Permutations and piecewise-constant approximation of continuous functions of n variables

1998 ◽  
Vol 50 (7) ◽  
pp. 1031-1044 ◽  
Author(s):  
N. P. Korneichuk
Keyword(s):  
Continuous Functions ◽  
Piecewise Constant ◽  
Piecewise Constant Approximation ◽  
Approximation Of Continuous Functions

scholarly journals On approximation of continuous functions by piecewise-constant ones in integral metrics

1998 ◽  
Vol 6 ◽  
pp. 128
Author(s):  
O.V. Chernytska
Keyword(s):  
Upper Bound ◽  
Best Approximation ◽  
Continuous Functions ◽  
Moduli Of Continuity ◽  
Piecewise Constant ◽  
Integral Metrics ◽  
The Best Approximation ◽  
Piecewise Constant Functions ◽  
Approximation Of Continuous Functions ◽  
Decreasing Functions

We obtain upper bound of the best approximation of the classes $H^{\omega} [a, b]$ by piecewise-constant functions over uniform split in metrics of $L_{\varphi}[a, b]$ spaces, which are generated by continuous non-decreasing functions $\varphi$ that are equal to zero in zero. We study the classes of functions $\varphi$, for which the obtained bound is exact for all convex moduli of continuity.

ReLU Networks Are Universal Approximators via Piecewise Linear or Constant Functions

2020 ◽  
Vol 32 (11) ◽  
pp. 2249-2278
Author(s):  
Changcun Huang
Keyword(s):  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Piecewise Linear Approximation ◽  
Piecewise Constant ◽  
Deep Layers ◽  
Univariate Function ◽  
Arbitrary Precision ◽  
Piecewise Constant Approximation ◽  
Neural Unit ◽  
Universal Approximators

This letter proves that a ReLU network can approximate any continuous function with arbitrary precision by means of piecewise linear or constant approximations. For univariate function [Formula: see text], we use the composite of ReLUs to produce a line segment; all of the subnetworks of line segments comprise a ReLU network, which is a piecewise linear approximation to [Formula: see text]. For multivariate function [Formula: see text], ReLU networks are constructed to approximate a piecewise linear function derived from triangulation methods approximating [Formula: see text]. A neural unit called TRLU is designed by a ReLU network; the piecewise constant approximation, such as Haar wavelets, is implemented by rectifying the linear output of a ReLU network via TRLUs. New interpretations of deep layers, as well as some other results, are also presented.

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