We prove by means of the Berezin symbols some theorems for the (L)-summability method for sequences and series. Also, we prove a new Tauberian type theorem for (L)-summability. | Turk J Math (2018) 42: 2417 – 2422 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On the summability methods of logarithmic type and the Berezin symbol Ulaş YAMANCI∗, Department of Statistics, Faculty of Arts and Sciences, Süleyman Demirel University, Isparta, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: We prove by means of the Berezin symbols some theorems for the (L) -summability method for sequences and series. Also, we prove a new Tauberian type theorem for (L) -summability. Key words: (L) -summability, Berezin symbol, (e) -convergence, compact operator, Tauberian type theorem, Dirichlet space, diagonal operator 1. Introduction In this article, by applying a new functional analytic approach based on the so-called the Berezin symbol technique, we prove the following results (see [3, 4]). Also, we give a new Tauberian type theorem for (L) summable sequences of complex numbers. Recall that a sequence (an )n≥0 of complex numbers an is said to be summable to a finite number ζ by the logarithmic method (L) (or (L) -summable to ζ ) if ∞ ∑ an n+1 x n +1 n=0 converges in the open interval (0, 1) and ∞ ∑ 1 an n+1 lim − x = ζ. log (1 − x) n=0 n + 1 x→1− The series ∞ ∑ n=0 an is (L)-summable to ζ if the sequence of partial sums s := (sn )n≥0 (where sn = n ∑ ak ) is k=0 (L) -summable to ζ. Theorem 1 If (ak )k≥0 converges to ζ , then (ak )k≥0 (L) -converges to ζ . Theorem 2 If the series ∞ ∑ ak converges to ζ , then k=0 ∞ ∑ ak is (L) -summable to ζ . k=0 ∗Correspondence: ulasyamanci@ 2010 AMS Mathematics Subject Classification: 40D09 2417 This work is licensed under a Creative Commons Attribution International License. YAMANCI/Turk J Math Before beginning the presentation, we recall some basic definitions and notations. Recall that in [6], Karaev introduced the notions of an (e) -convergent sequence .
