zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
$\lambda $-statistical convergence. (English) Zbl 0953.40002
This paper deals with a generalization of the Cesàro mean summability of the sequences $$ [C,1]:=\{x=(x_n): \text{ there is } L \in \bbfR\text{ such that } \lim _{n \to \infty }{1 \over n} \sum _{k=1}^n {}x_k - L{}=0 \}$$ to $$[V,\lambda ]:= \{x=(x_n): \text{ there is } L \in \bbfR \text{ such that } \lim_{n \to \infty }{1 \over \lambda _n} \sum _{k \in I_n} {}x_k - L{}=0{}\}$$ for some interval $I_n$. Comparisons of statistical convergence and $\lambda $-statistical convergence for a sequence $x=(x_n)$ defined by using limits $$\lim _{n \to \infty }{1 \over n} {}\{ k \leq n : {}x_n - L{} \geq \varepsilon \}=0 \quad \text{and}\quad\lim _{n \to \infty }{1 \over \lambda _n} {}\{k \in I_n : {}x_n - L{} \geq \varepsilon \}= 0$$ are given.
Reviewer: Ondrej Kováčik (Žilina)

40A05Convergence and divergence of series and sequences
40C05Matrix methods in summability
Full Text: EuDML
[1] CONNOR J. S.: The statistical and strong p-Cesáro convergence of sequences. Analysis 8 (1988), 47-63. · Zbl 0653.40001
[2] CONNOR J. S.: On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull. 32 (1989), 194-198. · Zbl 0693.40007 · doi:10.4153/CMB-1989-029-3
[3] FAST H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244. · Zbl 0044.33605 · eudml:209960
[4] FREEDMAN A. R.-SEMBER J. J.-RAPHAEL M.: Some Cesáro type summability spaces. Proc. London Math. Soc. 37 (1978), 508-520. · Zbl 0424.40008 · doi:10.1112/plms/s3-37.3.508
[5] FRIDY J. A.: On statistical convergence. Analysis 5 (1985), 301-313. · Zbl 0588.40001
[6] FRIDY J. A.-ORHAN C.: Lacunary statistical convergence. Pacific. J. Math. 160 (1993), 43-51. · Zbl 0794.60012 · doi:10.2140/pjm.1993.160.43
[7] KOLK E.: The statistical convergence in Banach spaces. Acta Comment. Univ. Tartu 928 (1991), 41-52.
[8] LEINDLER L.: Über die de la Vallée-Pousinsche Summierbarkeit allgemeiner Orthogonalreihen. Acta Math. Acad. Sci. Hungar. 16 (1965), 375-387. · Zbl 0138.28802 · doi:10.1007/BF01904844
[9] ŠALÁT T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. · Zbl 0437.40003 · eudml:34081