188 
Proceedings of the Royal Society of Edinburgh. [Sess. 
This is the required cardinal function. It is the only analytic function 
having the above tabular values which has no singularities in the finite 
part of the ^c-plane and no oscillations of period less than 2. 
Example 2. — Suppose that the given tabular values of the function 
f{x) are as follows : — 
f{a) = 0, J\a + w) = \, f{a+‘ho) = l, /{a + 3w?) = 0, /(a + 4?(;) = - 1, .... 
f{a -w)= - 1, /(a - 2tv) = - 1, f{a - Siv) = 0, f(a - 4:iv) = 1, . . . . , 
so that by (4) the cardinal function is in this case 
sin I - - a) I r ^ ^ 
^ ) \_x- a + iv X- a + 10 x -a- 
1 
1 
2io X- a + 2io x-a- iiv 
+ 
1 
a; - a + x 
- a-bw J 
Now remembering that 
cot 
{x - a - to) 3iv 
3w 
3w I 1 
7T ] X - a 
1 1 1 
+ ^ + 
w X- a + 2io x-a - ^'10 x-a-\-biv 
+ 
i- no ) 
and 
7t{x -a + io) _3io { 1 
7T 1 a? 
cot 
3w 
1 
1 
+ 
a + w X - a + 4:10 x- a-2w x - a + 7io 
1 
X - a-bio 
+ 
} 
we see that this cardinal function is * 
/ -(a; - a) I 
1 . 
-sm 
, Trix-a + w) , irix-a-w) 
cot - cot ^ ^ - ' 
3io ow 
or 
1 . 7t{x - a) . 2tt 
— o sm sm 
S to 3 , 
. 7t(x - a + to) . 7t(x - a- w) ^ 
sm 5 sm ^ 
3to 6w 
so, making use of the identity 
sin 3x = - 4 sin x sin sin (^x " ^ > 
we obtain the cardinal function corresponding to the above tabular values 
in the simple form 
2 , '7t{x - a) 
sm — 5 • 
J3 
It will be noticed that in Example 1 the tabular values of the function 
tend to the limit zero for infinite values of the argument, whereas in 
Example 2 they do not tend to the limit zero. 
* It is not in general permissible to alter the order of the terms in a conditionally 
convergent series : but it may readily be proved that in the present case the value of the 
sum is not altered by the particular rearrangement which is made. 
