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Denote by Fkn the projection of xken on 3fTl (k = 0, ± 1 , · · · ; n = 1, 2, · · ·)· These functions span 2fTL Unfortunately each Fkn is merely an equivalence class of functions; choose some definite representative so that Fkn is now a function in the strict sense. Let J(eix) be the smallest closed subspace of 3C containing all the vectors Fkn(eix). I shall prove that J is a measurable range function, and that 2fTC = Îftlj. It is obvious that 9ΠΧ is contained in 9fïlj, whether or not J is measurable.

23) lim (F(reix), (0 < r < 1). The scalar boundary value theorem produces a boundary function F in i/p3e having the φ^ as Fourier coefficients. 24) thus provides an alternate function-theoretic definition of HP3Q, in the same way as for scalar functions. 2. The following definition is fundamental for our subject. Definition. A range function J = J(eix) is a function on the circle taking values in the family of closed subspaces of 3C.

Now let 9TC = g «if2 where g is a measurable unitary function. If P\ is the projection on S\H2, then the projections on 9fTC\ = S\3K are evidently q'P\q~l} where q and its inverse operate by multiplication. 23) Vt = - Γe~iU diqP^q-1) = - q Γ e~iU dP^qr1 = •'-co Q'T^r1. 24) Vtf = q> Ttir1'/) = ( ί ' ^(Γ 1 ) · Ttf. 25) At = q-Ttq-1. 25) is called a coboundary. 25) is a cocycle for each unitary function q is assured by its derivation, but it is easy to verify (A, B, C) directly. The final step of the proof is to show that every cocycle is a coboundary.

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