Good-irreducible inner functions on a polydisc
Annales de l'Institut Fourier, Tome 29 (1979) no. 2, pp. 185-210.

Une formule explicite est développée pour les fonctions de classe de Nevanlinna qui sont “suffisamment rationnelles” sur la frontière et elle est alors utilisée pour déduire l’unicité de la factorisation de telles fonctions intérieures. Une généralisation d’un théorème de Frostman est présentée et les résultats ci-dessus sont alors appliqués pour construire des fonctions intérieures irréductibles et/ou “bonnes” sur un polydisque.

An explicit formula is developed for Nevanlinna class functions whose behaviour at the boundary is “sufficiently rational” and is then used to deduce the uniqueness of the factorization of such inner functions. A generalization of a theorem of Frostman is given and the above results are then applied to the construction of good and/or irreducible inner functions on a polydisc.

@article{AIF_1979__29_2_185_0,
     author = {Sawyer, Eric T.},
     title = {Good-irreducible inner functions on a polydisc},
     journal = {Annales de l'Institut Fourier},
     pages = {185--210},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {2},
     year = {1979},
     doi = {10.5802/aif.746},
     mrnumber = {82a:32011},
     zbl = {0381.32007},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.746/}
}
TY  - JOUR
AU  - Sawyer, Eric T.
TI  - Good-irreducible inner functions on a polydisc
JO  - Annales de l'Institut Fourier
PY  - 1979
SP  - 185
EP  - 210
VL  - 29
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.746/
DO  - 10.5802/aif.746
LA  - en
ID  - AIF_1979__29_2_185_0
ER  - 
%0 Journal Article
%A Sawyer, Eric T.
%T Good-irreducible inner functions on a polydisc
%J Annales de l'Institut Fourier
%D 1979
%P 185-210
%V 29
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.746/
%R 10.5802/aif.746
%G en
%F AIF_1979__29_2_185_0
Sawyer, Eric T. Good-irreducible inner functions on a polydisc. Annales de l'Institut Fourier, Tome 29 (1979) no. 2, pp. 185-210. doi : 10.5802/aif.746. http://www.numdam.org/articles/10.5802/aif.746/

[1]P. R. Ahern, Singular sets of inner functions, Indiana Univ. Math. J., 21 (1971), 147-155. | MR | Zbl

[2]A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta. Math., 81 (1949), 239-255. | MR | Zbl

[3]R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier, Grenoble, 20, 1 (1970), 37-76. | Numdam | MR | Zbl

[4]O. Frostman, Potentiel d'équilibre et capacité des ensembles..., Lunds Univ. Mat. Sem. d., (1935), 1-118. | JFM | Zbl

[5]L. L. Helms, Introduction to Potential Theory, Wiley-Interscience, 1969. | MR | Zbl

[6]E. A. Nordgren, Composition operators, Can. J. Math., 20 (1968), 442-449. | MR | Zbl

[7] W. Rudin, Function Theory in Polydiscs, Benjamin, 1969. | MR | Zbl

[8] W. Rudin and P. R. Ahern, Factorizations of bounded holomorphic functions, Duke Math. J., 39 (1972), 767-777. | MR | Zbl

[9]W. Rudin and E. L. Stout, Boundary properties of functions of several complex variables, J. Math. Mech., 14 (1965), 991-1006. | MR | Zbl

[10]J. V. Ryff, Subordinate Hp functions, Duke Math. J., 33 (1966), 347-354. | MR | Zbl

[11]E. Sawyer, Inner functions on polydiscs, Ph. D. thesis, McGill University (1977).

[12]M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co. | Zbl

[13]A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, 1959. | Zbl

Cité par Sources :