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Remarks on de la Vallée Poussin means and convex conformal maps of the circle

1958; Mathematical Sciences Publishers; Volume: 8; Issue: 2 Linguagem: Inglês

10.2140/pjm.1958.8.295

1945-5844

Georg Pólya, I. J. Schoenberg,

Point processes and geometric inequalities

Introduction.The aims of the present remarks are similar to those pursued by L. Fejer in several papers in the early nineteen thirties and well described by the title of one of his paper: Gestaltlίckes ύber die Partialsummen und ihre Mittelwerte bei der Fourierreihe und der Potenzreihe.However, the means which we use to realize these aims are different.Fejer discovered the remarkable behavior of certain Cesaro means, especially that of the third Cesaro means for even or odd functions of certain simple basic shapes.In what follows we show that the de la Vallee Poussin means possess such shape-preserving properties to a much higher degree thanks to their variation diminishing character.Before stating our results, we have to explain a few concepts.Variation diminishing Transformations on the Circle.If a 19 a 2 , , a n s a finite sequence of real numbers we shall denote by v{a) or v(a v ) the number of variations of sign in the terms of this sequence.By the number v c (a) of cyclic variations of sign of our sequence we mean the following: If all a v -0 we set v c (a) = 0.If α^O we set v o (a) = v(a i9 a ι+19, a n9 a 19 α 2 , , α f _i, α 4 ) .