Statistical gap Tauberian theorems in metric spaces

Abstract By using the concept of statistical convergence we present statistical Tauberian theorems of gap type for the Cesaro, Euler–Borel family and the Hausdorff families applicable in arbitrary metric spaces. In contrast to the classical gap Tauberian theorems, we show that such theorems exist in the statistical sense for the convolution methods which include the Taylor and the Borel matrix methods. We further provide statistical analogs of the gap Tauberian theorems for the Hausdorff methods and provide an explanation as to how the Tauberian rates over the gaps may differ from those of the classical Tauberian theorems.

[1]  Peter Cass,et al.  Classical and modern methods in summability , 2000 .

[2]  M. K. Khan,et al.  Statistical extensions of some classical Tauberian theorems , 2000 .

[3]  N. Levinson.  Gap and Density Theorems , 1940 .

[4]  K. Zeller,et al.  On Borel’s method of summability , 1960 .

[5]  J. Littlewood,et al.  A Further Note on the Converse of Abel's Theorem , 1926 .

[6]  Lückenumkehrsätze und Lückenperfektheit , 1956 .

[7]  Characterizations of density Tauberian theorems , 1998 .

[8]  H. Fast,et al.  Sur la convergence statistique , 1951 .

[9]  Tauberian Theorems for Integrals II , 1972 .

[10]  N. Bingham.  TAUBERIAN THEOREMS AND THE CENTRAL LIMIT THEOREM , 1981 .

[11]  E. Landau.  Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie , 1930 .

[12]  Über das Eulersche Summierungsverfahren , 1922 .

[13]  J. A. Fridy,et al.  A MATRIX CHARACTERIZATION OF STATISTICAL CONVERGENCE , 1991 .

[14]  N. Bingham.  Tauberian Theorems for Summability Methods of Random‐Walk Type , 1984 .

[15]  J. A. Fridy,et al.  ON STATISTICAL CONVERGENCE , 1985 .

[16]  J. E. Littlewood,et al.  The Converse of Abel's Theorem on Power Series , 1911 .

[17]  J. Littlewood,et al.  Theorems concerning the summability of series by boreľs exponential method , 1916 .

[18]  G. H. Hardy,et al.  Tauberian Theorems Concerning Power Series and Dirichlet's Series whose Coefficients are Positive* , 1914 .

[19]  Paul Turán,et al.  On a new method of analysis and its applications , 1984 .

[20]  D. Gaier.  Der allgemeine Lückenumkehrsatz für das Borel-Verfahren , 1965 .

[21]  G. Lorentz.  Direct Theorems on Methods of Summability , 1949, Canadian Journal of Mathematics.

[22]  G. Hardy.  Theorems Relating to the Summability and Convergence of Slowly Oscillating Series , 1910 .

[23]  M. K. Khan,et al.  Tauberian theorems via statistical convergence , 1998 .

[24]  E. Landau,et al.  Über die Bedeutung einiger neuen Grenzwertsätze der Herren Hardy und Axer , 2022 .

[25]  Robert Schmidt,et al.  Über divergente Folgen und lineare Mittelbildungen , 1925 .

[26]  D. Gaier.  On the Coefficients and the Growth of Gap Power Series , 1966 .

[27]  N. Bingham.  Tauberian theorems for Jakimovski and Karamata-Stirling methods , 1988 .

[28]  A. J. Stam,et al.  Tauberian theorems for limitation methods admitting a central limit theorem , 1976 .

[29]  Amnon Jakimovski,et al.  A generalization of the Lototsky method of summability. , 1959 .