Necessary and Sufficient Tauberian Conditions under which Convergence follows from (A r,δ ,p,q;1,1), (A r,*,p,*;1,0) and (A *,δ ,*,q;0,1) Summability Methods of Double Sequences
Kitapta bölümler > Diğer · 2022
Yazarlar: Çağla Kambak, İbrahim ÇANAK
Sequence Space Theory with Applications, Diğer, Uluslararası, 2022, ss 75-94, ,
Özet
We say that https://www.w3.org/1998/Math/MathML"> ( x m n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_16.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is https://www.w3.org/1998/Math/MathML"> ( A r , δ , p , q ; 1,1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_17.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> summable to https://www.w3.org/1998/Math/MathML"> l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_18.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if https://www.w3.org/1998/Math/MathML"> ( σ m , n r , δ , p , q ( x ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_19.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> has a finite limit https://www.w3.org/1998/Math/MathML"> l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_20.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . It is known that if https://www.w3.org/1998/Math/MathML"> l i m m , n → ∞ x m n = l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_21.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ( x m n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_22.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is bounded, then https://www.w3.org/1998/Math/MathML"> l i m m , n → ∞ σ m , n r , δ , p , q ( x ) = l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_23.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . But the converse of this implication is not true in general. Our aim is to obtain necessary and sufficient conditions for https://www.w3.org/1998/Math/MathML"> ( A r , δ , p , q ; 1,1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_24.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> summability under which the converse of this implication holds. Similarly, we also examine https://www.w3.org/1998/Math/MathML"> ( A r , * , p , * ; 1,0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_25.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ( A * , δ , * , q ; 0,1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003178200/13bf4b00-f595-4bbe-9be7-58bd31e5928b/content/math0_26.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> summability methods and obtain Tauberian theorems for these summability methods.