New Results on the SSIE with an Operator of the form FΔ⊂E+Fx′ Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method

Given any sequence a=(an)n≥1 of positive real numbers and any set E of complex sequences, we can use Ea to represent the set of all sequences y=(yn)n≥1 such that y/a=(yn/an)n≥1∈E. In this paper, we use the spaces w∞, w0 and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that n−1∑k=1nykn is bounded and tends to zero, and such that y−le∈w0, for some scalarl . These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE FΔ⊂E+Fx′, where E is a linear space of sequences, F=c0, c, ℓ∞, w0, w or w∞, and F′=c0, c or ℓ∞. For instance, the solvability of the SSIE wΔ⊂w0+sxc relies on determining the set of all sequences x=xnn≥1∈U+ that satisfy the following statement. For every sequence y that satisfies the condition limn→∞n−1∑k=1nyk−yk−1−l=0, there are two sequences u and v, with y=u+v such that limn→∞n−1∑k=1nuk=0 and limn→∞vn/xn=L for some scalars l and L.

Stereographic Projection

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's Planisphere, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.

A Tauberian theorem for the product of Abel and Cesàro summability methods

In this paper, we prove a Tauberian theorem for the product of the Abel method and the Cesàro method of order α, which improves some classical Tauberian theorems for the Abel and Cesàro summability methods.

Summation methods applied to Voronovskaya-type theorems for the partial sums of Fourier series and for Fejér operators

We obtain Voronovskaya-type theorems for the partial sums of Fourier series using the second order Cesáro method of summation. Then we obtain two versions of Voronovskaya-type theorems for Fejér operators and finally we deduce an integral identity.

Cesàro Summable Sequence Spaces over the Non-Newtonian Complex Field

The spacesω0p,ωp, andω∞pcan be considered the sets of all sequences that are strongly summable to zero, strongly summable, and bounded, by the Cesàro method of order1with indexp. Here we define the sets of sequences which are related to strong Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine theβ-duals of the new spaces and characterize matrix transformations on them into the sets of⁎-bounded,⁎-convergent, and⁎-null sequences of non-Newtonian complex numbers.

Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces

We study the spacesw0p,wp, andw∞pof sequences that are strongly summable to 0, summable, and bounded with indexp≥1by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaceswpinto the spacesw∞1,w1, andw01. We also give estimates for the operator norm and the Hausdorff measure of noncompactness of such operators. Finally we apply our results to characterize the classes of compact bounded linear operators fromw0pandwpintow01andw1.

Approximation of a generalized Lipschitz class function by Euler - Cesàro means of Fourier series

In this paper, I have taken product of two summability methods, Euler and Cesaro; and establish a new theorem on the degree of approximation of the function f belonging to W(Lp, ?(t)) classes by Euler - Cesaro method. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7190 BIBECHANA 9 (2013) 151-158

SOME NOTES ON MATRIX TRANSFORMS OF SUMMABILITY DOMAINS OF CESÀRO MATRICES

In this paper sufficient conditions for a matrix M = (mnk ) (mnk are Cesàro numbers As n‐k, s ∈ C if k ≤ n and mnk= 0 if k > n) to be a transform from the summability domain of the Cesàro method Cα into the summability domain of another Cesàro method Cβ , where α, β ∈ C\{-1, -2,…}, are found.

SEVERAL THEOREMS ON Λ–SUMMABLE SERIES

We prove several propositions on λ‐summable series by Cesàro method (C, 1) or by weighted mean methods N, which are also often called Riesz methods P = (R, pn ).