Fágaðar nálganir og fjölmættisfræði - verkefni lokið
Fréttatilkynning verkefnisstjóra
Nálgun á föllum með margliðum er stórt rannsóknasvið í stærðfræðigreiningu með ótal hagnýtingum. Til þess að nálgunarverkefni séu vel skilgreind, þarf að hafa til taks staðla sem gefa mælikvarða á gæði nálgananna sem um ræðir. Í þessu verkefni er nálgun á fáguð föll af mörgum breytistærðum rannsökuð.
Gæði nálgananna eru sett fram með vigtum sem skilgreindar eru með fjölundirþýðum föllum og fást þá bæði vegnir eframarksstaðlar og vegnir ferviksstaðlar. Þannig verða til flokkar af margliðum sem litið er á sem heil fáguð föll sem uppfylla ákveðin vaxtarskilyrði og flokkar af fjölundirþýðum föllum sem skilgreina möguleg vaxtarskilyrði. Tilvistarfræðum Hörmanders fyrir lausnir á Cauchy-Riemann-jöfnum er beitt til þess að rannsaka sambandið milli þessara tveggja ólíku tegunda af flokkum falla. Niðurstöðurnar hjálpa okkur að skilja betur samspil fágaðra falla og fjölmættisfræðinnar, sem er mun sveigjanlegri. Þetta mun hjálpa okkur að afhjúpa tengslin milli mismunandi sviða stærðfræðinnar og hagnýtinga á þeim.
English:
Approximation of functions by polynomials is a major field of research in mathematical analysis with numerous applications. In order to present a well posed approximation method, norms have to be defined which reflect the quality of the approximations at hand. In this research project approximation of holomorphic functions of several complex variables is investigated. The quality of the approximations is presented in terms of weights which are defined by plurisubharmonic functions and then both supremum norms and weighted square norms appear. This leads to classes of polynomials which are viewed as entire functions satisfying possible growth conditions. Hörmander's existence theory for solutions of the Cauchy-Riemann equations system are applied to investigate the connection between these two different classes of functions.
Information on how the results will be applied:
The results help us understand better the interplay between the rigid world of holomorphic functions and the more flexible world of pluripotential theory. This will help us unveil the connections between different complex mathematical structures and their applications.
A list of the project’s outputs:
[1] Masanori Adachi, Séverine Biard. On Levi flat hypersurfaces with transversely affine foliation. Math. Z. 301 (2022), no. 1, 373–383. 32V40 (32D15 32M25 32T15). doi.org/10.1007/s00209-021-02927-z
[2] Master’s thesis: Eggert Karl Hafsteinsson. Random Polynomials and Convex Bodies. (2022). Supervisor: Benedikt Magnússon and Ragnar Sigurðsson. skemman.is/handle/1946/41548
[3] Masanori Adachi, Séverine Biard, Judith Brinkschulte. A residue formula for meromorphic connections and applications to stable sets of foliations. J. Geom. Anal. 33 (2023), no. 10. doi.org/10.1007/s12220-023-01385-9
[4] Giovanni Domenico Di Salvo, Tyson Ritter, Erlend F. Wold. Families of Proper Holomorphic Embeddings and Carleman-type Theorems with parameters. J. Geom. Anal. 33 (2023), no. 3. doi.org/10.1007/s12220-022-01110-y
[5] Preprint: Benedikt Steinar Magnússon, Álfheiður Edda Sigurðardóttir, Ragnar Sigurðsson, Bergur Snorrason. Polynomials with exponents in compact convex sets and associated weighted extremal functions - Fundamental results. (2023). arxiv.org/abs/2305.04779. Under review in Annales Polonici Mathematici.
[6] Preprint: Benedikt Steinar Magnússon, Ragnar Sigurðsson, Bergur Snorrason. Polynomials with exponents in compact convex sets and associated weighted extremal functions - The Bernstein-Walsh-Siciak theorem. (2023). arxiv.org/abs/2306.02486. Under review in Complex Analysis and its Synergies.
[7] Preprint: Benedikt Steinar Magnússon, Álfheiður Edda Sigurðardóttir, Ragnar Sigurðsson. Polynomials with exponents in compact convex sets and associated weighted extremal functions - The Siciak-Zakharyuta theorem. (2023). arxiv.org/abs/2305.08260. Under review in Complex Analysis and its Synergies.
[8] Master’s thesis: Arngunnur Einarsdóttir. Convergence of power series in several complex variables. (2023). Supervisors: Benedikt Magnússon and Ragnar Sigurðsson. skemman.is/handle/1946/45833
[9] Preprint: Séverine Biard, Jujie Wu. Equivalence between VMO functions and Zero Lelong numbers functions. (2024). arxiv.org/abs/2403.03568
[10] Manuscript: Bergur Snorrason. Polynomials with exponents in compact convex sets and associated weighted extremal functions - Generalized product property. (2024)
Heiti verkefnis: Fágaðar
nálganir og fjölmættisfræði - Holomorphic
approximation and pluripotential theory
Verkefnisstjórar: Ragnar Sigurðsson og Benedikt
Steinar Magnússon,
Háskóla Íslands, Tyson Ritter, University of Stavanger og Severine
Michele Jeanine Biard, Háskóla Íslands
Tegund styrks: Verkefnisstyrkur
Styrktímabil: 2022
Fjárhæð styrks kr. 56.249.750
Tilvísunarnúmer Rannsóknasjóðs: 207236
