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 → $\backslash mathbb\{C\}$

(where D is the open unit disk in the complex plane $\backslash mathbb\{C\}$) that extend to a continuous function on the closure of D. That is,

: $A(\backslash mathbf\{D\})\; =\; H^\backslash infty(\backslash mathbf\{D\})\; \backslash cap\; C(\backslash overline\{\backslash mathbf\{D\}\}),$

where {{math|H^∞(D)}} denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space).

When endowed with the pointwise addition {{nobr|(f + g)(z) {{=}} f(z) + g(z)}} and pointwise multiplication {{nobr|(fg)(z) {{=}} f(z)g(z),}} 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

: $\backslash |f\backslash |\; =\; \backslash sup\backslash big\backslash \{|f(z)|\; \backslash mid\; z\; \backslash in\; \backslash mathbf\{D\}\backslash big\backslash \}\; =\; \backslash max\backslash big\backslash \{|f(z)|\; \backslash mid\; z\; \backslash in\; \backslash overline\{\backslash mathbf\{D\}\}\backslash big\backslash \},$

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.