Disk algebra

In functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions In functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions where D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D. That is, where H ∞ ( D ) {displaystyle H^{infty }(mathbf {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, (f+g)(z)=f(z)+g(z),and pointwise multiplication, this set becomes an algebra over C, since if ƒ and g belong to the disk algebra then so do ƒ + g and ƒg. Given the uniform norm, 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.