# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
On the Riesz almost convergent sequences space. (English) Zbl 1250.46005
Summary: The purpose of this paper is to introduce new spaces $\widehat{f}$ and $\widehat{f}_0$ that consist of all sequences whose Riesz transforms of order one are in the spaces $f$ and $f_0$, respectively. We also show that $\widehat{f}$ and $\widehat{f}_0$ are linearly isomorphic to the spaces $f$ and $f_0$, respectively. The $\beta$- and $\gamma$-duals of the spaces $\widehat{f}$ and $\widehat{f}_0$ are computed. Furthermore, the classes $(\widehat{f} : \mu)$ and $(\mu : \widehat{f})$ of infinite matrices are characterized for any given sequence space $\mu$ and determine the necessary and sufficient conditions on a matrix $A$ to satisfy $B_R - \text{core}(Ax) \subseteq K - \text{core}(x)$, $B_R - \text{core}(A_R) \subseteq st - \text{core}(x)$ for all $x \in \ell_\infty$.

##### MSC:
 46A45 Sequence spaces 40C05 Matrix methods in summability
##### Keywords:
sequence spaces; Riesz method
Full Text:
##### References:
 [1] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, UK, 2000. · Zbl 0954.40001 [2] B. Altay and F. Ba\csar, “The fine spectrum and the matrix domain of the difference operator \Delta on the sequence space lp, (0<p<1),” Communications in Mathematical Analysis, vol. 2, no. 2, pp. 1-11, 2007. · Zbl 1173.47021 [3] M. Ba\csarir, “On some new sequence spaces and related matrix transformations,” Indian Journal of Pure and Applied Mathematics, vol. 26, no. 10, pp. 1003-1010, 1995. · Zbl 0855.40005 [4] C. Aydın and F. Ba\csar, “Some generalizations of the sequencespace arp,” Iranian Journal of Science and Technology, vol. 20, no. 2, pp. 175-190, 2006. [5] M. Kiri\cs\cci and F. Ba\csar, “Some new sequence spaces derived by the domain of generalized difference matrix,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1299-1309, 2010. · Zbl 1201.40001 · doi:10.1016/j.camwa.2010.06.010 [6] M. \cSengönül and F. Ba\csar, “Some new Cesàro sequence spaces of non-absolute type which include the spaces c0 and c,” Soochow Journal of Mathematics, vol. 31, no. 1, pp. 107-119, 2005. [7] H. Polat and F. Ba\csar, “Some Euler spaces of difference sequences of order m,” Acta Mathematica Scientia. Series B, vol. 27, no. 2, pp. 254-266, 2007. · Zbl 1246.46007 · doi:10.1016/S0252-9602(07)60024-1 [8] E. Malkowsky, M. Mursaleen, and S. Suantai, “The dual spaces of sets of difference sequences of order m and matrix transformations,” Acta Mathematica Sinica, vol. 23, no. 3, pp. 521-532, 2007. · Zbl 1123.46007 · doi:10.1007/s10114-005-0719-x [9] B. Altay and F. Ba\csar, “Certain topological properties and duals of the domain of a triangle matrix in a sequence space,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 632-645, 2007. · Zbl 1152.46003 · doi:10.1016/j.jmaa.2007.03.007 [10] M. Kiri\cs\cci and F. Ba\csar, “Some new sequence spaces derived by the domain of generalized difference matrix,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1299-1309, 2010. · Zbl 1201.40001 · doi:10.1016/j.camwa.2010.06.010 [11] G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167-190, 1948. · Zbl 0031.29501 · doi:10.1007/BF02393648 [12] D. Butković, H. Kraljević, and N. Sarapa, “On the almost convergence,” in Functional Analysis, II, vol. 1242 of Lecture Notes in Mathematics, pp. 396-417, Springer, Berlin, Germany, 1987. · Zbl 0633.40001 · doi:10.1007/BFb0072446 [13] J. A. Fridy and C. Orhan, “Statistical limit superior and limit inferior,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3625-3631, 1997. · Zbl 0883.40003 · doi:10.1090/S0002-9939-97-04000-8 [14] J. S. Connor, “The statistical and strong p-Cesàro convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47-63, 1988. · Zbl 0653.40001 [15] S. L. Mishra, B. Satapathy, and N. Rath, “Invariant means and \sigma -core,” The Journal of the Indian Mathematical Society. New Series, vol. 60, no. 1-4, pp. 151-158, 1994. · Zbl 0882.40004 [16] K. Kayaduman and H. \cCo\cskun, “On the \sigma (A)-summability and \sigma (A)-core,” Demonstratio Mathematica, vol. 40, no. 4, pp. 859-867, 2007. · Zbl 1140.40002 [17] M. Mursaleen, “On some new invariant matrix methods of summability,” Quarterly Journal of Mathematics, vol. 24, pp. 77-86, 1983. · Zbl 0539.40006 · doi:10.1093/qmath/34.1.77 [18] K. Kayaduman and M. \cSengönül, “On the Cesàro almost convergent sequences spaces,” under comminication. [19] F. Móricz and B. E. Rhoades, “Some characterizations of almost convergence for single and double sequences,” Publications De L’institut Mathématique, Nouvelle série tome, vol. 48, no. 62, pp. 61-68, 1990. · Zbl 0732.40006 [20] J. A. Sıddıqı, “Infinite matrices summing every almost periodic sequence,” Pacific Journal of Mathematics, vol. 39, pp. 235-251, 1971. · Zbl 0229.42005 · doi:10.2140/pjm.1971.39.235 [21] F. Ba\csar, “Matrix transformations between certain sequence spaces of Xp and lp,” Soochow Journal of Mathematics, vol. 26, no. 2, pp. 191-204, 2000. · Zbl 0960.40002 [22] F. Ba\csar and R. \cColak, “Almost-conservative matrix transformations,” Turkish Journal of Mathematics, vol. 13, no. 3, pp. 91-100, 1989. · Zbl 0970.40500 [23] B. Kuttner, “On dual summability methods,” Proceedings of the Cambridge Philosophical Society, vol. 71, pp. 67-73, 1972. · Zbl 0224.40004 [24] G. G. Lorentz and K. Zeller, “Summation of sequences and summation of series,” Proceedings of the Cambridge Philosophical Society, vol. 60, pp. 67-73, 1972. · Zbl 0125.03301 [25] G. Das, “Sublinear functionals and a class of conservative matrices,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 15, no. 1, pp. 89-106, 1987. · Zbl 0632.46008