scholarly journals One counterexample for convex approximation of function with fractional derivatives, r>4

2018 ◽  
pp. 53-56
Author(s):  
T. Petrova
Keyword(s):  
Convex Function ◽  
Algebraic Polynomial ◽  
Fractional Derivatives ◽  
Nondecreasing Function ◽  
Degree Of Approximation ◽  
Monotone Approximation ◽  
Convex Approximation ◽  
Positive Functions ◽  
Approximation Of A Function ◽  
Approximation Of Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.

Some negative results for the interpolation monotone approximation of functions having a fractional derivative

2020 ◽  
pp. 122-127
Author(s):  
T. O. Petrova ◽  
I. P. Chulakov
Keyword(s):  
Fractional Derivative ◽  
Convex Function ◽  
Nondecreasing Function ◽  
Degree Of Approximation ◽  
Approximation Of Functions ◽  
Monotone Approximation ◽  
Convex Approximation ◽  
Negative Results ◽  
Positive Functions ◽  
Approximation Of A Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

scholarly journals On the Degree of Approximation of a Function by Nörlund Means of its Fourier Laguerre Series

2020 ◽  
Vol 1 ◽  
pp. 65-70
Author(s):  
Suresh Kumar Sahani ◽  
Vishnu Narayan Mishra ◽  
Narayan Prasad Pahari
Keyword(s):  
Degree Of Approximation ◽  
End Point ◽  
Laguerre Series ◽  
Nörlund Means ◽  
The Given ◽  
Approximation Of A Function ◽  
Approximation Of Function

In this paper, we have proved the degree of approximation of function belonging to L[0, ∞) by Nörlund Summability of Fourier-Laguerre series at the end point x = 0. The purpose of this paper is to concentrate on the approximation relations of the function in L[0, ∞) by Nörlund Summability of Fourier- Laguerre series associate with the given function motivated by the works [3], [9] and [13].  

scholarly journals Approximation of a Function Belonging to Lip ((α, r) Class by Np,q. C1 Summability Method of its Fourier Series

2014 ◽  
Vol 14 (2) ◽  
pp. 117-122 ◽  
Author(s):  
JP Kushwaha ◽  
BP Dhakal
Keyword(s):  
Fourier Series ◽  
Science And Technology ◽  
Degree Of Approximation ◽  
Summability Method ◽  
Approximation Of A Function

In this paper, an estimate for the degree of approximation of a function belonging to Lip(α, r) class by product summability method Np.q.C1 of its Fourier series has been established. DOI: http://dx.doi.org/10.3126/njst.v14i2.10424 Nepal Journal of Science and Technology Vol. 14, No. 2 (2013) 117-122

Performance characteristics of the right heart bypass preparation

1960 ◽  
Vol 198 (3) ◽  
pp. 552-560 ◽  
Author(s):  
Fred S. Grodins ◽  
William H. Stuart ◽  
Rodney L. Veenstra
Keyword(s):  
Venous Pressure ◽  
Self Regulation ◽  
Degree Of Approximation ◽  
Linear Functions ◽  
Regression Equations ◽  
Right Heart ◽  
Pump Output ◽  
Steady State Responses ◽  
The Right ◽  
Positive Functions

In eight dogs under morphine-barbital anesthesia, the right ventricle was replaced by a sigmamotor pump in order to study steady-state responses of circuit pressures to arbitrary variations in ‘right heart’ output. In the systemic circuit, it was found that arterial pressure (PAS, mm Hg) was a positive and venous pressure (PVS, mm Hg) a negative function of pump output (Q, cc/min). In the pulmonary circuit, both arterial (PAP) and left atrial (PLA) pressures were positive functions of pump output. Despite considerable variations in detail between animals, individual animal pressure-flow curves could all be represented to a reasonable degree of approximation by linear functions within the range of the data. Average regression equations for the eight animals were: PAS = 18.3 + 0.0639Q, PVS = 20.2 – 0.0124Q, (PAS – PVS) = 0.0756Q – 1.59 PAP = 11.2 + 0.0151Q, PLA = 7.0 + 0.00208Q, and (PAP – PLA) - 3.9 + 0.0154Q. These results were interpreted in terms of a quantitative theory of mechanical self-regulation in the cardiovascular system.

scholarly journals Degree of Approximation of Functionsf~∈HωClass by the(Np·E1)Means in the Hölder Metric

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Vishnu Narayan Mishra ◽  
Kejal Khatri
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Hölder Metric ◽  
Approximation Of A Function

A new estimate for the degree of approximation of a functionf˜∈Hωclass by(Np·E1)means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001). Some corollaries have also been deduced from our main theorem.

scholarly journals On Caputo–Fabrizio Fractional Integral Inequalities of Hermite–Hadamard Type for Modified h -Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xiaobin Wang ◽  
Muhammad Shoaib Saleem ◽  
Kiran Naseem Aslam ◽  
Xingxing Wu ◽  
Tong Zhou
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Convex Function ◽  
Convex Functions ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Applied Mathematics ◽  
Fractional Integral Operator ◽  
Real Numbers ◽  
Special Means

The theory of convex functions plays an important role in engineering and applied mathematics. The Caputo–Fabrizio fractional derivatives are one of the important notions of fractional calculus. The aim of this paper is to present some properties of Caputo–Fabrizio fractional integral operator in the setting of h -convex function. We present some new Caputo–Fabrizio fractional estimates from Hermite–Hadamard-type inequalities. The results of this paper can be considered as the generalization and extension of many existing results of inequalities and convex functions. Moreover, we also present some application of our results to special means of real numbers.

Convex Polynomial Approximation in the Uniform Norm: Conclusion

2005 ◽  
Vol 57 (6) ◽  
pp. 1224-1248 ◽  
Author(s):  
K. A. Kopotun ◽  
D. Leviatan ◽  
I. A. Shevchuk
Keyword(s):  
Convex Function ◽  
Polynomial Approximation ◽  
Uniform Norm ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
Type Estimate ◽  
Open Problems ◽  
Open Questions

AbstractEstimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

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