scholarly journals Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
José A. Adell ◽  
A. Lekuona
Keyword(s):  
Piecewise Linear ◽  
Bernstein Polynomials ◽  
Positive Linear Operator ◽  
Linear Operators ◽  
Linear Functions ◽  
Closed Form Expression ◽  
Piecewise Linear Functions ◽  
Form Expression ◽  
Best Constant ◽  
Quantitative Estimates

We give accurate estimates of the constantsCn(A(I),x)appearing in direct inequalities of the form|Lnf(x)-f(x)|≤Cn(A(I),x)ω2  f;σ(x)/n,f∈A(I),x∈I,  and  n=1,2,…,whereLnis a positive linear operator reproducing linear functions and acting on real functionsfdefined on the intervalI,A(I)is a certain subset of such functions,ω2(f;·)is the usual second modulus off, andσ(x)is an appropriate weight function. We show that the size of the constantsCn(A(I),x)mainly depends on the degree of smoothness of the functions in the setA(I)and on the distance from the pointxto the boundary ofI. We give a closed form expression for the best constant whenA(I)is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.

Modelling the single-electron transistor with piecewise linear functions

2009 ◽  
Author(s):  
Arturo Sarmiento-Reyes ◽  
Luis Hernandez-Martinez ◽  
Miguel Angel Gutierrez de Anda ◽  
Francisco Javier Castro Gonzalez
Keyword(s):  
Piecewise Linear ◽  
Linear Functions ◽  
Single Electron ◽  
Single Electron Transistor ◽  
Piecewise Linear Functions

Mesh duality and Legendre duality

1990 ◽  
Vol 428 (1875) ◽  
pp. 351-377 ◽  
Keyword(s):  
Piecewise Linear ◽  
Voronoi Tessellation ◽  
Tangent Plane ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Dual Transformation ◽  
Delaunay Mesh ◽  
Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

scholarly journals Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 166 ◽  
Author(s):  
Andreas Griewank ◽  
Andrea Walther
Keyword(s):  
Piecewise Linear ◽  
Linear Representation ◽  
Local Minimizer ◽  
Descent Method ◽  
Linear Functions ◽  
Steepest Descent Method ◽  
Piecewise Linear Functions ◽  
Piecewise Linearization ◽  
Algorithmic Differentiation ◽  
Concave Part

For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f ˇ and a concave part f ^ , including a pair of generalized gradients g ˇ ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f ˇ and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g ˇ and − g ^ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization.

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