On the transpose of simple sets of polynomials effective in Faber regions

Nassif, M.

doi:10.1090/S0002-9939-1970-0262514-8

On the transpose of simple sets of polynomials effective in Faber regions
HTML articles powered by AMS MathViewer

by M. Nassif

Proc. Amer. Math. Soc. 25 (1970), 209-219

DOI: https://doi.org/10.1090/S0002-9939-1970-0262514-8

PDF | Request permission

Abstract:

The effectiveness properties, in a Faber region, of the transpose of simple absolutely monic sets of polynomials effective in the same region, are investigated in the present paper. A lower bound is calculated for an inevitable normalizing factor which ensures the finiteness of the Cannon function of the transpose of the normalized set in the region considered. Yet, except in the case of a circle with centre origin, no value of the normalizing factor can ensure the effectiveness of the transpose in the region considered.

References

Georg Faber, Über polynomische Entwickelungen, Math. Ann. 57 (1903), no. 3, 389–408 (German). MR 1511216, DOI 10.1007/BF01444293
W. F. Newns, On the representation of analytic functions by infinite series, Philos. Trans. Roy. Soc. London Ser. A 245 (1953), 429–468. MR 54717, DOI 10.1098/rsta.1953.0003
W. F. Newns, The product of two simple sets effective in a Faber region, Indag. Math. 22 (1960), 187–191. Nederl. Akad. Wetensch. Proc. Ser. A 63. MR 0110817
W. F. Newns, The range of effectiveness of a basic set, Proc. London Math. Soc. (3) 18 (1968), 745–767. MR 232002, DOI 10.1112/plms/s3-18.4.745
J.-M. Whittaker, Sur les Séries de Base de Polynomes Quelconques, Gauthier-Villars, Paris, 1949 (French). Avec la collaboration de C. Gattegno. MR 0032769