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Sets in N with vanishing global extremal function and polynomial approximation

Józef Siciak (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Let Γ be a non-pluripolar set in N . Let f be a function holomorphic in a connected open neighborhood G of Γ . Let { P n } be a sequence of polynomials with deg P n d n ( d n < d n + 1 ) such that lim sup n | f ( z ) - P n ( z ) | 1 / d n < 1 , z Γ . We show that if lim sup n | P n ( z ) | 1 / d n 1 , z E , where E is a set in N such that the global extremal function V E 0 in N , then the maximal domain of existence G f of f is one-sheeted, and lim sup n f - P n K 1 d n < 1 for every compact set K G f . If, moreover, the sequence { d n + 1 / d n } is bounded then G f = N .If E is a closed set in N then V E 0 if and only if each series of homogeneous polynomials j = 0 Q j , for which some subsequence { s n k } ...

Singular integrals with highly oscillating kernels on product spaces

Elena Prestini (2000)

Colloquium Mathematicae

We prove the L 2 ( 2 ) boundedness of the oscillatory singular integrals P 0 f ( x , y ) = D x e i ( M 2 ( x ) y ' + M 1 ( x ) x ' ) ο v e r x ' y ' f ( x - x ' , y - y ' ) d x ' d y ' for arbitrary real-valued L functions M 1 ( x ) , M 2 ( x ) and for rather general domains D x 2 whose dependence upon x satisfies no regularity assumptions.

Singular sets of separately analytic functions

Zbigniew Błocki (1992)

Annales Polonici Mathematici

We complete the characterization of singular sets of separately analytic functions. In the case of functions of two variables this was earlier done by J. Saint Raymond and J. Siciak.

Singularité de séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres

Driss Essouabri (1997)

Annales de l'institut Fourier

Soit P [ X 1 , ... , X n ] un polynôme. On appelle série de Dirichlet associée à P la fonction : s Z ( P ; s ) = m * n P ( m ) - s ( s ) . Dans cet article nous étudions l’existence et les propriétés du prolongement méromorphe d’une telle série sous l’hypothèse qu’il existe B ] 0 , 1 [ tel que : i) P ( x ) + quand | | x | | + et x [ B , + [ n et ii) d ( Z ( P ) , [ B , + [ n ) > 0 Z ( P ) = { z n | P ( z ) = 0 } . Cette hypothèse est probablement optimale et en tout cas contient strictement toutes les classes de polynômes déjà traitées antérieurement. Sous cette hypothèse nos principaux résultats sont : l’existence du prolongement méromorphe au plan...

Solution d'une conjecture de C. Berenstein - A. Yger et invariants de contact à l'infini

Michel Hickel (2001)

Annales de l’institut Fourier

Soient k un corps commutatif et I = ( p 1 , , p m ) k n [ X ] un idéal de l’anneau des polynômes k [ X 1 , , X n ] (éventuellement I = k n [ X ] ). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme p , élément de la clôture intégrale I ¯ de l’idéal I , on a une représentation p m = 1 i m p i q i , avec max deg ( q i p i ) m deg p + m d 1 d m , d i = deg p i , 1 i m .

Solutions to a class of polynomially generalized Bers–Vekua equations using Clifford analysis

Min Ku, Uwe Kähler, Paula Cerejeiras (2012)

Archivum Mathematicum

In this paper a class of polynomially generalized Vekua–type equations and of polynomially generalized Bers–Vekua equations with variable coefficients defined in a domain of Euclidean space are discussed. Using the methods of Clifford analysis, first the Fischer–type decomposition theorems for null solutions to these equations are obtained. Then we give, under some conditions, the solutions to the polynomially generalized Bers–Vekua equation with variable coefficients. Finally, we present the structure...

Currently displaying 961 – 980 of 1395