On the preservation of periodic monotonicity. Ruscheweyh, S. T. & Salinas, L. C. *Constructive Approximation*, 8(2):129-140, 1992. doi abstract bibtex A 2π-periodic continuous real function f is said to be periodically monotone if it has the following property: there exist number t1≤t2≤t3colon equalst1+2π such that f is nonincreasing for t1≤t2 and nondecreasing in t2≤t≤t3. For any 2π-periodic, integrable real function g with ∫02π|g(t|dt<∞) we define Mathematical expressiong is said to be periodic monotonicity preserving (g∈PMP) if f*g is periodically monotone whenever f is periodically monotone. This class of functions was introduced by I. J. Schoenberg in 1959. In the present paper we give an explicit description of the members in PMP. It turns out that an old necessary condition due to Loewner is (essentially) also sufficient. Our result extends to noncontinuous periodically monotone functions, solves Schoenberg's problem about the preservation of convex curves, and even improves on the present knowledge concerning properties of cyclic variation diminishing transforms. © 1992 Springer-Verlag New York Inc.

@article{10.1007/BF01238264,
abstract = "A 2π-periodic continuous real function f is said to be periodically monotone if it has the following property: there exist number t1≤t2≤t3{colon equals}t1+2π such that f is nonincreasing for t1≤t2 and nondecreasing in t2≤t≤t3. For any 2π-periodic, integrable real function g with ∫02π|g(t|dt<∞) we define {Mathematical expression}g is said to be periodic monotonicity preserving (g∈PMP) if f*g is periodically monotone whenever f is periodically monotone. This class of functions was introduced by I. J. Schoenberg in 1959. In the present paper we give an explicit description of the members in PMP. It turns out that an old necessary condition due to Loewner is (essentially) also sufficient. Our result extends to noncontinuous periodically monotone functions, solves Schoenberg's problem about the preservation of convex curves, and even improves on the present knowledge concerning properties of cyclic variation diminishing transforms. © 1992 Springer-Verlag New York Inc.",
number = "2",
year = "1992",
title = "On the preservation of periodic monotonicity",
volume = "8",
keywords = "AMS classification: 42A45, 42A85 , Convex curves , Periodic monotoxicity , Variation diminishing transforms",
pages = "129-140",
doi = "10.1007/BF01238264",
journal = "Constructive Approximation",
author = "Ruscheweyh, Stephan T. and Salinas, Luís C."
}

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