How is the absolute summability of a series defined?
Absolute summability is defined in a similar manner for methods involving matrix transformations of series into sequences. If the summation method is defined by a transformation of the series (2) into a series
When is a summation method preserves absolute convergence?
A summation method is said to preserve absolute convergence of a series if it absolutely sums each absolutely convergent series. If each such series is summable by this method to a sum equal to the sum to which it converges, then the method is called absolutely regular.
When is a system said to be absolutely summable?
The “absolutely summable” refers to the use of absolute values in the summation. BIBO stability A system is said to be bounded input-bounded output stable (BIBO stable or just stable) if the output signal is bounded for all input signals that are bounded.
Why is the Borel summation method absolute summability?
In the case of integral summation methods, absolute summability is distinguished by the requirement of absolute convergence of the corresponding integrals. Thus, for the Borel summation method the integral must be absolutely convergent.
Where did the idea of absolute summability come from?
The concept of absolute summability was introduced by E. Borel for one of his methods, but was stated differently from its modern formulation: absolute summability was distinguished by imposing the condition for each p = 0, 1, …. Absolute summability was initially used in the study of the summability of power series outside the disc of convergence.
When is absolute convergence the same as absolute summability?
In the particular case in which the method A corresponds to the identity transformation of a sequence into a sequence (or of a series into a series), absolute summability of a series is the same as absolute convergence. The supplementary requirements are suitably modified for non-matrix summation methods.