On the preservation of periodic monotonicity

Abstract

A 2π-periodic continuous real functionf is said to beperiodically monotone if it has the following property: there exist numbert ₁≤t ₂≤t ₃≔t ₁+2π such thatf is nonincreasing fort ₁≤t ₂ and nondecreasing int ₂≤t≤t ₃. For any 2π-periodic, integrable real functiong with ∫ ^2π₀ |g(t|dt<∞) we define

$$(f * g)(x): = \frac{1}{{2\pi }}\int_0^{2\pi } {f(t)} g(x - t)dt$$

g is said to beperiodic monotonicity preserving (g∈PMP) iff*g is periodically monotone wheneverf 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 inPMP. 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.