M A J O R A N T F U N C T I O N S . A N A L O G U E

O F T H E G I B B S ' P H E N O M E N O N *

M. Y a . Z i n g e r UDC517.512.6

Let ~ n be the c l a s s of a lgebra ic polynomials P(x) of degree no higher than n subject to the condition max IP(x) l = 1. One of the definitions of ma jo ran t function Mn(x) cons is t s of the following [o31

M~ (xo) = sup I p ' (xo)i, -- or < x o < -~ ~ . P(x)~ n

The p r o p e r t i e s of the functions {Mn(X) } were studied p rev ious ly by V. A. Markov, S. N. Be rnsh - tein, E. V. Voronovskii , and numerous other ma themat i c i ans . The list of well-known p rope r t i e s of the functions {Mn(x) } is given in par t one of the pape r .

The asympto t ic points {~, i} a r e studied, at which Mn(x ) = Mn_l(x), and the l imiting constants of growth of functions {Mn(x)} a r e found at these points as n ~ ~ .

I t is shown that in the neighborhoods (0; e) and (1 - e ; 1), e > 0, the boundary of in te rva l (0, 1) of the sequence of functions {Mn(x)}~= 1 has a p roper ty analogous to the Gibbs ' phenomenon for t r igonomet r i c s e r i e s . E s t i m a t e s for the Gibbs ' constant a re found.

*The comple te paper is deposited with VINITI as No. 3784-71.

Trans la t ed f r o m Siberski i Matemat icheski i Zhurnal , Vol. 13, No. 3, p. 724, May-June, 1972. Original a r t i c le submit ted November 17, 1970.

0 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

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