On Certain Series to Series Transformation and Analytic Continuations by Matrix Methods

Abstract

In this research paper, we proved some general theorems on the absolute convergence transformation of matrix which is expressed in terms of preserving transformation under the very general conditions. This works is motivated by the works of [3],[12] and [14].

On a New Application of Positive and Decreasing Sequences to Double Fourier Series Associated with (N, p^(1)_m, p^(2)_n)

“…There are several results on summability by Nörlund means of Fourier series. The authors [2,4,6,7,9,13,14] have looked into it. This encourages us to research it in both more generalized and particular cases.…”

“…Paudel et al [12] studied sequence and generalized sequence space. Sahani et al [13] studied series-to-series transformations and analytical continuations using matrix methods. In this paper, we have studied about the double Fourier series.…”

Critical Analyzing on Some New Application of Almost Decreasing Sequence to Legendre Series Associated with [B] Sum

“…Most eminent mathematical analysts have focused their creative energy on the theory of series. The late 19th and early 20th centuries are credited with being the eras when adequate methods were able to associate with them certain values that could be called their sums in a reasonable manner through a process closely associated with Cauchy's concept of convergence (see [13][14][15][16][17][18][19][20]). [11]); Consider a sequence { 𝑈 𝑚 } of 𝑚 𝑡ℎ partial sums corresponding to the infinite series ∑ 𝑢 𝑚 .…”

Properties of Infinite Matrices and Sequence Spaces

“…Several researchers have studied the distinctive features of sequence to a sequence, series to sequence, and series to series transformations matrices; for a thorough description, (see Dienes,1950) and (Cooke,1931 and2022). The Silverman-Toeplitz theorem gives us the necessary and sufficient conditions for a matrix to represent a standard method, a method we can then use to generate regular transformations.…”

“…The Silverman-Toeplitz theorem gives us the necessary and sufficient conditions for a matrix to represent a standard method, a method we can then use to generate regular transformations. Dienes (1950) and Cooke (1931 and2022) and others have studied the unique properties between sequence to sequence, series to sequence, and series to series transformations matrices. Hill has defined the sequence-toseries transition in 1939.…”