Let $B$ be the open unit ball in ▫${\Bbb C}^2$▫. This paper deals with the analog of Hartogs' separate analyticity theorem for CR functions on the sphere ▫$bB$▫. We prove such a theorem for functions in ▫$C^\infty (bB)$▫: If ▫$a, b \in \overline B$▫, ▫$a \ne b$▫ and if ▫$f \in C^\infty (bB)$▫ extends holomorphically into ▫$B$▫ along any complex line passing through either ▫$a$▫ or ▫$b$▫, then ▫$f$▫ extends holomorphically through ▫$B$▫. On the other hand, for each ▫$k \in \Bbb{N}$▫ there is a function ▫$f \in C^k(bB)$▫ which extends holomorphically into ▫$B$▫ along any complex line passing through either ▫$a$▫ or ▫$b$▫ yet ▫$f$▫ does not extend holomorphically through ▫$B$▫. More generally, in the paper we obtain a fairly complete description of pairs of points ▫$a, b \in {\Bbb C}^2$, $a \ne b$▫, such that if ▫$f \in C^\infty (bB)$▫ extends holomorphically into ▫$B$▫ along every complex line passing through either ▫$a$▫ or ▫$b$▫ that meets ▫$B$▫, then ▫$f$▫ extends holomorphically through ▫$B$▫.
COBISS.SI-ID: 16521561
We prove a bilinear embedding theorem for Schrödinger operators with nonnegative potentials. The embedding, acting on the cartesian product of ▫$L^p({\mathbb R}^n)$▫ and its dual, involves estimates that are independent of the dimension ▫$n$▫ and linear in terms of ▫$p$▫. This feature is achieved by means of a particular Bellman function which satisfies three crucial properties. Connections with known results on the Heisenberg group as well as with results for the Hilbert transform along the parabola are also explored. We believe our approach is quite universal in the sense that one could apply it to a whole range of Riesz transforms arising from various differential operators.
COBISS.SI-ID: 16214873
We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in ▫$\mathbb{C}^n$▫ by Lempert and by Lárusson and Sigurdsson.
COBISS.SI-ID: 16436057
Polynomial geometric interpolation by parametric curves has become one of the standard techniques for interpolation of geometric data. An obvious generalization leads to rational geometric interpolation schemes, which are a much less investigated research topic. The aim of this paper is to present a general framework for Hermite geometric interpolation by rational Bézier spatial curves. In particular, cubic ▫$G^2$▫ and quartic ▫$G^3$▫ interpolations are analyzed in detail. Systems of nonlinear equations are derived in a simplified form, and the existence of admissible solutions is studied. For the cubic case, geometric conditions implying solvability of the nonlinear system are also stated. The asymptotic analysis is done in both cases, and optimal approximation orders are proved. Numerical examples are given, which confirm the theoretical results.
COBISS.SI-ID: 16449369
Let ▫$\mathbb{B}$▫ be the open unit ball in ▫$\mathbb{C}^2$▫ and let ▫$a, b, c$▫ be three points in ▫$\mathbb{C}^2$▫ which do not lie in a complex line, such that the complex line through ▫$a, b$▫ meets ▫$\mathbb{B}$▫ and such that if one of the points ▫$a, b$▫ is in ▫$\mathbb{B}$▫ and the other in ▫$\mathbb{C}^2 \setminus \overline {\mathbb{B}}$▫ then ▫$\langle a\vert b \rangle \not = 1$▫ and such that at least one of the numbers ▫$\langle a\vert c\rangle$▫, ▫$\langle b \vert c \rangle$▫ is different from ▫$1$▫. We prove that if a continuous function ▫$f$▫ on ▫$b\mathbb{B}$▫ extends holomorphically into ▫$\mathbb{B}$▫ along each complex line which meets ▫$\{ a, b, c\}$▫, then ▫$f$▫ extends holomorphically through ▫$\mathbb{B}$▫. This generalizes the recent result of L. Baracco who proved such a result in the case when the points ▫$a, b, c$▫ are contained in ▫$\mathbb{B}$▫. The proof is quite different from the one of Baracco and uses the following one-variable result, which we also prove in the paper: Let ▫$\Delta $▫ be the open unit disc in ▫$\mathbb{C}$▫. Given ▫$\alpha \in \Delta$▫ let ▫$\mathcal {C}_\alpha$▫ be the family of all circles in ▫$\Delta$▫ obtained as the images of circles centered at the origin under an automorphism of ▫$\Delta$▫ that maps ▫$0$▫ to ▫$\alpha$▫. Given ▫$\alpha , \beta \in \Delta$▫, ▫$\alpha \not = \beta$▫, and ▫$n \in \mathbb{N}$▫, a continuous function ▫$f$▫ on ▫$\overline {\Delta }$▫ extends meromorphically from every circle ▫$\Gamma \in \mathcal{C}_\alpha \cup \mathcal{C}_\beta$▫ through the disc bounded by ▫$\Gamma$▫ with the only pole at the center of ▫$\Gamma$▫ of degree not exceeding ▫$n$▫ if and only if ▫$f$▫ is of the form ▫$f(z) = a_0(z) + a_1(z){\overline z} +\cdots + a_n(z){\overline z}^n$▫ ▫$(z \in \Delta)$▫ where the functions ▫$a_j, \; 0 \leq j \leq n$▫, are holomorphic on ▫$\Delta $▫.
COBISS.SI-ID: 16364633