Real and Complex Analysis (Rudin)

234 REAL AND COMPLEX ANALYSIS
Iffis real, formula 11.5(2) shows that p[n is the real part of
1 In e il + z
-2 -il- f(t) dt,
1t -n e - z
(3)
which is a holomorphic function of z = rei9 in U, by Theorem 10.7. Hence p[n
is harmonic in U. Since linear combinations (with constant coefficients) of harmonic
functions are harmonic, we see that the following is true:
11.7 Theorem Iff E Ll(T) then the Poisson integral p[n is a harmonic function
in U.
The following theorem shows that Poisson integrals of continuous functions
behave particularly well near the boundary of U.
11.8 Theorem Iff E C(T) and if Hfis defined on the closed unit disc 0 by
i {f(ei~
(Hf)(re ~ = p[n(rei!l)
if r = 1,
ifO ~ r < 1,
(1)
then Hf E C(O).
PROOF Since P,(t) > 0, formula 11.5(3) shows, for every 9 E C(T), that
so that
I P[g](rei~ I ~ IlgiiT (0 ~ r < 1), (2)
IIHgilu = IIgllT (g E C(T». (3)
(As in Sec. 5.22, we use the notation IIgllE to denote the supremum of I 9 I on
the set E.) If
N
g(ei~ = L c. ei•9 (4)
.= -N
is any trigonometric polynomial, it follows from 11.5(1) that
N
(H g)(rei~ = L C n rlnlei.9, (5)
n= -N
so that Hg E C(U).
Finally, there are trigonometric polynomials gk such that Ilgk - fllT- 0
as k- 00. (See Sec. 4.24.) By (3), it follows that
IIH9k - Hfllu = IIH(gk - f)llu- 0 (6)
as k- 00. This says that the functions Hg k E C(O) converge, uniformly on 0,
to Hf Hence Hf E C(O).
IIII