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Modified Stancu operators based on inverse Polya Eggenberger distribution
Journal of Inequalities and Applications volume 2017, Article number: 57 (2017)
Abstract
In this paper, we construct a sequence of modified Stancu-Baskakov operators for a real valued function bounded on \([0,\infty)\), based on a function \(\tau(x)\). This function \(\tau(x)\) is infinite times continuously differentiable on \([0,\infty)\) and satisfy the conditions \(\tau (0)=0,~\tau ^{\prime}(x)>0\) and \(\tau^{\prime\prime}(x)\) is bounded for all \(x\in {}[0,\infty)\). We study the degree of approximation of these operators by means of the Peetre K-functional and the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja-type theorems are also established in terms of the first order Ditzian-Totik modulus of smoothness.
1 Introduction
In 1923, Eggenberger and Pólya [1] were the first to introduce Pólya-Eggenberger distribution. After that, in 1969, Johnson and Kotz [2] gave a short discussion of Pólya-Eggenberger distribution.
The Pólya-Eggenberger distribution X [2] is defined by
The inverse Pólya-Eggenberger distribution N is defined by
In 1970, Stancu [3] introduced a generalization of the Baskakov operators based on inverse Pólya-Eggenberger distribution for a real valued bounded function on \([0,\infty)\), defined by
where α is a non-negative parameter which may depend only on \(n\in \mathbb{N}\) and \(a^{[n,h]}=a(a-h)(a-2h)\cdots(a-(n-1)h), a^{[0,h]}= 1\) is known as a factorial power of a with increment h. For \(\alpha=0\), the operator (1.3) reduces to Baskakov operators [4].
In 1989, Razi [5] studied convergence properties of Stancu-Kantorovich operators based on Pólya-Eggenberger distribution. Very recently, Deo et al. [6] introduced a Stancu-Kantorovich operators based on inverse Pólya-Eggenberger distribution and studied some of its convergence properties. For some other relevant research in this direction we refer the reader to [7–9].
Now, for \(\alpha=\frac{1}{n}\), we get a special case of Stancu-Baskakov operators (1.3) defined as
where \((a)_{n}:= a^{[n,-1]}=a(a+1)\cdots(a+(n-1))\) is called the Pochhammer symbol.
For the Lupas operator, given by
let \(\mu_{n,m}(x)= L_{n}(t^{m};x), m\in\mathbb{N}\cup\{0\}\) be the mth order moment.
Lemma 1
For the function \(\mu_{n,m}(x)\), we have \(\mu_{n,0}(x)=1\) and we have the recurrence relation
where \(\mu_{n,m}^{\prime}(x)\) is the derivative of \(\mu_{n,m}(x)\).
Proof
On differentiating \(\mu_{n,m}(x)\) with respect to x, the proof of the recurrence relation easily follows; hence the details are omitted. □
Remark 1
From Lemma 1, we have
The values of the Stancu-Baskakov operators (1.4) for the test functions \(e_{i}(t)=t^{i}\), \(i=0,1,2\), are given in the following lemma.
Lemma 2
[10]
The Stancu-Baskakov operators (1.4) verify:
-
(i)
\(V_{n}^{\langle \frac{1}{n}\rangle }(1;x)= 1\),
-
(ii)
\(V_{n}^{\langle \frac{1}{n}\rangle }(t;x)= \frac{n x}{n-1}\),
-
(iii)
\(V_{n}^{\langle \frac{1}{n}\rangle }(t^{2};x)= \frac{n^{2}}{(n-1)(n-2)} [x^{2}+\frac{x(x+1)}{n}+\frac{1}{n}(1-\frac{1}{n})x ]\).
-
(iv)
\(V_{n}^{\langle \frac{1}{n}\rangle }(t^{3};x)= \frac{n^{3}}{(n-1)(n-2)(n-3)} [\frac{(n+1)(n+2)}{n^{2}}x^{3}+\frac{3(2n^{2}+n-1)}{n^{3}}x^{2}+\frac {(2n-1)(3n-1)}{n^{4}}x ]\)
-
(v)
\(V_{n}^{\langle \frac{1}{n}\rangle }(t^{4};x)= \frac{n^{4}}{(n-1)(n-2)(n-3)(n-4)} [\frac{(n+1)(n+2)(n+3)}{n^{3}}x^{4}+\frac{6(n+1)(n+2)(2n-1)}{n^{4}}x^{3}+\frac{6(6n^{3}+n^{2}-4n+1)}{n^{5}}x^{2}+ \frac {26n^{2}-27n+7}{n^{5}}x ]\).
Proof
The identities (i)-(iii) are proved in [10], hence we give the proof of the identity (iv). The identity (v) can be proved in a similar manner.
We have
where \(B(nx,n)\) is the Beta function.
Therefore using Remark 1, we get
Now, by a simple calculation, we get the required result. □
As a consequence of Lemma 2, we obtain the following.
Lemma 3
For the Stancu-Baskakov operator (1.4), the following equalities hold:
-
(i)
\(V_{n}^{\langle \frac{1}{n}\rangle }((t-x);x)= \frac{x}{n-1}\),
-
(ii)
\(V_{n}^{\langle \frac{1}{n}\rangle }((t-x)^{2};x)= \frac{2nx(x+1)+(2x-1)x}{(n-1)(n-2)} \),
-
(iii)
\(V_{n}^{\langle \frac{1}{n}\rangle }((t-x)^{4};x)= \frac{1}{n(n-1)(n-2)(n-3)(n-4)} [ 12n(n^{2}-13n+2)x^{3}(x+1)+12n(n^{2}+8n-13)x^{2}(x+1)+(26n^{2}+48n-22)x(x+1)+(29-75n)x ]\).
Let \(0\leq r_{n}(x)\leq1\) be a sequence of continuous functions for each \(x\in[0,1]\) and \(n\in\mathbb{N}\). Using this sequence \(r_{n}(x)\), for any \(f\in C[0,1]\), King [11] proposed the following modification of the Bernstein polynomial for a better approximation:
Gonska et al. [12] introduced a sequence of King-type operators \(D_{n}^{\tau}: C[0,1]\rightarrow C[0,1]\) defined as
where \(\tau\in C[0,1]\) such that \(\tau(0)=0, \tau(1)=1\) and \(\tau^{\prime}(x)>0\) for each \(x\in[0,1]\) and studied global smoothness preservation, the approximation of decreasing and convex functions, the validity of a Voronovskaja-type theorem and a recursion formula generalizing a corresponding result for the classical Bernstein operators.
Motivated by the above work, in the present paper we introduce modified Stancu-Baskakov operators based on a function \(\tau(x)\) and obtain the rate of approximation of these operators with the help of Peetre’s K-functional and the Ditzian-Totik modulus of smoothness. Also, we prove a quantitative Voronovskaja-type theorem by using the first order Ditzian-Totik modulus of smoothness.
Throughout this paper, we assume that C denotes a constant not necessarily the same at each occurence.
2 Modified Stancu-Baskakov operators
Let \(\tau(x)\) be continuously differentiable ∞ times on \([0, \infty)\), such that \(\tau(0)=0, \tau^{\prime}(x)>0\) and \(\tau^{\prime \prime}(x)\) is bounded for all \(x\in[0,\infty)\). We introduce a sequence of Stancu-Baskakov operators for \(f\in C_{B}[0, \infty)\), the space of all continuous and bounded functions on \([0,\infty)\), endowed with the norm \(\| f \|= \sup_{x\in[0,\infty)}| f(x)|\), by
where
Lemma 4
The operator defined by (2.1) satisfies the following equalities:
-
(i)
\(V_{n}^{\langle \frac{1}{n},\tau\rangle }(1;x)= 1\),
-
(ii)
\(V_{n}^{\langle \frac{1}{n},\tau\rangle }(\tau(t);x)= \frac{n \tau(x)}{n-1}\),
-
(iii)
\(V_{n}^{\langle \frac{1}{n},\tau\rangle }(\tau^{2}(t);x)= \frac{n^{2}}{(n-1)(n-2)} [\tau^{2}(x)+\frac{\tau(x)(\tau(x)+1)}{n}+\frac{1}{n}(1- \frac{1}{n})\tau(x) ]\),
-
(iv)
\(V_{n}^{\langle \frac{1}{n}\rangle }(\tau^{3}(t);x)= \frac{n^{3}}{(n-1)(n-2)(n-3)} [\frac{(n+1)(n+2)}{n^{2}}\tau^{3}(x)+\frac {3(2n^{2}+n-1)}{n^{3}}\tau^{2}(x)+\frac{(2n-1)(3n-1)}{n^{4}}\tau(x) ]\),
-
(v)
\(V_{n}^{\langle \frac{1}{n}\rangle }(\tau^{4}(t);x)= \frac{n^{4}}{(n-1)(n-2)(n-3)(n-4)} [\frac{(n+1)(n+2)(n+3)}{n^{3}}\tau^{4}(x)+\frac{6(n+1)(n+2)(2n-1)}{n^{4}}\tau^{3}(x) +\frac{6(6n^{3}+n^{2}-4n+1)}{n^{5}}\tau^{2}(x)+ \frac{26n^{2}-27n+7}{n^{5}}\tau(x) ]\).
Proof
The proof of lemma is straightforward on using Lemma 2. Hence we omit the details. □
Let the mth order central moment for the operators given by (2.1) be defined as
Lemma 5
For the central moment operator \(\mu_{n,m}^{\tau}(x)\), the following equalities hold:
-
(i)
\(\mu_{n,1}^{\tau}(x)= \frac{\tau(x)}{n-1}\),
-
(ii)
\(\mu_{n,2}^{\tau}(x)= \frac{2n\phi^{2}_{\tau}(x)+(2\tau(x)-1)\tau(x)}{(n-1)(n-2)} \),
-
(iii)
\(\mu_{n,4}^{\tau}(x)= \frac{1}{n(n-1)(n-2)(n-3)(n-4)} [ 12n(n^{2}-13n+2)\tau^{2}(x)\phi^{2}_{\tau}(x)+12n(n^{2}+8n-13)\tau(x)\phi ^{2}_{\tau}(x)+(26n^{2}+48n-22)\phi^{2}_{\tau}(x)+(29-75n)\tau(x) ]\),
where \(\phi_{\tau(x)}^{2}(x)=\tau(x)(\tau(x)+1)\).
Proof
Using the definition (2.1) of the modified Stancu-Baskakov operators and Lemma 4, the proof of the lemma easily follows. Hence, the details are omitted. □
Let
For \(f\in C_{B}[0,\infty)\) and \(\delta>0\), the Peetre K-functional [13] is defined by
where
From [14], Proposition 3.4.1, there exists a constant \(C > 0\) independent of f and δ such that
where \(\omega_{2}\) is the second order modulus of smoothness of \(f\in C_{B}[0,\infty)\) and is defined as
In the following, we assume that \(\inf_{x\in[0,\infty)}\tau^{\prime}(x)\geq a, a\in\mathbb{R}^{+}:=(0,\infty)\).
Next, we recall the definitions of the Ditzian-Totik first order modulus of smoothness and the K-functional [15]. Let \(\phi_{\tau}(x):= \sqrt {\tau(x)(1 + \tau(x))}\) and \(f\in C_{B}[0,\infty)\). The first order modulus of smoothness is given by
Further, the appropriate K-functional is defined by
where \(W_{\phi_{\tau} }[0,\infty)=\{g:g\in AC_{\mathrm{loc}}[0,\infty),\|\phi _{\tau} g^{\prime}\|<\infty\}\) and \(g\in AC_{\mathrm{loc}}[0,\infty)\) means that g is absolutely continuous on every interval \([a,b]\subset[0,\infty)\). It is well known [15], p.11, that there exists a constant \(C>0\) such that
Theorem 1
If \(f\in C_{B}[0,\infty)\), then
Proof
By the definition of the modified Stancu-Baskakov operators (2.1) and using Lemma 4 we have
for every \(x\in[0,\infty)\). Hence the required result is immediate. □
Theorem 2
Let \(f\in C_{B}[0,\infty)\). Then, for \(n\geq3\), there exists a constant \(C>0\) such that
on each compact subset of \([0,\infty)\).
Proof
Let U be a compact subset of \([0,\infty)\). For each \(x\in U\), first we define an auxiliary operator as
Now, using Lemma 4, we have
Let \(g\in W^{2}\), \(x\in U\) and \(t\in[0, \infty)\). Then by Taylor’s expansion, and using results in [16], p.32, we get
Now, applying the operator \(V_{n}^{*\langle \frac{1}{n},\tau\rangle }(\cdot;x)\) to both sides of the above equality, we get
Again, for each \(x\in U\), we have
Now, using the definition of the auxiliary operators, Theorem 1 and inequality (2.7), for each \(x\in U\) we have
Let \(C=\max (4,\frac{4}{a^{2}}, \frac{4}{a^{3}}\|\tau^{\prime \prime} \| )\), we get
Taking the infimum on the right side of the above inequality over all \(g\in W^{2}\) and for all \(x\in U\), we have
using equation (2.2), we get the required result. □
Theorem 3
Let \(f\in C_{B}[0,\infty)\). Then for every \(x\in[0,\infty )\), and \(n\geq3\) we have
Proof
For any \(g\in W_{\phi_{\tau}}[0,\infty)\), by Taylor’s expansion, we have
Applying the operator \(V_{n}^{\langle \frac{1}{n},\tau\rangle }(\cdot;x)\) on both sides of the above equality, we get
From [16], we have
and
Now, from equations (2.12)-(2.13) and using the Cauchy-Schwarz inequality, we obtain
Thus, for \(f\in C_{B}[0,\infty)\) and any \(g\in W_{\phi_{\tau}}[0,\infty)\), we have
Taking the infimum on the right side of the above inequality over all \(g\in W_{\phi_{\tau}}[0, \infty)\), we get
Finally, using equation (2.3), the theorem is immediate. □
Theorem 4
For any \(f\in C^{2}[0,\infty)\) and \(x\in[0,\infty)\), the following inequality hold:
Proof
Let \(f\in C^{2}[0,\infty)\) and \(x,t\in{}[0,\infty)\). Then by Taylor’s expansion, we have
Hence,
Applying \(V_{n}^{\langle \frac{1}{n},\tau\rangle }\) to both sides of the above relation, we get
For \(g\in W_{\phi_{\tau}[0,\infty)}\), we have
Using the inequality
we can write
Therefore,
Now combining equations (2.16)-(2.17), applying Lemma 3 and the Cauchy-Schwarz inequality, we get
This completes the proof of the theorem. □
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The first author is thankful to The Ministry of Human Resource and Development, India, for the financial support to carry out the above work.
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Deshwal, S., Agrawal, P. & Araci, S. Modified Stancu operators based on inverse Polya Eggenberger distribution. J Inequal Appl 2017, 57 (2017). https://doi.org/10.1186/s13660-017-1328-9
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DOI: https://doi.org/10.1186/s13660-017-1328-9