If y F, the Cauchy theorem (together with the fact that g vanishes at infinity, by
(iii ) shows that each y, can be replaced by any sufficiently close y
without chang-
ing (7). It föllows, for each t 6 Rn, that ey%tG (t) i s constant in each of the connect-
ed open sets V^ and VT
Also, it föllows from (iii' ) and the uniform continuity of g that y g is a con-
tinuous map of V into L 1 , so that G (t) is (for each t 6 Rn) a continuous function
of ã on F. In conjuction with the preceding paragraph it now föllows that eymtG it)
i s independent of y, on F. Thus (6) is proved.
The Plancherel theorem [lO] implies that each G is in L 2 , but our hypotheses
do not imply that G is in L . To remedy this, let ø = ö be a nonnegative function
in L which vanishes outside a ball of radius r with center at the origin of Rnf and
whose integral is 1. Define
(8) h{z) = J ø(î)
ßae - î)Üî (z€Rn + if).
Then h = ø * g
where * indicates convolution. If Ç and Ø are the Fourier trans-
forms of h and ø, then Ç = Ø(ú . This has two consequences: first, (6) becomes
(9) HU) = e-*'*H0U) Uer,
e?) ,
second, y —• g is a continuous map of V into L , by (iii ' ) and the uniform contin-
uity of g; by the Plancherel theorem, the same is true of y G ; since Ø 6 £ 2 , the
Schwarz inequality now implies that y —- » Ç is á continuous map of V into Ll.
In particular, the Fourier Inversion formula can be applied to h . In view of (9),
the result i s
(10) Ä(z)= f Ç0{ß)â1ae**Üß {z£Rn + iV).
We can now apply our lemma, with H' in place of ut V in place of K, and con-
clude from part (a) that y —* Ç is a continuous map of W into Ll (where, we recall,
W i s the convex hüll of F), which implies that formula (10) defines h a s a continuous
function on Rn + iW which is holomorphic in Rn + iW. From part (b) of the lemma we
conclude that the supremum of \h\ over Rn + iW is the same as that over Rn + iV.
Let us now write h in place of A, to indicate the dependence of h on ö9 hence
on r. As r —* 0, we see from (8) and the uniform continuity of g that h —* g uni-
formly on Rn + iV. The conclusion of the preceding paragraph (applied to the difference
of two functions h rather than to h) shows that \h } is a uniform Cauchy sequence on
Rn + i E The limit of this sequence furnishes the desired extehsion of g, and the
proof of the theorem is complete,
The set W in the conclusion of the theorem cannot be replaced by any larger one.
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