Colloquium

H¹(R) is a Banach algebra which has better mapping properties under singular integrals than L¹(R). We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence v_n, and introduce a Banach algebra Q that properly lies between H¹ and L¹, and use it to show that the approximate identity sequence v_n satisfies

c (1 + ln n) ≤ ||v_n||_H¹ ≤ Cn^½.

We identify the maximal ideal space of H¹ and give the appropriate versions of Wiener's ideal theorem and Wiener's Tauberian theorem.