Piecewise monotone functions

1983 ◽  
pp. 19-40
Author(s):  
Chris Preston
Keyword(s):  
Monotone Functions ◽  
Piecewise Monotone

On Comonotone Approximation

1983 ◽  
Vol 26 (2) ◽  
pp. 220-224 ◽  
Author(s):  
R. K. Beatson ◽  
D. Leviatan
Keyword(s):  
Jackson Type ◽  
Monotone Functions ◽  
Piecewise Monotone ◽  
Comonotone Approximation

AbstractJackson type theorems are obtained for the comonotone approximation of piecewise monotone functions by polynomials.

scholarly journals Asymptotic behaviour of iterated piecewise monotone maps

1988 ◽  
Vol 8 (1) ◽  
pp. 111-131 ◽  
Author(s):  
Jürgen Willms
Keyword(s):  
Finite Number ◽  
Asymptotic Behaviour ◽  
Disjoint Union ◽  
Monotone Functions ◽  
Connected Component ◽  
Piecewise Monotone ◽  
Closed Intervals ◽  
Monotone Maps ◽  
Typical Point ◽  
Topological Sense

AbstractIn this paper the asymptotic behaviour of piecewise monotone functions f: I → I with a finite number of discontinuities is studied (where I ⊆ ℝ is a compact interval). It is shown that there is a finite number of f-almost-invariant subsets C1,…, Cr, R1,…, Rs, where each Ci is a disjoint union of closed intervals and each Rj is a Cantor-like subset of I, such that if x is a ‘typical’ point in I (in a topological sense) then exactly one of the following three possibilities will happen:(1) {fn (x)}n ≥ 0 eventually ends up in some Ci.(2) {fn (x)}n ≥ 0 is attracted to some Rj.(3) {fn (x): n ≥ 0} is contained in an open, invariant set Z ⊆ I, which is such that for each n ≥ 1 fn is monotone and continuous on each connected component of Z.Moreover, f acts topologically transitively on each Ci and minimally on each Rj. Furthermore, it is shown how the sets C1,…, Cr, R1,…, Rs can be constructed. Finally, our results are applied to some examples.

Coapproximation of piecewise-monotone functions by polynomials

1981 ◽  
Vol 30 (6) ◽  
pp. 920-924
Author(s):  
A. S. Shvedov
Keyword(s):  
Monotone Functions ◽  
Piecewise Monotone
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