Background from analysis Background from computability theory Results References Computable Aspects of Inner Functions Timothy H. McNicholl mcnichollth@my.lamar.edu Department of Mathematics Lamar University March 30, 2007 / Graduate Student Seminar, GWU Timothy H. McNicholl Computable Aspects of Inner Functions
Background from analysis Background from computability theory Results References Outline Background from analysis 1 The class H ∞ ( D ) Some types of functions in H ∞ ( D ) Some types of inner functions Factorization Frostman’s Theorem Background from computability theory 2 Computability over the natural numbers Computability over uncountable spaces: Type-Two Effectivity Theory Statement of results 3 References 4 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Outline Background from analysis 1 The class H ∞ ( D ) Some types of functions in H ∞ ( D ) Some types of inner functions Factorization Frostman’s Theorem Background from computability theory 2 Computability over the natural numbers Computability over uncountable spaces: Type-Two Effectivity Theory Statement of results 3 References 4 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem The class H ∞ ( D ) D = df { z ∈ C : | z | < 1 } H ∞ ( D ) is the set of all bounded analytic functions f : D → C . For f ∈ H ∞ ( D ) , let � f � ∞ = sup {| f ( z ) | : z ∈ D } . H ∞ ( D ) is a Banach space under � � ∞ . Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Outline Background from analysis 1 The class H ∞ ( D ) Some types of functions in H ∞ ( D ) Some types of inner functions Factorization Frostman’s Theorem Background from computability theory 2 Computability over the natural numbers Computability over uncountable spaces: Type-Two Effectivity Theory Statement of results 3 References 4 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Kinds of functions in H ∞ ( D ) Q ∈ H ∞ ( D ) is outer if there is a positive measurable φ : ∂ D → R such that log φ ∈ L 1 ( ∂ D ) and � 1 � π e it + z � e it − z log φ ( e it ) dt Q ( z ) = λ exp . 2 π − π for some λ ∈ ∂ D . u ∈ H ∞ ( D ) is inner if lim z → z 0 | u ( z ) | = 1 for almost all z 0 ∈ ∂ D . Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Outline Background from analysis 1 The class H ∞ ( D ) Some types of functions in H ∞ ( D ) Some types of inner functions Factorization Frostman’s Theorem Background from computability theory 2 Computability over the natural numbers Computability over uncountable spaces: Type-Two Effectivity Theory Statement of results 3 References 4 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Singular functions Definition A function s ∈ H ∞ ( D ) is singular if there is a finite positive Borel measure on ∂ D , µ , that is singular with respect to Lebesgue measure and such that � π e it + z � � s ( z ) = exp − e it − z d µ ( t ) − π Theorem If s is singular, then: s is inner. 1 s ( 0 ) is a positive real number. 2 s has no zeros. 3 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Blaschke products Definition Let A = { a n } ∞ n = 0 be a sequence of points in D − { 0 } . The product ∞ | a n | a n − z B A , k ( z ) = df z k � a n 1 − a n z n = 0 is called a Blaschke product . We abbreviate B A , 0 with B A . Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Definition Let A = { a n } ∞ n = 0 be a sequence of points in D − { 0 } . The series ∞ � Σ A = df ( 1 − | a n | ) n = 0 is called the Blaschke sum of A . The inequality ∞ � ( 1 − | a n | ) < ∞ n = 0 is called the Blaschke condition . Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Theorem Let A = { a n } ∞ n = 0 be a sequence of points in D − { 0 } . If A satisfies the Blaschke condition, then B A , k is an inner 1 function. If A satisfies the Blaschke condition, then the terms of A 2 are precisely the zeros of B A . Furthermore, the number of times a zero of B A appears in A is its multiplicity. If A does not satisfy the Blaschke condition, then B A ≡ 0 . 3 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Outline Background from analysis 1 The class H ∞ ( D ) Some types of functions in H ∞ ( D ) Some types of inner functions Factorization Frostman’s Theorem Background from computability theory 2 Computability over the natural numbers Computability over uncountable spaces: Type-Two Effectivity Theory Statement of results 3 References 4 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Definition N is the class of all f ∈ H ∞ ( D ) such that � π log + | f ( re i θ ) | d θ < ∞ sup 0 < r < 1 − π Theorem (Canonical Factorization Theorem) If f ∈ N, then there exist λ , F, B, S 1 , and S 2 such that f ( z ) = λ F ( z ) B ( z ) S 1 ( z ) S 2 ( z ) where λ ∈ ∂ D , B is a (possibly finite) Blaschke product, and S 1 , S 2 are singular functions. Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Corollary (Factorization of Inner Functions) If u is an inner function, then there exist unique λ u , b u , s u such that u = λ u b u s u , λ u ∈ ∂ D , b u is a (possibly finite) Blaschke product, and s u is a singular function. Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem Outline Background from analysis 1 The class H ∞ ( D ) Some types of functions in H ∞ ( D ) Some types of inner functions Factorization Frostman’s Theorem Background from computability theory 2 Computability over the natural numbers Computability over uncountable spaces: Type-Two Effectivity Theory Statement of results 3 References 4 Timothy H. McNicholl Computable Aspects of Inner Functions
H ∞ ( D ) Background from analysis Some types of functions in H ∞ ( D ) Background from computability theory Some types of inner functions Results Factorization References Frostman’s Theorem For each closed K ⊆ D and each positive measure σ on K , let U σ : D → D be defined by the equation � 1 U σ ( z ) = log | z − ζ | d σ ( ζ ) . K Definition Let F ⊆ D be closed. We say that F has zero capacity if for every positive measure on F , σ , with σ � = 0, U σ is not bounded on any neighborhood of F . Otherwise, we say that F has positive capacity . If U is an arbitrary subset of D , then we say that U has positive capacity just in case it has a closed subset with positive capacity; otherwise, we say that it has zero capacity. Timothy H. McNicholl Computable Aspects of Inner Functions
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