Rannsóknir í tvinnfallagreiningu með áherslu á fjölmættisfræði - verkefni lokið
Fréttatilkynning verkefnisstjóra
Verkefnið hefur að mestu leyti snúist um fágaðar framlengingar og nálganir með fáguðum föllum. Annars vegar höfum við skoðað framlengingar yfir í heil föll í tengslum við Radon-ummyndunina [1] og hins vegar mat á því hversu langt er hægt að framlengja fáguð föll í tengslum við nálganir með margliðum [8,9].
Tilvist á nálgunum með fáguðum föllum er í raun spurning um
það hvort hlutrúm fágaðra falla sé þétt í tilteknu fallarúmi. Við höfum skoðað
þetta verkefni þegar undirliggjandi mengi er lokað og ótakmarkað [2,6] og þegar
undirliggjandi mengi er þjappað [2,3]. Í flestum tilvikum hafa fjölundirþýð
föll gegnt einhverju hlutverki, annað hvort sem vigt í L2-fallarúmi
[2,3,4,5] eða sem tæming fyrir undirliggjandi mengi [1,7,8,9].
English:
The project was mostly centered around holomorphic extensions and holomorphic approximations. On the one hand, we have looked at the extensions to entire functions in connection with the Radon transform [1] and on the other hand how far holomorphic functions on a neighborhood of compact set can be extended in relation to polynomial approximations on the compact set [8,9]. The existence of approximation is in principle a question about the density of holomorphic function in a given function space. We have worked on this when the underlying set is closed and unbounded [2,6] and when the underlying set is compact [2,3]. In most cases, plurisubharmonic functions have played some role, either as a weight in a L2 space [2,3,4,5] or as an exhaustion for the underlying set [1,7,8,9].
[1] Bergh, J., & Sigurdsson, R. (2019). Siciak's homogeneous extremal functions, holomorphic extension and a generalization of Helgason's support theorem. Annales Polonici Mathematici, 123(1), 61–70. https://doi.org/10.4064/ap190128-22-7
[2] Biard, S., Fornæss, J. E., & Wu, J. (2019a). Weighted L2 version of Mergelyan and Carleman approximation. arXiv:1910.05777 [math]. http://arxiv.org/abs/1910.05777
[3] Biard, S., Fornæss, J. E., & Wu, J. (2019b). Weighted-L2 polynomial approximation in C. arXiv:1805.11756 [math]. http://arxiv.org/abs/1805.11756
[4] Biard, S., & Straube, E. J. (2017). L2-Sobolev theory for the complex Green operator. International Journal of Mathematics, 28(9), 1740006, 31. https://doi.org/10.1142/S0129167X17400067
[5] Biard, S., & Straube, E. J. (2019). Estimates for the complex Green operator: symmetry, percolation, and interpolation. Transactions of the American Mathematical Society, 371(3), 2003–2020. https://doi.org/10.1090/tran/7385
[6] Magnússon, B. S., & Wold, E. F. (2016). A Characterization of Totally Real Carleman Sets and an Application to Products of Stratified Totally Real Sets. Mathematica Scandinavica, 118(2), 285–290. https://doi.org/10.7146/math.scand.a-23690
[7] Sigurdsson, R., & Snæbjarnarson, A. S. (2019). Monge–Ampère measures of pluri-subharmonic exhaustions associated to the Lie norm of holomorphic maps. Annales Polonici Mathematici, 123(1), 481–504. https://doi.org/10.4064/ap181001-1-4
[8] Snæbjarnarson, A. (2019a). Polynomial approximation on Stein manifolds and the Monge-Ampère operator. Doctoral Thesis at the University of Iceland, Reykjavik, Iceland.
Snæbjarnarson, A. (2019b). Rapid polynomial approximation on Stein manifolds. Annales Polonici Mathematici, 122(1), 81–100. https://doi.org/10.4064/ap180711-13-11
Heiti
verkefnis: Rannsóknir í tvinnfallagreiningu með áherslu á fjölmættisfræði/Research
in Complex Analysis with Emphasis on Pluripotential Theory
Verkefnisstjóri: Ragnar Sigurðsson, Raunvísindastofnun HÍ
Tegund styrks: Verkefnisstyrkur
Styrktímabil: 2015-2017
Fjárhæð styrks: 22,326 millj. kr. alls
Tilvísunarnúmer Rannís: 152572
