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Shape Preserving Properties of Parametric Szasz Type Operators on Unbounded Intervals

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A peer-reviewed article of this preprint also exists.

Hui Dong  †,
Qiulan Qi  *,†

  † These authors contributed equally to the work.

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03 August 2023

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04 August 2023

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Abstract
In this paper, we investigate some elementary properties of two kinds modified Szasz type basis functions, depending on non-negative parameters. We study some shape preserving properties of these operators concerning monotonicity, convexity, starlikeness, semi-additivity and smoothness.
Keywords: 
Subject: Computer Science and Mathematics  -   Mathematics

1. Introduction

The famous Bernstein operator has become powerful tool for curve and surface design and representation because of its good shape-preserving properties. It has been effectively applied to computer aided geometric design. In recent years, the shape-preserving properties of various operators have been deeply studied [1,2,3,4,5,6,7,8,9,10].
In order to be more effectively applied to CAGD on infinite intervals, two kinds of Sz a ´ sz operators based on non-negative parameters were introduced [11,12]. The aim of this paper is to show some shape preserving properties of these parametric Sz a ´ sz type operators. The complete structure of the manuscript constitutes five sections. The rest of this paper is constructed as follows. In Section 2, the fundamental facts are summarized for use in the sequel. In Section 3, we shall prove that the operators preserve monotonicity, convexity, starlikeness and semi-additivity. In Section 4, we investigate the preservation of smoothness. Finally, in Section 5, some conclusions are provided.

2. Fundamental Properties of the Basis Functions and the Operators

In this section, some basic facts that will be used in the following sections are given.
Let C [ 0 , ∞ ) denote the space of continuous functions on [ 0 , ∞ ) , C B [ 0 , ∞ ) be the space of bounded functions in the space of C [ 0 , ∞ ) endowed with the norm ∥ f ∥ = sup x ∈ [ 0 , ∞ ) | f ( x ) | . Let A i ( i = 0 , 1 , 2 , 3 ) denote the subclasses of C [ 0 , ∞ ) :
A 0 : = { f ∈ C [ 0 , ∞ ) : f ( 0 ) = 0 } ;
A 1 : = { f ∈ A 0 : f ( x ) is convex on [ 0 , ∞ ) } ;
A 2 : = { f ∈ A 0 : x − 1 f ( x ) is increasing on ( 0 , ∞ ) } ;
A 3 : = { f ∈ A 0 : f ( x ) is super-additive on [ 0 , ∞ ) } ,
where f ( x ) is said to be super-additive on [ 0 , ∞ ) , if for any x 1 , x 2 ∈ [ 0 , ∞ ) , f ( x 1 + x 2 ) ≥ f ( x 1 ) + f ( x 2 ) . In addition, by [9 Theorem 5], we find that A 1 ⊂ A 2 ⊂ A 3 . On the other hand, A i * ( i = 1 , 2 , 3 ) stand for the other three subsets:
A 1 * : = { f ∈ A 0 : f ( x ) is concave on [ 0 , ∞ ) } ;
A 2 * : = { f ∈ A 0 : x − 1 f ( x ) is decreasing on ( 0 , ∞ ) } ;
A 3 * : = { f ∈ A 0 : f ( x ) is semi-additive on [ 0 , ∞ ) } ,
where f ( x ) is said to be semi-additive on [ 0 , ∞ ) , if for any x 1 , x 2 ∈ [ 0 , ∞ ) , f ( x 1 + x 2 ) ≤ f ( x 1 ) + f ( x 2 ) .
The parametric Sz a ´ sz type operators are defined by [11,12],
S n μ ( f ; x ) = ∑ k = 0 ∞ f ( k n ) s n , k ( x ) ,
where
s n , k ( x ) = e − n α ( x ) ( n α ( x ) ) k k ! ,
(1) when α ( x ) = α 1 ( x ) = μ x n ( 1 − e μ n ) , the operators S n μ ( f ; x ) preserve the functions 1 and e − μ x ( μ > 0 ) ;
(2) when α ( x ) = α 2 ( x ) = 2 μ x n ( e 2 μ n − 1 ) , the operators S n μ ( f ; x ) preserve the functions 1 and e 2 μ x ( μ > 0 ) .
Remark 1.  [ 11 L e m m a 3 ; 12 R e m a r k 1 ] lim n → ∞ α ( x ) = x .
Remark 2.  [ 11 L e m m a 2 ; 12 L e m m a 2.1 , L e m m a 2.2 ] For x ∈ [ 0 , ∞ ) , μ > 0 , S n μ ( 1 ; x ) = 1 , S n μ ( t ; x ) = α ( x ) .
Remark 3.  [ 11 T h e o r e m 3 ; 12 T h e o r e m 3.1 ] For f ( x ) ∈ C * [ 0 , ∞ ) : = { f ∈ C [ 0 , ∞ ) : lim n → ∞ f ( x ) exists and is finite}, one has the sequence { S n μ ( f ; x ) } converges to f ( x ) uniformly.
For μ > 0 , the new basis functions s n , k ( x ) are definied by
s n , k ( x ) = e − n α ( x ) ( n α ( x ) ) k k ! , k = 0 , 1 , ⋯ .
Figures 1–3 show the Sz a ´ sz type basis with α ( x ) = α 1 ( x ) of degree 4, Figure 4–6 show the Sz a ´ sz type basis with α ( x ) = α 2 ( x ) of degree 4.Preprints 81490 i001
The new basis functions s n , k ( x ) hold the following properties.
( 1 ) . Non-negativity: For x ∈ [ 0 , ∞ ) , μ > 0 , s n , k ( x ) ≥ 0 .
( 2 ) . Partition of unity: ∑ k = 0 ∞ s n , k ( x ) = 1 .
( 3 ) . Properties at the endpoint:
s n , k ( 0 ) = 1 , k = 0 ; 0 , k ≠ 0 .
( 4 ) . Integrals:
∫ 0 ∞ s n , k ( x ) d x = 1 − e − μ n μ , α ( x ) = α 1 ( x ) ; e 2 μ n − 1 2 μ , α ( x ) = α 2 ( x ) .
( 5 ) . Derivative: s n , k ′ ( x ) = n α ( x ) x [ s n , k − 1 ( x ) − s n , k ( x ) ] , x ∈ ( 0 , ∞ ) .
( 6 ) . Maximum value: s n , k ( x ) has only one local maximum at x = k ( 1 − e − μ n ) μ and k ( e 2 μ n − 1 ) 2 μ for α 1 ( x ) and α 2 ( x ) respectively.
For the operators S n μ ( f ; x ) , we can deduce the following geometric properties from those of the Sz a ´ sz basis functions.
( 1 ) . Normativity and boundedness: S n μ ( 1 ; x ) = 1 .
( 2 ) . Endpoint interpolation: S n μ ( f ; 0 ) = f ( 0 ) .
( 3 ) . Linearity: For all real numbers λ 1 and λ 2 , and functions f ( x ) , g ( x ) , one has
S n μ ( λ 1 f + λ 2 g ; x ) = λ 1 S n μ ( f ; x ) + λ 2 S n μ ( g ; x ) .

3. Shape Preservation

First, let us consider the monotonicity and convexity for the operators S n μ ( f ; x ) .
Theorem 3.1.(Monotonicity) Let f ∈ C [ 0 , ∞ ) , if f ( x ) is monotonically increasing (or decreasing) on [ 0 , ∞ ) , for n ∈ N , so are all the operators S n μ ( f ; x ) .
Proof. We write
d d x S n μ ( f ; x ) = n α ( x ) x e − n α ( x ) ∑ k = 0 ∞ f ( k + 1 n ) − f ( k n ) ( n α ( x ) ) k k ! .
Noting that if f ( x ) is monotonically increasing, then the derivative of the operators S n μ ( f ; x ) is nonnegative on [ 0 , ∞ ) , and so S n μ ( f ; x ) is monotonically increasing.
Similarly, we see that if f ( x ) is monotonically decreasing on [ 0 , ∞ ) , so are the operators S n μ ( f ; x ) .
Theorem 3.2.(Convexity) Let f ( x ) ∈ C [ 0 , ∞ ) , if f ( x ) is convex (or concave), so are all the operators S n μ ( f ; x ) .
Proof. Now let us take the second order derivative of S n μ ( f ; x ) . It follows from ( 1 ) that
d 2 d x 2 S n μ ( f ; x ) = n α ( x ) x 2 e − n α ( x ) ∑ k = 0 ∞ f ( k + 2 n ) − 2 f ( k + 1 n ) + f ( k n ) ( n α ( x ) ) k k ! .
If f ( x ) is convex on C [ 0 , ∞ ) , all second order derivative of S n μ ( f ; x ) in ( 2 ) is nonnegative, which implies the convexity of S n μ ( f ; x ) .
Similarly, we see that if f ( x ) is concave on C [ 0 , ∞ ) , so are the operators S n μ ( f ; x ) .
Next, we turn to the starlikeness and semi-additivity.
Theorem 3.3.(Starlikeness) If f ( x ) ∈ A 2 * (or A 2 ), then for n ∈ N , S n μ ( f ; x ) ∈ A 2 * (or A 2 ).
Proof.
d d x ( x − 1 S n μ ( f ; x ) ) = − x − 2 S n μ ( f ; x ) + x − 1 d d x S n μ ( f ; x ) = − x − 2 S n μ ( f ; x ) + x − 2 α ( x ) e − n α ( x ) ∑ k = 0 ∞ n f ( k + 1 n ) − f ( k n ) ( n α ( x ) ) k k ! ,
where
∑ k = 0 ∞ n f ( k + 1 n ) − f ( k n ) ( n α ( x ) ) k k ! = n f ( 1 n ) − n f ( 0 n ) + n f ( 2 n ) − n f ( 1 n ) n α ( x ) + n f ( 3 n ) − n f ( 2 n ) ( n α ( x ) ) 2 2 ! + ⋯ = n f ( 1 n ) + n 2 f ( 2 n ) − n f ( 1 n ) n α ( x ) + n 2 f ( 2 n ) n α ( x ) + 2 n 3 f ( 3 n ) − n 2 f ( 2 n ) ( n α ( x ) ) 2 2 ! + n 3 f ( 3 n ) ( n α ( x ) ) 2 2 ! + ⋯ .
Let
I = x − 2 α ( x ) e − n α ( x ) ∑ k = 1 ∞ n k + 1 f ( k + 1 n ) − n k f ( k n ) ( n α ( x ) ) k k ! ,
which yields,
d d x ( x − 1 S n μ ( f ; x ) ) = − x − 2 S n μ ( f ; x ) + x − 2 α ( x ) e − n α ( x ) ∑ k = 1 ∞ n k f ( k n ) ( n α ( x ) ) k − 1 ( k − 1 ) ! + I = − x − 2 S n μ ( f ; x ) + x − 2 e − n α ( x ) ∑ k = 1 ∞ f ( k n ) ( n α n ( x ) ) k k ! + I = − x − 2 S n μ ( f ; x ) + x − 2 S n μ ( f ; x ) − x − 2 e − n α ( x ) f ( 0 n ) + I = x − 2 α ( x ) e − n α ( x ) ∑ k = 1 ∞ n k + 1 f ( k + 1 n ) − n k f ( k n ) ( n α ( x ) ) k k ! .
Noting that if x − 1 f ( x ) is decreasing on ( 0 , ∞ ) , then I ≤ 0 , i.e. the derivative of x − 1 S n μ ( f ; x ) is negative on ( 0 , ∞ ) , and so x − 1 S n μ ( f ; x ) is decreasing.
Similarly, we see that if x − 1 f ( x ) is increasing on ( 0 , ∞ ) , then I ≥ 0 , so is x − 1 S n μ ( f ; x ) .
Theorem 3.4.(Semi-additivity) If f ( x ) ∈ A 3 * (or A 3 ), then for n ∈ N , S n μ ( f ; x ) ∈ A 3 * (or A 3 ).
Proof. We only take α ( x ) = α 1 ( x ) as an example to prove the semi-additivity, the case α ( x ) = α 2 ( x ) is similar. ∀ x , y ∈ [ 0 , ∞ ) , ∀ n ∈ N ,
S n μ ( f ; x + y ) = e − μ ( x + y ) 1 − e − μ n ∑ k = 0 ∞ f ( k n ) μ ( x + y ) 1 − e − μ n k k ! = e − μ ( x + y ) 1 − e − μ n ∑ k = 0 ∞ f ( k n ) μ 1 − e − μ n k k ! ∑ j = 0 k k j x j y k − j = e − μ ( x + y ) 1 − e − μ n ∑ j = 0 ∞ ∑ k = j ∞ f ( k n ) μ 1 − e − μ n k x j y k − j j ! ( k − j ) ! ,
let K = k − j , then k = K + j ,
S n μ ( f ; x + y ) = e − μ x 1 − e − μ n · e − μ y 1 − e − μ n ∑ j = 0 ∞ ∑ K = 0 ∞ f ( K + j n ) μ x 1 − e − μ n j j ! · μ y 1 − e − μ n K K ! .
If f ( x ) is semi-additive on [ 0 , ∞ ) , one has,
S n μ ( f ; x + y ) ≤ e − μ x 1 − e − μ n · e − μ y 1 − e − μ n ∑ j = 0 ∞ ∑ K = 0 ∞ f ( K n ) μ x 1 − e − μ n j j ! · μ y 1 − e − μ n K K ! + e − μ x 1 − e − μ n · e − μ y 1 − e − μ n ∑ j = 0 ∞ ∑ K = 0 ∞ f ( j n ) μ x 1 − e − μ n j j ! · μ y 1 − e − μ n K K ! = S n μ ( f ; x ) + S n μ ( f ; y )
which implies S n μ ( f ; x ) is semi-additive on [ 0 , ∞ ) . Similarly, if f ( x ) is super-additive on [ 0 , ∞ ) , so is S n μ ( f ; x ) , and thus, the proof is completed.

4. Preservation of Smoothness

For t > 0 , the continuous modulus is defined as [13]:
ω ( f ; t ) = s u p { | f ( x ) − f ( y ) | : | x − y | ≤ t , x , y ∈ [ 0 , ∞ ) } .
A function ω ( t ) on [ 0 , 1 ] s called a modulus of continuity if ω ( t ) is continuous, nondecreasing, semi-additive, and lim t → 0 + ω ( t ) = ω ( 0 ) = 0 . [ 4 ]
Lemma 4.1. [14] For any continuous ω ( t ) (not identical to 0), there exists a concave continuous modulus ω * ( t ) such that for t > 0 , one has ω ( t ) ≤ ω * ( t ) ≤ 2 ω ( t ) , where the constant 2 can not be any smaller.
Lemma 4.2. Let { L n ( f ; x ) ; n = 0 , 1 , ⋯ } be a sequence of linear positive operators from C ( I ) to C ( I ) , where I is a finite or infinite interval, and L n ( 1 ; x ) = 1 , L n ( t ; x ) = C n ( x ) x , 0 ≤ C n ( x ) ≤ 1 . If g ( x ) is a convave, monotonically increasing and continuous function, then L n ( g ; x ) ≤ g ( x ) .
Remark 4. We can get Lemma 4.2 by imitating the proof of the Lemma in Ref [10], here we omit the details.
Theorem 4.1. For f ( x ) ∈ C [ 0 , ∞ ) , ∀ n ∈ N , one has ω ( S n μ ; δ ) ≤ S n μ ( ω ; δ ) .
Proof. As before, we only prove the case α ( x ) = α 1 ( x ) , the case α ( x ) = α 2 ( x ) is similar. Since
e − μ x 1 − e − μ n ∑ j = 0 ∞ μ x 1 − e − μ n j j ! = 1 ,
for h > 0 , from (3), we write
S n μ ( f ; x + h ) − S n μ ( f ; x ) ≤ e − μ x 1 − e − μ n · e − μ h 1 − e − μ n ∑ j = 0 ∞ ∑ k = 0 ∞ μ x 1 − e − μ n j j ! · μ h 1 − e − μ n k k ! f ( k + j n ) − f ( j n ) ≤ e − μ x 1 − e − μ n · e − μ h 1 − e − μ n ∑ j = 0 ∞ ∑ k = 0 ∞ μ x 1 − e − μ n j j ! · μ h 1 − e − μ n k k ! ω ( f ; k n ) , ( 4 )
then the desired result is obtained.
Theorem 4.2.
(i) For f ∈ H ω ( A ) = { f : ω ( f ; x ) ≤ A ω ( x ) } , then S n μ ( f ; x ) ∈ H ω ( 2 A ) , ∀ n ∈ N ;
(ii) If lim n → ∞ S n μ ( f ; x ) = f ( x ) , then f ∈ H ω ( A ) ⇔ S n μ ( f ; x ) ∈ H ω ( A ) , here n is big enough.
Proof. (i) As before, we only prove the case α ( x ) = α 1 ( x ) , the case α ( x ) = α 2 ( x ) is similar.
∀ x 1 , x 2 ∈ I , x 1 > x 2 , from (4), and using Lemma 4.1, we have
S n μ ( f ; x 1 ) − S n μ ( f ; x 2 ) ≤ e − μ ( x 1 − x 2 ) 1 − e − μ n ∑ j = 0 ∞ ω ( f ; j n ) μ ( x 1 − x 2 ) 1 − e − μ n j j ! ≤ A L n ( ω ; x 1 − x 2 ) ≤ A L n ( ω * ; x 1 − x 2 ) ≤ A ω * ( x 1 − x 2 ) ≤ 2 A ω ( x 1 − x 2 ) ,
here we use Lemma 4.2 for ω * ( t ) .
(ii) If l i m n → ∞ S n μ ( f ; x ) = f ( x ) , then
lim n → ∞ S n μ ( f ; x 1 ) − S n μ ( f ; x 2 ) = f ( x 1 ) − f ( x 2 ) ,
we have f ∈ H ω ( A ) ⇔ S n μ ( f ; x ) ∈ H ω ( A ) , here n is big enough.
Remark 5. If f ∈ L i p K α = { f : ω ( f ; δ ) ≤ K δ α , K ≥ 0 , α ∈ ( 0 , 1 ] } , then S n μ ( f ; x ) ∈ L i p K α here n is big enough.
Theorem 4.3. For any modulus of continuity ω ( t ) , S n μ ( ω ; t ) is also a modulus of continuity.
Proof. For any modulus of continuity ω ( t ) , S n μ ( ω ; t ) is continuous, nondecreasing, and
lim t → 0 + S n μ ( ω ; t ) = S n μ ( ω ; 0 ) = ω ( 0 ) = 0 .
Combining the semi-additivity of ω ( t ) and S n μ ( ω ; t ) , we deduce that S n μ ( ω ; t ) is a modulus of continuity.

5. Conclusions

In this paper, some fundamental facts of two kinds of parametric Sz a ´ sz type operators are presented. The shape preserving properties, such as linearity, monotonicity, convexity, starlikeness, semi-additivity are examined. Finally, with the help of the continuous modulus, the preservation of smoothness are discussed. We think that these newly operators are appliable to CAGD. For future work, we propose to obtain some inverse theorems of these kinds of operators.

Author Contributions

Conceptualization, H.D. and Q.Q.; methodology, H.D.; software, H.D.; validation, H.D.; formal analysis, H.D. and Q.Q.; investigation, H.D. and Q.Q.; resources, H.D. and Q.Q.; data curation, H.D.; writing—original draft preparation, H.D.; writing—review and editing, H.D. and Q.Q.; visualization, H.D. ; supervision, Q.Q.; project administration, Q.Q.; funding acquisition, Q.Q. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors extend their appreciation to the Science and Technology Project of Hebei Education Department (No. ZD2019053) and Science Foundation of Hebei Normal University (No. L2020203). The authors are grateful to the responsible Editor and anonymous reviewers for their valuable comments and suggestions, which have greatly improved this paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Bloom, W. R.; Elliott, D. The modulus of continuity of the remainder in the approximation of Lipschitz functions. Journal of Approximation Theory 1981, 31, 59–66. [Google Scholar] [CrossRef]
  2. Brown, B. M.; Elliott, D.; Paget, D. F. Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function. Journal of Approximation Theory 1987, 49, 196–199. [Google Scholar] [CrossRef]
  3. Chang, G. Z.; Davis, P. J. The convexity of Bernstein polynomials over triaugles. Journal of Approximation Theory 1984, 40, 11–28. [Google Scholar] [CrossRef]
  4. Li, Z. K. Bernstein ploynomials and modulus of continuity. Journal of Approximation Theory 2000, 102, 171–174. [Google Scholar] [CrossRef]
  5. Khan, M. K.; Vecchia, B. D.; Fassih, A. On the monotonicity of positive linear operators. Journal of Approximation Theory 1998, 92, 22–37. [Google Scholar] [CrossRef]
  6. Carbone, I. Shape proserving properties of some positive linear operators on unbounded intervals. Journal of Approximation Theory 1998, 93, 140–156. [Google Scholar] [CrossRef]
  7. Adell, J. A.; Palomares, A. P. Best constants in preservation inequalities concerning the first modulus and Lipschitz classes for Bernstein-type operators. Journal of Approximation Theory 1998, 93, 128–139. [Google Scholar] [CrossRef]
  8. Cal, J. D. L.; Cácomo, J. On certain best constants for Bernstein-type operators. Journal of Approximation Theory 2001, 113, 189–206. [Google Scholar] [CrossRef]
  9. Bruckner, A. M.; Ostrow, E. Some function classes related to the class of convex functions. Pacific Journal of Mathematics 1962, 12, 1203–1215. [Google Scholar] [CrossRef]
  10. Hou, X. Q.; Xue, Y. C. On the property of some linear positive operators preserving the class Λω(A). Journal of Ningxia University (Natural Sci. Edition) 1995, 16, 11–16. (in Chinese). [Google Scholar]
  11. Acar, T.; Aral, A.; Gonska, H. On Sza´sz-Mirakyan operators preserving e2ax(a>0). Mediterranean Journal of Mathematics 2017, 14(6), 1–14. [Google Scholar]
  12. Huang, J. Y.; Qi, Q. L.; Yang, G. Approximation properties of a modified Sza´sz type operators. Pure Mathematics 2022, 12(5), 803–813. [Google Scholar] [CrossRef]
  13. Ditzian, Z.; Totik, V. Moduli of Smoothness; Springer-Verlag, New Tork, 1987.
  14. Chen, W. Z. Approximation Theory of Operators; Xiamen: Xiamen University Press, China, 1989. [Google Scholar]
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