Lesson plan
Disk algebra
In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions : ƒ : D → \mathbb{C} (where D is the open unit disk in the complex plane \mathbb{C} ) that extend to a continuous function on the closure of D. That is, : A(\mathbf{D}) = H^\infty(\mathbf{D}) \cap C(\overline{\mathbf{D}}), where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space). When endowed with the pointwise addition and pointwise multiplication this set becomes an algebra over C, since if f and g belong to the disk algebra, then so do f + g and fg. Given the uniform norm : \|f\| = \sup\big\{|f(z)| \mid z \in \mathbf{D}\big\} = \max\big\{|f(z)| \mid z \in \overline{\mathbf{D}}\big\}, by construction, it becomes a uniform algebra and a commutative Banach algebra. By construction, the disc algebra is a closed subalgebra of the Hardy space H ∞ . In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H ∞ can be radially extended to the circle almost everywhere.
Article summary
In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions : ƒ : D → \mathbb{C} (where D is the open unit disk in the complex plane \mathbb{C} ) that extend to a continuous function on the closure of D. That is, : A(\mathbf{D}) = H^\infty(\mathbf{D}) \cap C(\overline{\mathbf{D}}), where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space). When endowed with the pointwise addition and pointwise multiplication this set becomes an algebra over C, since if f and g belong to the disk algebra, then so do f + g and fg. Given the uniform norm : \|f\| = \sup\big\{|f(z)| \mid z \in \mathbf{D}\big\} = \max\big\{|f(z)| \mid z \in \overline{\mathbf{D}}\big\}, by construction, it becomes a uniform algebra and a commutative Banach algebra. By construction, the disc algebra is a closed subalgebra of the Hardy space H ∞ . In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H ∞ can be radially extended to the circle almost everywhere.
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Discussion questions
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