The methods of Borel, Abel and Cesaro, Voronoi & Norlund part III

We now come to the last part of the present series regarding summability methods. The methods we shall presently consider is due to Norlund and are known as Norlund methods. The method of Cesaro means which we shall consider later is an elementary case of these. A nearly equivalent definition of these had already been given by Voronoi which is why we mention him here.

Say we have a sequence of weights, (w_k) where k runs from 0 to n, with only the restriction that each term is non-negative and, for our convenience, w_0 is never zero.

We consider the sequence of partial sums of the form s_n = a_0 + a_1 + a_2 + ... + a_n

and construct the sequence t_n such that,

t_n = [(w_n).(s_0) + (w_(n-1)).(s_1) + (w_(n-2)).(s_2) + ... + (w_0).(s_n)] / (w_0 + w_1 + w_2 + ... + w_n)

and consider the limiting value of this new sequence as the sum, i.e-

a_0 + a_1 + a_2 + ... = s , (N, w_k)

if,

lim_{n -> infinity} t_n = s

Of course, an immediate concern here is where regularity holds given the freedom we have with assigning the weights and thus the generality of this method. However, it may be shown that regularity holds given,

w_n / (w_0 + w_1 + w_2 + ... + w_n) -> 0

and that any two such regular Norlund methods are consistent with each other, i.e- summing a series to the same sum given both limits exist.

Indeed, for a regular Norlund method, an Abelian theorem holds which draws the connection between Abel's method and Norlund's.

i.e-

If (N, w_k) is regular and a_0 + a_1 + a_2 + ... + a_n = s , (N, w_k) then the series given by,

f(x) = a_0 + (a_1).x + (a_2).x^2 + ... + (a_n).x^n + ...

has a positive radius of convergence and f(x) is analytic and regular for x between 0 (inclusive) and 1.

lim_{x -> 1^-} f(x) = s

Which means that if f(x) does converge for x between 0 and 1 the (N, w_k) method gives the same sum as the A method.

For w_k = 1 for all k we have a special case we call Cesaro summation to (C, 1) and with,

w_k = (n + m - 1)_C_(m - 1) 

we have the more general (C, k) method of Cesaro. (note that the notation n_C_k stands for the binomial coefficient nCk)

The (C, 1) method for instance, sums the series,

1 - 1 + 1 -1 + ... to 1/2 by averaging the partial sums when taking the 'Cesaro limit'.