1914-15.] Expansions of the Interpolation-Theory. 
189 
§ 6. Direct proof of the properties of the cardinal function. 
Let C(x) denote the cardinal function associated with a given function 
f(x), so that 
f{a + 7'iv) sin I ~{x- a- w) I 
( w ) 
C(x)=2 
~ {X- a — rtv) 
Then we can prove the characteristic properties of this function directly. 
1°. C(x) is cotabular with f(x). 
sin I — — a — rw) i 
( w ) 
For the expression 
— a — rw) 
has the value unity when 
IV 
x->{a-{-rw), and has the value zero when x has any other one of the 
values a, a-{-w, a — w, a-\-2w, . . . From this it follows at once that 
C{a + nv)=/{a + rw) (r = 0, ±1, ±2, ±3, . . . .), 
which establishes the property of cotabularity. 
2°. C(x) has no singularities in the finite part of the x-plane. 
For a singularity at any point would give rise to a failure of con- 
vergence of the series (3) at that point : but its convergence, for the class 
of functions f(x) considered, can readily be deduced from its mode of origin 
as a sum of residues. 
3°. When C(x) is analysed into periodic constituents by Fourier's 
integral-theorem, all constituents of period less than 2w are absent. 
For if we resolve the function 
sm 
w ) 
(5) 
(where c denotes any constant) into periodic constituents by Fourier’s 
integral-theorem, we have 
-{x - c) i ^ sin I -(/X - c) I cos | \{x - /x) | 
dX I (7/x — 
-{x - c) 
-{[X - c) 
writing y for 
7t(m - 
w 
w , 
= —^\ dX 
sin y cos \Xx — Xc 
dy 
Xiv 
y 
