Hardy Space - Factorization Into Inner and Outer Functions (Beurling)

Factorization Into Inner and Outer Functions (Beurling)

For 0 < p ≤ ∞, every non-zero function ƒ in Hp can be written as the product ƒ = Gh where G is an outer function and h is an inner function, as defined below (Rudin 1987, Thm 17.17). This "Beurling factorization" allows the Hardy space to be completely characterized by the spaces of inner and outer functions.

One says that G(z) is an outer (exterior) function if it takes the form

for some complex number c with |c| = 1, and some positive measurable function φ on the unit circle such that log φ is integrable on the circle. In particular, when φ is integrable on the circle, G is in H1 because the above takes the form of the Poisson kernel (Rudin 1987, Thm 17.16). This implies that

for almost every θ.

One says that h(z) is an inner (interior) function if and only if |h(z)| ≤ 1 on the unit disk and the limit

exists for almost all θ and its modulus is equal to 1. In particular, h is in H∞. The inner function can be further factored into a form involving a Blaschke product.

The function f, decomposed as f = Gh, is in Hp if and only if the positive function φ belongs to Lp(T), where φ is the function in the representation of the outer function G.

Let G be an outer function represented as above from a function φ on the circle. Replacing φ by φα, α > 0, a family (G_α) of outer functions is obtained, with the properties:

G₁ = G, G_α+β = G_α G_β and |G_α| = |G|α almost everywhere on the circle.

It follows that whenever 0 < p, q, r < ∞ and 1/r = 1/p + 1/q, every function f in Hr can be expressed as the product of a function in Hp and a function in Hq. For example: every function in H1 is the product of two functions in H2; every function in Hp, p < 1, can be expressed as product of several functions in some Hq, q > 1.