Intrinsic interpolation, near-circularity and maximal convergence
Let E be compact and connected with capE>0 and connected complement Ω=ℂ¯∖E, let gΩ(z,∞) be the Green’s function of Ω with pole at infinity and let Eσ≔{z∈Ω:gΩ(z,∞)<logσ}∪E,1<σ<∞, be the Green domains with boundaries Γσ. Let f be holomorphic on E and let ρ(f) denote the maximal parameter o...
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| Published in: | Journal of approximation theory Vol. 312; p. 106201 |
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| Abstract | Let E be compact and connected with capE>0 and connected complement Ω=ℂ¯∖E, let gΩ(z,∞) be the Green’s function of Ω with pole at infinity and let Eσ≔{z∈Ω:gΩ(z,∞)<logσ}∪E,1<σ<∞, be the Green domains with boundaries Γσ. Let f be holomorphic on E and let ρ(f) denote the maximal parameter of holomorphy of f and let pnn∈N be a sequence of polynomials converging maximally to f on E. If σ, 1<σ<ρ(f)<∞, is fixed and if mn(σ) denotes the number of interpolation points of pn to f in Eσ with normalized counting measure μσ,n, then there exists a subset Λ⊂N such that mn(σ)=n+o(n)asn∈Λ,n→∞,μσ,n|Ê+μσ,n|Ω⟶∗μEasn∈Λ,n→∞, where μσ,n=μσ,n|E+μσ,n|Ω, μσ,n|Ê denotes the balayage measure of μσ,n|E onto the boundary of E and μE is the equilibrium measure of E. Moreover, there exists a sequence σnn∈Λ converging to σ such that the closed curves γn=(f−pn)(Γσn) do not pass through the point 0 and the winding numbers Indγn(0) satisfy Indγn(0)=mn(σn)=n+o(n)asn∈Λ,n→∞. |
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| AbstractList | Let E be compact and connected with capE>0 and connected complement Ω=ℂ¯∖E, let gΩ(z,∞) be the Green’s function of Ω with pole at infinity and let Eσ≔{z∈Ω:gΩ(z,∞)<logσ}∪E,1<σ<∞, be the Green domains with boundaries Γσ. Let f be holomorphic on E and let ρ(f) denote the maximal parameter of holomorphy of f and let pnn∈N be a sequence of polynomials converging maximally to f on E. If σ, 1<σ<ρ(f)<∞, is fixed and if mn(σ) denotes the number of interpolation points of pn to f in Eσ with normalized counting measure μσ,n, then there exists a subset Λ⊂N such that mn(σ)=n+o(n)asn∈Λ,n→∞,μσ,n|Ê+μσ,n|Ω⟶∗μEasn∈Λ,n→∞, where μσ,n=μσ,n|E+μσ,n|Ω, μσ,n|Ê denotes the balayage measure of μσ,n|E onto the boundary of E and μE is the equilibrium measure of E. Moreover, there exists a sequence σnn∈Λ converging to σ such that the closed curves γn=(f−pn)(Γσn) do not pass through the point 0 and the winding numbers Indγn(0) satisfy Indγn(0)=mn(σn)=n+o(n)asn∈Λ,n→∞. |
| ArticleNumber | 106201 |
| Author | Blatt, Hans-Peter |
| Author_xml | – sequence: 1 givenname: Hans-Peter surname: Blatt fullname: Blatt, Hans-Peter email: hans.blatt@ku.de organization: Katholische Universität Eichstätt-Ingolstadt, Mathematisch-Geographische Fakultät, 85071 Eichstätt, Germany |
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| Cites_doi | 10.1007/BF02559510 10.7146/math.scand.a-10741 10.1007/s00365-022-09564-7 |
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| Keywords | Weak convergence Interpolation 30E10 Condenser Complex approximation 41A10 Near-circularity Equilibrium measure Maximal convergence 41A05 |
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| References | Carleson (b7) 1964; 15 Ahlfors (b1) 1979 Blatt, Grothmann (b6) 2019 Rudin (b12) 1987 Blatt (b4) 2022; 56 Pommerenke (b10) 1975 Bagby (b2) 1967; 17 Grothmann (b9) 1996; 34 Saff, Totik (b13) 1997; vol. 316 Behnke, Sommer (b3) 1962 Ransford (b11) 1995 Walsh (b15) 1969; 20 Trefethen (b14) 1981; 31 Blatt (b5) 2024 Cornea (b8) 1991; 28 Blatt (10.1016/j.jat.2025.106201_b4) 2022; 56 Pommerenke (10.1016/j.jat.2025.106201_b10) 1975 Blatt (10.1016/j.jat.2025.106201_b5) 2024 Walsh (10.1016/j.jat.2025.106201_b15) 1969; 20 Grothmann (10.1016/j.jat.2025.106201_b9) 1996; 34 Trefethen (10.1016/j.jat.2025.106201_b14) 1981; 31 Carleson (10.1016/j.jat.2025.106201_b7) 1964; 15 Bagby (10.1016/j.jat.2025.106201_b2) 1967; 17 Blatt (10.1016/j.jat.2025.106201_b6) 2019 Behnke (10.1016/j.jat.2025.106201_b3) 1962 Rudin (10.1016/j.jat.2025.106201_b12) 1987 Cornea (10.1016/j.jat.2025.106201_b8) 1991; 28 Ransford (10.1016/j.jat.2025.106201_b11) 1995 Saff (10.1016/j.jat.2025.106201_b13) 1997; vol. 316 Ahlfors (10.1016/j.jat.2025.106201_b1) 1979 |
| References_xml | – volume: 20 year: 1969 ident: b15 article-title: Interpolation and approximation by rational functions in the complex domain publication-title: Amer. Math. Soc. Colloq. Pub. – volume: 34 start-page: 103 year: 1996 end-page: 117 ident: b9 article-title: Distribution of interpolation points publication-title: Ark. Mat. – start-page: 155 year: 2019 end-page: 169 ident: b6 article-title: Interpolation characteristics of maximal polynomial approximants to rational functions publication-title: Ann. Polon. Math. – volume: 31 start-page: 344 year: 1981 end-page: 367 ident: b14 article-title: Near-circularity of the error curve in complex Chebyshev approximation publication-title: JAT – volume: vol. 316 year: 1997 ident: b13 publication-title: Logarithmic Potentials with External Fields – year: 1962 ident: b3 article-title: Theorie der analytischen Funktionen einer komplexen Veränderlichen – year: 1995 ident: b11 article-title: Potential Theory in the Complex Plane publication-title: London Mathematical Society Student Texts – year: 1987 ident: b12 article-title: Real and Complex Analysis – volume: 17 start-page: 315 year: 1967 end-page: 329 ident: b2 article-title: The modulus of a plane condenser publication-title: J. Math. Mech. – year: 2024 ident: b5 article-title: Near-Circularity in Capacity and Maximally Convergent Polynomials – volume: 28 start-page: 829 year: 1991 end-page: 836 ident: b8 article-title: An identity theorem for logarithmic potentials publication-title: Osaka J. Math. – year: 1975 ident: b10 article-title: Univalent Functions – volume: 15 start-page: 167 year: 1964 end-page: 175 ident: b7 article-title: Mergelyan’s theorem on uniform polynomial approximation publication-title: Math. Scand. – year: 1979 ident: b1 article-title: Complex Analysis – volume: 56 start-page: 505 year: 2022 end-page: 535 ident: b4 article-title: Maximal convergence and interpolation on unconnected sets publication-title: Constr. Approx. – start-page: 155 year: 2019 ident: 10.1016/j.jat.2025.106201_b6 article-title: Interpolation characteristics of maximal polynomial approximants to rational functions publication-title: Ann. Polon. Math. – volume: 28 start-page: 829 year: 1991 ident: 10.1016/j.jat.2025.106201_b8 article-title: An identity theorem for logarithmic potentials publication-title: Osaka J. Math. – volume: vol. 316 year: 1997 ident: 10.1016/j.jat.2025.106201_b13 – year: 1995 ident: 10.1016/j.jat.2025.106201_b11 article-title: Potential Theory in the Complex Plane – volume: 20 year: 1969 ident: 10.1016/j.jat.2025.106201_b15 article-title: Interpolation and approximation by rational functions in the complex domain publication-title: Amer. Math. Soc. Colloq. Pub. – year: 1979 ident: 10.1016/j.jat.2025.106201_b1 – year: 1962 ident: 10.1016/j.jat.2025.106201_b3 – volume: 34 start-page: 103 year: 1996 ident: 10.1016/j.jat.2025.106201_b9 article-title: Distribution of interpolation points publication-title: Ark. Mat. doi: 10.1007/BF02559510 – volume: 15 start-page: 167 year: 1964 ident: 10.1016/j.jat.2025.106201_b7 article-title: Mergelyan’s theorem on uniform polynomial approximation publication-title: Math. Scand. doi: 10.7146/math.scand.a-10741 – volume: 17 start-page: 315 issue: 4 year: 1967 ident: 10.1016/j.jat.2025.106201_b2 article-title: The modulus of a plane condenser publication-title: J. Math. Mech. – year: 1975 ident: 10.1016/j.jat.2025.106201_b10 – year: 1987 ident: 10.1016/j.jat.2025.106201_b12 – volume: 31 start-page: 344 year: 1981 ident: 10.1016/j.jat.2025.106201_b14 article-title: Near-circularity of the error curve in complex Chebyshev approximation publication-title: JAT – volume: 56 start-page: 505 year: 2022 ident: 10.1016/j.jat.2025.106201_b4 article-title: Maximal convergence and interpolation on unconnected sets publication-title: Constr. Approx. doi: 10.1007/s00365-022-09564-7 – year: 2024 ident: 10.1016/j.jat.2025.106201_b5 |
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| SubjectTerms | Complex approximation Condenser Equilibrium measure Interpolation Maximal convergence Near-circularity Weak convergence |
| Title | Intrinsic interpolation, near-circularity and maximal convergence |
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