Quantitative-Voronovskaya and Grüss-Voronovskaya type theorems for Szász-Durrmeyer type operators blended with multiple Appell polynomials
Neer and Agrawal Journal of Inequalities and Applications
Quantitative-Voronovskaya and Grüss-Voronovskaya type theorems for Szász-Durrmeyer type operators blended with multiple Appell polynomials
Trapti Neer
Purshottam Narain Agrawal
In this paper, we establish a link between the Szász-Durrmeyer type operators and multiple Appell polynomials. We study a quantitative-Voronovskaya type theorem in terms of weighted modulus of smoothness using sixth order central moment and Grüss-Voronovskaya type theorem. We also establish a local approximation theorem by means of the Steklov means in terms of the first and the second order modulus of continuity and Voronovskaya type asymtotic theorem. Further, we discuss the degree of approximation by means of the weighted spaces. Lastly, we find the rate of approximation of functions having a derivative of bounded variation.
Steklov mean; first and second order modulus of continuity; weighted modulus of continuity; Grüss-Voronovskaya type theorem; functions of bounded variation
1 Introduction
For f ∈ C(R+) and x ∈ R+ (R+ = [, ∞)), Szász [] introduced the well-known operators
Sn(f ; x) = e–nx
k=
∞ (nx)k
k!
f
k
n
such that Sn(|f |; x) < ∞. Several generalizations of Szász operators have been introduced in
the literature and authors have studied their approximation properties. In [], the author
considered Baskakov-Szász type operators and studied the rate of convergence for
absolutely continuous functions having a derivative equivalent with a function of bounded
variation. In [], the authors introduced the q-Baskakov-Durrmeyer type operators and
studied the rate of convergence and the weighted approximation properties. In [] the
authors proposed the β-operators based on q-integers and established some direct theorems
by means of modulus of continuity and also studied the weighted approximation and
better approximation using King type approach. For exhaustive literature on approximation
by linear positive operators one can refer to [–] and the references therein.
Pn(f ; x) =
e–nx ∞
g() k=
k
pk(nx)f n .
A(t, t)ex(t+t) = k= k∞= pkk,k!k(x!) tk tk ,
∞
where A is given by
A(t, t) = k= k∞= kak!k,k! tk tk ,
∞
with A(, ) = a, = .
For g(u) = , these operators reduce to Szász-Mirakjan operators ().
A set of polynomials {pk,k (x)}k∞,k= with degree k + k for k, k ≥ is called multiple
polynomial system (multiple PS) and a multiple PS is called multiple Appell if it is
generated by the relation
Now let us recall some results on multiple Appell polynomials []. Let g(z) = n∞= anzn,
g() = , be an analytic function in the disc |z| ≤ r, r > and pk(x) be the Appell
polynomials having the generating function g(u)eux = k∞= pk(x)uk , with g() = and pk(x) ≥ ,
∀x ∈ R+.
Jakimovski and Leviatan [] proposed a generalization of Szász-Mirakjan operators by
means of the Appell polynomials as follows:
()
()
()
()
()
Theorem . For multiple PS, {pk,k (x)}k∞,k=, the following statements are equivalent:
(a) {pk,k (x)}k∞,k= is a set of multiple Appell polynomials.
∞
(b) There exists a sequence {ak,k }k,k= with a, = such that
(c) For every k + k ≥ , we have
pk,k (x) =
∞ ∞
k
r= r= r
k ak–r,k–r xr+r .
r
pk,k (x) = kpk–,k (x) + kpk,k–(x).
Varma [] defined a sequence of linear positive operators for any f ∈ C(R+), by
Kn(f ; x) =
e–nx
∞
A(, ) k= k=
∞ pk,k ( nx ) f
k!k!
k + k ,
n
provided A(, ) = , aAk(,,k) ≥ for k, k ∈ N, and the series () and() converge for |t| < R,
|t| < R (R, R > ), respectively.
For α > , ρ > and f : R+ → R, being integrable function, Paˇltaˇnea [] defined a
modification of the Szász operators by
Lρα(f ; x) =
∞
k=
sα,k(x)
∞ αρe–αρt(αρt)(k)ρ–
(k)ρ
f (t) dt + e–αxf (), x ∈ R.
+
Motivated by [], for f ∈ CE(R+), the space of all continuous functions satisfying |f (t)| ≤
Keat (t ≥ ) for some positive constant K and a, we propose an approximation method by
linking the operators () and the multiple Appell polynomials by
nx f (),
∞ nρe–nρt(nρt)(k+k)ρ–
and establish a quantitative Voronovskaya type theorem, a Grüss Voronovskaya type
theorem, a local approximation theorem by means of the Steklov mean, a Voronovskaya type
asymptotic theorem and error estimates for several weighted spaces. Lastly, we study the
rate of convergence of functions having a derivative of bounded variation.
2 Basic results
In order to prove the main results of the paper, we shall need the following auxiliary results.
Lemma . For Kn(ti; x), i = , , , , , we have
(i) Kn(; x) = ,
(ii) Kn(t; x) = x + At (,nA)(+,At) (, ) ,
(iii) Kn t; x = x + nx + A(, ) At (, ) + At (, )
+ nA(, ) At (, ) + At (, ) + Att (, ) + Att (, ) + Att (, ) ,
(iv) Kn t; x = x + nx + A(, ) At (, ) + At (, ) + nx
+ A(, ) At (, ) + At (, ) + Att (, ) + Att (, ) + Att (...truncated)