_𝑛,𝑞 to (4.10) we obtain [𝑛]𝑀_𝑛,𝑞𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥)[𝑛]_𝑞𝑀_𝑛,𝑞()+1𝑡−𝑥;𝑥2𝑓^′′(𝑥)[𝑛]_𝑞𝑀_𝑛,𝑞(𝑡−𝑥)²;𝑥+[𝑛]_𝑞𝑀_𝑛,𝑞𝑟(𝑡;𝑥)(𝑡−𝑥)²;𝑥. (4.11) By the Cauchy-Schwartz inequality, we have 𝑀_𝑛,𝑞𝑟(𝑡;𝑥)(𝑡−𝑥)²;𝑥≤𝑀_𝑛,𝑞𝑟²(𝑡;𝑥);𝑥𝑀_𝑛,𝑞(𝑡−𝑥)⁴;𝑥. (4.12) Observe that 𝑟²(𝑥;𝑥)=0. Then it follows from Corollary 3.5 that lim_𝑛→∞𝑀_𝑛,𝑞𝑟²(𝑡;𝑥);𝑥=𝑟²(𝑥;𝑥)=0. (4.13) Now from (4.12), (4.13), and Lemma 2.5 we get immediately lim_𝑛→∞[𝑛]_𝑞𝑀_𝑛,𝑞𝑟(𝑡;𝑥)(𝑡−𝑥)²;𝑥=0. (4.14) The proof is completed.