By M. Mursaleen
This brief monograph is the 1st ebook to concentration solely at the examine of summability equipment, that have develop into lively components of study in recent times. The booklet offers uncomplicated definitions of series areas, matrix adjustments, ordinary matrices and a few designated matrices, making the cloth available to mathematicians who're new to the topic. one of the middle goods coated are the evidence of the major quantity Theorem utilizing Lambert's summability and Wiener's Tauberian theorem, a few effects on summability assessments for singular issues of an analytic functionality, and analytic continuation via Lototski summability. nearly summability is brought to end up Korovkin-type approximation theorems and the final chapters function statistical summability, statistical approximation, and a few purposes of summability tools in fastened element theorems.
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Additional info for Applied Summability Methods
E. e. X0 D W0 D 0/: In fact, the converse also holds in the present case and we have the following theorem. 1. d. 1 ! 4) iD1 Proof. 4) are well known (cf. ). 1). d. Let 1 X 1 e m nDmC1 n=m Xns : P Then Ym C Zm ! 0, as m ! 1, Ym and Zm are independent and symmetric. P Therefore it follows easily from the Levy’s inequality [57, p. 247] that Zm ! 0. Y1 ; : : : ; Ym / are independent and Ym C Zm ! , Zm ! 1/ obtain by Lemma 3 of  that Ym ! e. m Xm / ! 0, we have by Lemma 3 of  that Xms =m !
4) F is nowhere dense in `1 , dense in itself, and closed and therefore perfect. (5) The method is not strong in spite of the fact that it contains certain classes of matrix methods for bounded sequences. ˛ > 0/ to its F -limit. , the method F cannot be expressed in the form of a matrix method. C; 1/. C; 1/ can be replaced in this definition by any other regular matrix method A satisfying certain conditions. (9) Since c F `1 , we have `1 D `1 F c D `1 . That is, the -dual of F is `1 , where stands for ˛, ˇ, and .
Cnk zk D kD0 1 X zk kD0 1 X ! 1 ˛t j < 1. 5) Then, for jt j Ä min. 5), the results of the lemma. 6. 1 ˇ kD0 for some ˛ < 1; ˇ < 1 and ˛ C ˇ > 0. 6) 28 3 Summability Tests for Singular Points Proof. t /. t / < 1 ; 1 ˇ 2˛ D 0I ˆ : t W jt C ˛Cˇ j > j 1 ˛ j ; 1 ˇ 2˛ < 0: 1 ˇ 2˛ 1 ˇ 2˛ In each case t D 1 is on the boundary of D and D contains all points of the closed unit disk except t D 1. 1 ˛t /2 k ; > 0 such that for jt j Ä provided t " D. ˛; ˇ/ < 1. 7) has radius of convergence one. 3 Tests for Singular Points 29 for jt j Ä 2 .