Composite sequence transformations

Summary The aim of this work is to introduce the new concept of composite sequence transformations and to show, by very simple examples and theorems, that it can be useful in accelerating the convergence of sequences. Generalizations of classical transformations and results are obtained.Work performed under the Nato Research Grant 027.81.Presented at the International Conference on Numerical Analysis, Munich, March 19–21, 1984     

Quasilinear sequence transformations

We give a new characterization of quasilinear sequence transformations; we prove that any quasilinear transformation can be represented by its kernel. This approach is new and allows one to give a general result of convergence acceleration and tools for the construction of new quasilinear transformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Interpolation between sequence transformations

Levin's sequence transformation [1] and a structurally very similar sequence transformation [4] behave quite differently in convergence acceleration and summation processes. In particular, it was found recently that Levin's transformation fails completely in the case of the strongly divergent Rayleigh-Schrödinger and renormalized perturbation expansions for the ground state energies of anharmonic oscillators, whereas the structurally very similar sequence transformation gives very good results [14,17]. For a more detailed investigation of these phenomena, a sequence transformation is constructed which — depending on a continuous parameter — is able to interpolate between Levin's transformation and the other sequence transformation. Some numerical examples, which illustrate the properties of the interpolating sequence transformation, are presented.     

Some vector sequence transformations with applications to systems of equations

First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the -algorithm is recovered and a nonlinear analog of the CGS method is obtained.     

Vector sequence transformations: Methodology and applications to linear systems

In this paper, a methodology for the construction of various vector sequence transformations is formulated leading to a unified presentation of the subject and to new results. The connections to the general interpolation problem and to projections are discussed. Various particular cases are examined in more details. Applications to the solution of systems of linear equations will end the paper and, in particular, their relation with Lanczos method will be studied. Some numerical examples will be given.     

On matrix transformations and sequence spaces

In this paper are given results on the spacesw _τ (μ) andc _τ (μ, μ′) the second one generalizing the well-known spacec _∞ (μ) of sequences that are strongly bounded. Then we deal with matrix transformations into these spaces. These results generalize those given in [7].     

Sum of sequence spaces and matrix transformations

Summary We are interested in the study of the sum <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>E+F$ and the product $E*F$, when $E$ and $F$ are of the form $s_{\xi}$, or $s_{\xi}^{\circ}$, or $s_{\xi}^{(c)}$. Then we deal with the identities $(E+F) (\Delta^{q}) \eg E$ and $(E+F) (\Delta^{q}) \eg F$. Finally we consider matrix transformations in the previous sets and study the identities $\big((E^{p_{1}}+F^{p_{2}}) (\Delta^{q}),s_{\mu}\big) \eg S_{\alpha^{p_{1}}\pl \beta^{p_{2}},\mu}$ and $\big(E+F(\Delta^{q}),s_{\gamma}\big) \eg S_{\beta,\gamma}$.     

Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness

The spaces and introduced by Ayd?n and Ba?ar [C. Ayd?n, F. Ba?ar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3) (2004) 677-693] can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero, summable, and bounded by the Cesàro method of order 1. Here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable, summable to zero, or bounded by the strong Cesàro method of order 1 with index p?1. We determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences.