On the acceleration of convergence by regular matrix methods

Abstract

Regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. The results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

Some alternative methods of convergence acceleration

“…First we notice that there exists no convergent sequence, converging faster or μ-faster than any sequence of ϕ. It is shown in [3], that if a sequence y = (y n ) ∈ c converges faster than x = (x n ) ∈ c \ ϕ, then y converges also μ-faster than x. But the converse assertion is not valid.…”

“…It is shown in [3] that every sequence x ∈ c \ ϕ has the limit speed of convergence and for each x ∈ c \ ϕ there exists an element μ ∈ μ x , which is not the limit speed of x. We say that the limit speed of convergence μ * = (μ * n ) of a sequence y is higher than the limit speed of convergence λ * = (λ * n ) of a sequence x if the ratio λ * n /μ * n is upper-bounded, but not lower-bounded.…”

“…But the converse assertion is not valid. We explain this assertion with the help of the following example from [3]:…”

“…If a sequence y converges weakly faster than x, then y converges weakly μ-faster than x, but the converse assertion is not valid (see [3]). Also it was proved in [3] (see Proposition 2.5) that for monotonic sequences the converse assertion holds. Thus the concept of μ-faster convergence is important for non-monotonic processes.…”

“…We describe the results, presented in [3]. A regular matrix A is said to be universally accelerating (see [11]) if (A n x) converges faster than x for every x ∈ c, and weakly accelerating if (A n x) converges weakly faster than x for every x ∈ c. We consider the convergence acceleration of sequences in the sense different from the classical concept.…”