SECTIONAL TRANSACTIONS.—A*. 455 
Lan is said to be absolutely summable (C,«) if &| S% —S%_, | is con- 
vergent, where S% is the «-th Cesaro mean of Sx =ad.+ ...+4n, 
The sum is then = S%, and the series is also absolutely summable (A). 
By employing absolute summability (C) more precise results may be 
obtained for Fourier series. In particular, (2) is necessary and sufficient 
for absolute summability (C) of an unspecified order. 
Dr. A. C. OrrorD.—Fourier and Hankel transforms (12.0). 
Two functions f(x) and F(x) are said to be Fourier cosine transforms of 
one another when they are connected by the formule 
(x) F(x) = (2) "cos xuf(u) du, 
° 
-00 
(2) f(x) = (2) ‘| cos xuF(u) du, 
° 
where the integrals may be either integrals of the classical kind or integrals 
in some generalised sense. 
More generally f(x) and F(x) are Hankel transforms of one another when 
they are connected by the relations 
F(x) = [vou Pema 
fe} 
ie | vowl (ou) F(u) du, 
(eo) 
where Jy(z) is Bessel’s function and R(v) > —}4. Whenv = — i, } these 
reduce to the cosine and sine transforms respectively. 
For simplicity we will first state the results for the special case of the cosine 
transform. We say that f(x) belongs to the class H if 
W 
(H) - \| (: — 4) cos xu f(u) du| < M, 
a w 
for all w and x, M being an absolute constant. This condition will obviously 
be satisfied when / (x) is absolutely integrable in (0, cc). We show that every 
function of H has a cosine transform which is bounded. More precisely 
we prove that, when f(x) belongs to H, the integral (1) is summable 
(C, 1) almost everywhere to F(x) and (2) is summable (C, 2) almost 
everywhere to f(x). 
Now consider the converse problem, Let F(x) be a bounded function 
and let it be such that the integral 
(2) (Pse ra, 
° 
TT 
which is known to exist in the (C, 1) sense, is uniformly summable (C, 1) 
to an indefinite integral. So that we can almost always write 
: co. 
(x) = (2)8 f \, sin (u) du. 
