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'.

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

The previous methods we encountered, our P definition and the B and B' definitions of Borel that the former led to are quite powerful methods capable of summing rapidly divergent and unruly series. Indeed, using the P method and a suitable auxiliary function one may give rise to further powerful methods. However, as Hardy notes in his Divergent Series, this power comes with a cost. In that these methods may be prone to failure in more subtle cases such as slowly diverging series or ones with oscillating terms. Prior to further discussion, we submit for your inspection the following method-

Summation by Abelian means

For 0 =< λ_0 < λ_1 < λ_2 < ... with (λ_n) tending towards infinity and

f(t) = sum_{n = 0}^{infinity} (a_n).exp [(λ_n)t] converges for all positive real values of t,

If,

lim_{t->0} f(t) = s

then,

a_0 + a_1 + a_2 + ... = s , (A, λ)

Summation by Abelian means (A, λ) embodies a whole class of methods (which may indeed be further generalised, see Hardy, Divergent Series, pp71-73) depending on the sequence (λ_n) chosen, though each such method can be shown to be regular, linear and stable. We are interested in the simplest case where λ_n = n, whereupon we have with;

f(x) = sum_{n = 0}^{infinity} (a_n).x^n

where x = e^-t and hence convergent for |x| <1

if,

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

(this notation implies the left hand limit)

then,

a_0 + a_1 + a_2 + ... = s , (A)

The 'A' stands in for Abel summation, which one might find a curious choice of name given Abel's attitude towards divergent series. Indeed, the present method is more fitting for the legacy of Euler or Poisson than Abel (we shall later see the intimate connection between a more powerful summation method attributed to Euler and Abel summation). However, it was Abel who employed a similar argument (under different conditions to do with limiting partial sums of course) for a theorem to do with Cauchy summation and the proper articulation has associated the method to his name.

The method is less powerful than the (B) and (B') methods discussed in the previous post, failing even to sum the geometric series outside its radius of convergence. However, it possesses the stability that Borel's methods in general do not and also generalise the Cesaro summation which we shall discuss in the next post.

Note: We may also mention in passing, the method of Lambert summation which also relies on a trick in the limit as with Borel's integral method but in a form more familiar with Abel summation.

We say,

a_0 + a_1 + a_2 + ... = s , (L)

if,

lim_{x -> 1^-} (1 - x). [sum_{n = 1}^{infinity} n.(a_n).(r^n) / (1 - x^n)] = s

Evaluating the limit (1 - x) / (1 - x^n) at each n shows how this method works.