Remarks on Functions Preserving Convergence of Infinite Series

Borsík, Ján; Červeňanský, Jaroslav; Šalát, Tibor

doi:10.2307/44152683

Cited by 3 publications

(7 citation statements)

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“…In this section we characterize many of the spaces F (A, B) in terms of wellknown and easily described classes of functions. Our results extend Theorem A, Theorem 2.6 and Theorem 2.10 in [1] and [2, Theorem 1]. In particular, the classes we consider include those of importance in [2, Theorem 1]: f ∈ F (cs, cs) if and only if f is linear near zero; and in [1, Theorem 2.10]: f ∈ F (cs, l 1 ) if and only if f is identically zero in a neighborhood of 0.…”

Section: Characterization Theoremssupporting

confidence: 87%

“…Specifically, if A and B are sets of real sequences we study the class of functions f such that the transformed sequence (f (a n )) belongs to B for each sequence (a n ) in A. A number of our results extend those in [1] and [2]. Some these were announced at the Summer Symposium in Real Analysis, XXII, at Santa Barbara.…”

Section: Introduction and Notationmentioning

confidence: 77%

“…This theorem is also obtained in [6] via purely real-variable arguments and an even simpler proof is found in [4]. Generalizations of convergence preserving functions are investigated by Borsík, Cerveňanský andŠalát in [1]. Their study entails the f -transform concept.…”

Section: Introduction and Notationmentioning

confidence: 98%

“…Given a function f : R → R the f -transform of the series ∞ n=1 a n is the series ∞ n=1 f (a n ). Three new classes of functions are defined in [1], in addition to convergence preserving functions. These classes are the four combinations obtained when the domain and range of the f -transform are chosen pairwise from the set of convergent and absolutely convergent series.…”

Section: Introduction and Notationmentioning

confidence: 99%

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Functions Preserving Sequence Spaces

Grinnell¹

1998

Real Analysis Exchange

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Let A and B be sets of real sequences. Let F (A, B) denote the set of functions f : R → R that preserve A and B in the sense that (f (an)) ∈ B for all sequences (an) ∈ A. These functions are generalizations of convergence preserving functions first introduced by Rado. We establish identities and inclusions for F (A, B) when A and B are l p -spaces and other well-known sequence spaces. We also characterize F (A, B) in terms of elementary classes of functions. Our characterizations are motivated by the work of Borsík,Červeňanský andŠalát.

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Section: Characterization Theoremssupporting

confidence: 87%

Section: Introduction and Notationmentioning

confidence: 77%

Section: Introduction and Notationmentioning

confidence: 98%

Section: Introduction and Notationmentioning

confidence: 99%

See 2 more Smart Citations

Functions Preserving Sequence Spaces

Grinnell¹

1998

Real Analysis Exchange

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“…Ján Borsík also studied functions with closed graphs [28], functions that preserve cauchy sequences and cauchy nets [7,34], functions that preserve convergence of infinite series [25], oscillations for quasicontinuity, almost continuity [33,40], convergences of functions and generalized continuities.…”

mentioning

confidence: 99%

Ján Borsík (1955 – 2019)

Frič

Holá²

2019

Tatra Mountains Mathematical Publications

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The sad news reached the participants of the 33rd International Summer Conference on Real Functions held in Ustka, Poland, only three days later. True, Ján struggled with cancer over a longer period, however, was still an active member of the "real functions community". Exactly one year ago, he headed/managed the 32nd ISCRF in Stará Lesná and served as an editor of the present issue of Tatra Mountains Mathematical Publications. True, the treacherous illness severely limited his life and he had to step down from his teaching duties and editorial responsibilities in last months. Even though the participants of the 33rd ISCRF could not participate in his funeral, they expressed their deep sorrow, and during the conference commemorated Professor Ján Borsík, his personal and research qualities, friendship and services to the conferrence. It is only natural to devote the present issue of TMMP to his memory. Ján Borsík was born on 11. 9. 1955 in Trstené, Slovakia. After completing his secondary education, in the period 1975-1980 he studied at the School of Natural Sciences, Comenius University in Bratislava, Slovakia. Already as a student, he participated in a scientific seminar devoted to real functions led by Professor TiborŠalát and remained faithful to real functions all his life. After his graduation, Janko Borsík obtained his RNDr. title in 1981 and, under the supervision of ProfessorŠalát, the scientific degree CSc. (equivalent to PhD). Finally, based on his Habilitationshrift, in 1999 he was promoted to Docent in Mathematics. Since 1980 until his premature death, he was affiliated with the Mathematical Institute of Slovak Academy of Sciences, Extension in Košice, as a Senior Scientific Fellow. For a long time Professor Ján Borsík lectured at theŠafárik University and the Technical University in Košice and at the Prešov University in Prešov, where his colleagues and students appreciated his scientific and pedagogical masterships very much. The same can be said about several PhD students and members of the broader Slovak mathematical community.

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Functions Preserving Some Types of Series

Borsík¹

2008

Journal of Applied Analysis

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Remarks on Functions Preserving Convergence of Infinite Series

Cited by 3 publications

References 4 publications

Functions Preserving Sequence Spaces

Functions Preserving Sequence Spaces

Ján Borsík (1955 – 2019)

Functions Preserving Some Types of Series

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