{"id":{"repo_id":"greece","oai_identifier":"oai:10442/0691"},"canonical_url":"https://search.dev.ndltd.org/etd/greece/oai:10442/0691","repository":{"repo_id":"greece","name":"Greek National Archive of PhD Theses","base_url":"https://phdtheses.ekt.gr/eadd_oai/request"},"display":{"title":"ΧΑΡΑΚΤΗΡΙΣΜΟΣ ΔΥΝΑΜΟΣΕΙΡΩΝ ΜΕ ΜΕΡΙΚΑ ΑΘΡΟΙΣΜΑΤΑ ΠΑΝΩ ΣΕ ΠΕΠΕΡΑΣΜΕΝΟ ΠΛΗΘΟΣ ΚΥΚΛΩΝ","abstract":"LET K BE THE CLASS OF TRIGONOMETRIC SERIES OF POWER TYPE, I.E. TAYLOR SERIES Σ C ZN FOR Z=E IX, WHOSE PARTIAL SUMS FOR ALL X IN E, WHERE E IS A NONDENUMERABLESUBSET OF (0,2Π) LIE ON A FINITE NUMBER OF CIRCLES (A PRIORI DEPENDINGON X) IN THE COMPLEX PLANE. THE MAIN RESULT OF THIS PAPER IS THAT FOR EVERY MEMBER OF THE CLASS K, THERE EXIST A COMPLEX NUMBER Ω, /Ω/=1, AND TWO POSITIVE INTEGERS Ν,Κ, N<K, SUCH THAT FOR THE COEFFICIENTS CN WE HAVE: CΜ+Λ(Κ-Ν)=CΜΩΛ, Μ=Ν, Ν+1,...,Κ-1, Λ=1,2,3,... . THUS, EVERY MEMBER OF THE CLASS K HAS (WITH MINOR MODIFICATIONS) A REPRESENTATION OF THE FORM: P(X)Σ EIKNX, N=0 WHERE P(X) IS A SUITABLE TRIGONOMETRIC POLYNOMIAL AND K A POSITIVE INTEGER. THE PROOF IS ELEMENTARY BUT RATHER LONG. THIS RESULT IS CLOSELY RELATED TO A THEOREM OF MARCINKIEWICZ AND ZYGMUND ON THE CIRCULAR STRUCTURE OF THE SET OF LIMIT POINTS OF THE SEQUENCE OF PARTIAL SUMS OF (C,1) SUMMABLE TAYLOR SERIES.","abstract_html":"LET K BE THE CLASS OF TRIGONOMETRIC SERIES OF POWER TYPE, I.E. TAYLOR SERIES Σ C ZN FOR Z=E IX, WHOSE PARTIAL SUMS FOR ALL X IN E, WHERE E IS A NONDENUMERABLESUBSET OF (0,2Π) LIE ON A FINITE NUMBER OF CIRCLES (A PRIORI DEPENDINGON X) IN THE COMPLEX PLANE. THE MAIN RESULT OF THIS PAPER IS THAT FOR EVERY MEMBER OF THE CLASS K, THERE EXIST A COMPLEX NUMBER Ω, /Ω/=1, AND TWO POSITIVE INTEGERS Ν,Κ, N&lt;K, SUCH THAT FOR THE COEFFICIENTS CN WE HAVE: CΜ+Λ(Κ-Ν)=CΜΩΛ, Μ=Ν, Ν+1,...,Κ-1, Λ=1,2,3,... . THUS, EVERY MEMBER OF THE CLASS K HAS (WITH MINOR MODIFICATIONS) A REPRESENTATION OF THE FORM: P(X)Σ EIKNX, N=0 WHERE P(X) IS A SUITABLE TRIGONOMETRIC POLYNOMIAL AND K A POSITIVE INTEGER. THE PROOF IS ELEMENTARY BUT RATHER LONG. THIS RESULT IS CLOSELY RELATED TO A THEOREM OF MARCINKIEWICZ AND ZYGMUND ON THE CIRCULAR STRUCTURE OF THE SET OF LIMIT POINTS OF THE SEQUENCE OF PARTIAL SUMS OF (C,1) SUMMABLE TAYLOR SERIES.","abstract_has_math":false,"creators":["Katsoprinakis, Emmanouil","Κατσοπρινάκης, Εμμανουήλ"],"institution":"University of Crete (UOC)","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1988,"date_issued":"1988","date_published":"1988","updated_at":"2026-07-24T02:25:24Z","subjects":["ΑΘΡΟΙΣΙΜΟΤΗΤΑ CESARO","Γωνιακή κατανομή","ΚΥΚΛΙΚΗ ΔΟΜΗ","ΜΕΡΙΚΑΑΘΡΟΙΣΜΑΤΑ","Σειρές Taylor","ΤΡΙΓΩΝΟΜΕΤΡΙΚΕΣ ΔΥΝΑΜΟΣΕΙΡΕΣ","Angular distribution","CESARO SUMMABILITY","CIRCULAR STRUCTURE","PARTIAL SUMS","Taylor series","TRIGONOMETRIC POWER SERIES","Φυσικές Επιστήμες","Μαθηματικά","Natural Sciences","Mathematics"],"languages":["gre"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.12681/eadd/0691"],"render_values":[{"text":"10.12681/eadd/0691","href":"https://doi.org/10.12681/eadd/0691","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10442/hedi/0691","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Katsoprinakis, Emmanouil","Κατσοπρινάκης, Εμμανουήλ"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["1988"]},{"key":"dc:publisher","label":"Institution","values":["University of Crete (UOC)","Πανεπιστήμιο Κρήτης"]},{"key":"dc:type","label":"Dc Type","values":["PhD Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ΑΘΡΟΙΣΙΜΟΤΗΤΑ CESARO","Γωνιακή κατανομή","ΚΥΚΛΙΚΗ ΔΟΜΗ","ΜΕΡΙΚΑΑΘΡΟΙΣΜΑΤΑ","Σειρές Taylor","ΤΡΙΓΩΝΟΜΕΤΡΙΚΕΣ ΔΥΝΑΜΟΣΕΙΡΕΣ","Angular distribution","CESARO SUMMABILITY","CIRCULAR STRUCTURE","PARTIAL SUMS","Taylor series","TRIGONOMETRIC POWER SERIES","Φυσικές Επιστήμες","Μαθηματικά","Natural Sciences","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["gre"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10.12681/eadd/0691","http://hdl.handle.net/10442/hedi/0691"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["LET K BE THE CLASS OF TRIGONOMETRIC SERIES OF POWER TYPE, I.E. TAYLOR SERIES Σ C ZN FOR Z=E IX, WHOSE PARTIAL SUMS FOR ALL X IN E, WHERE E IS A NONDENUMERABLESUBSET OF (0,2Π) LIE ON A FINITE NUMBER OF CIRCLES (A PRIORI DEPENDINGON X) IN THE COMPLEX PLANE. THE MAIN RESULT OF THIS PAPER IS THAT FOR EVERY MEMBER OF THE CLASS K, THERE EXIST A COMPLEX NUMBER Ω, /Ω/=1, AND TWO POSITIVE INTEGERS Ν,Κ, N<K, SUCH THAT FOR THE COEFFICIENTS CN WE HAVE: CΜ+Λ(Κ-Ν)=CΜΩΛ, Μ=Ν, Ν+1,...,Κ-1, Λ=1,2,3,... . THUS, EVERY MEMBER OF THE CLASS K HAS (WITH MINOR MODIFICATIONS) A REPRESENTATION OF THE FORM: P(X)Σ EIKNX, N=0 WHERE P(X) IS A SUITABLE TRIGONOMETRIC POLYNOMIAL AND K A POSITIVE INTEGER. THE PROOF IS ELEMENTARY BUT RATHER LONG. THIS RESULT IS CLOSELY RELATED TO A THEOREM OF MARCINKIEWICZ AND ZYGMUND ON THE CIRCULAR STRUCTURE OF THE SET OF LIMIT POINTS OF THE SEQUENCE OF PARTIAL SUMS OF (C,1) SUMMABLE TAYLOR SERIES.","ΕΣΤΩ Κ Η ΚΛΑΣΗ ΤΩΝ ΤΡΙΓΩΝΟΜΕΤΡΙΚΩΝ ΔΥΝΑΜΟΣΕΙΡΩΝ (ΣΕΙΡΩΝ TAYLOR) Σ CNEINX, N=0 ΠΟΥ ΕΧΟΥΝ ΤΑ ΜΕΡΙΚΑ ΤΟΥΣ ΑΘΡΟΙΣΜΑΤΑ ΠΑΝΩ Σ'ΕΝΑ ΠΕΠΕΡΑΣΜΕΝΟ ΠΛΗΘΟΣ ΚΥΚΛΩΝ ΤΟΥ ΜΙΓΑΔΙΚΟΥ ΕΠΙΠΕΔΟΥ, ΟΤΑΝ ΤΟ Χ ΑΝΗΚΕΙ Σ'ΕΝΑ ΥΠΕΡΑΡΙΘΜΗΣΙΜΟ ΥΠΟΣΥΝΟΛΟ Ε ΤΟΥ (0,2Π). ΤΟ ΚΥΡΙΟ ΑΠΟΤΕΛΕΣΜΑ ΑΥΤΗΣ ΤΗΣ ΕΡΓΑΣΙΑΣ ΕΙΝΑΙ Ο ΧΑΡΑΚΤΗΡΙΣΜΟΣ ΤΩΝ ΣΕΙΡΩΝ ΤΗΣ ΚΛΑΣΗΣ Κ. ΣΥΓΚΕΚΡΙΜΕΝΑ, ΑΠΟΔΕΙΚΝΥΟΥΜΕ ΟΤΙ ΚΑΘΕ ΣΕΙΡΑ ΤΗΣ ΚΛΑΣΗΣ Κ ΕΧΕΙ (ΕΚΤΟΣ ΑΠΟ ΤΕΤΡΙΜΜΕΝΕΣ ΑΛΛΑΓΕΣ) ΜΙΑ ΑΝΑΠΑΡΑΣΤΑΣΗ ΤΗΣ ΜΟΡΦΗΣ: Ρ(Χ) Σ EIKNX, N=0 ΟΠΟΥ Ρ(Χ) ΕΙΝΑΙ ΕΝΑ ΚΑΤΑΛΛΗΛΟ ΤΡΙΓΩΝΟΜΕΤΡΙΚΟ ΠΟΛΥΩΝΥΜΟ ΚΑΙ Κ ΕΝΑΣ ΘΕΤΙΚΟΣ ΑΚΕΡΑΙΟΣ. Η ΑΠΟΔΕΙΞΗ ΕΙΝΑΙ ΣΤΟΙΧΕΙΩΔΗΣ, ΑΛΛΑ ΚΑΠΩΣ ΕΚΤΕΤΑΜΕΝΗ. ΟΙ ΠΑΡΑΠΑΝΩ ΣΕΙΡΕΣ ΣΧΕΤΙΖΟΝΤΑΙ Μ'ΕΝΑ ΘΕΩΡΗΜΑ ΤΩΝ MARCINKIEWICZ ΚΑΙ ZYGMUND, ΠΟΥ ΑΝΑΦΕΡΕΤΑΙ ΣΤΗΝ ΚΥΚΛΙΚΗ ΔΟΜΗ ΤΟΥ ΣΥΝΟΛΟΥ ΤΩΝ ΣΗΜΕΙΩΝ ΣΥΣΣΩΡΕΥΣΗΣ ΤΗΣ ΑΚΟΛΟΥΘΙΑΣ ΤΩΝ ΜΕΡΙΚΩΝ ΑΘΡΟΙΣΜΑΤΩΝ ΤΩΝ ΣΕΙΡΩΝ TAYLOR ΠΟΥ ΕΙΝΑΙ (C,1) ΑΘΡΟΙΣΙΜΕΣ Σ'ΕΝΑ ΥΠΟΣΥΝΟΛΟ Ε ΤΟΥ (0,2Π)."]},{"key":"dc:title","label":"Title","values":["ΧΑΡΑΚΤΗΡΙΣΜΟΣ ΔΥΝΑΜΟΣΕΙΡΩΝ ΜΕ ΜΕΡΙΚΑ ΑΘΡΟΙΣΜΑΤΑ ΠΑΝΩ ΣΕ ΠΕΠΕΡΑΣΜΕΝΟ ΠΛΗΘΟΣ ΚΥΚΛΩΝ","CHARACTERIZATION OF POWER SERIES WITH PARTIAL SUMS LYING ON A FINITE NUMBER OF CIRCLES"]}]}],"canonical_facts":{"dc:creator":["Katsoprinakis, Emmanouil","Κατσοπρινάκης, Εμμανουήλ"],"dc:date":["1988"],"dc:description":["LET K BE THE CLASS OF TRIGONOMETRIC SERIES OF POWER TYPE, I.E. TAYLOR SERIES Σ C ZN FOR Z=E IX, WHOSE PARTIAL SUMS FOR ALL X IN E, WHERE E IS A NONDENUMERABLESUBSET OF (0,2Π) LIE ON A FINITE NUMBER OF CIRCLES (A PRIORI DEPENDINGON X) IN THE COMPLEX PLANE. THE MAIN RESULT OF THIS PAPER IS THAT FOR EVERY MEMBER OF THE CLASS K, THERE EXIST A COMPLEX NUMBER Ω, /Ω/=1, AND TWO POSITIVE INTEGERS Ν,Κ, N<K, SUCH THAT FOR THE COEFFICIENTS CN WE HAVE: CΜ+Λ(Κ-Ν)=CΜΩΛ, Μ=Ν, Ν+1,...,Κ-1, Λ=1,2,3,... . THUS, EVERY MEMBER OF THE CLASS K HAS (WITH MINOR MODIFICATIONS) A REPRESENTATION OF THE FORM: P(X)Σ EIKNX, N=0 WHERE P(X) IS A SUITABLE TRIGONOMETRIC POLYNOMIAL AND K A POSITIVE INTEGER. THE PROOF IS ELEMENTARY BUT RATHER LONG. THIS RESULT IS CLOSELY RELATED TO A THEOREM OF MARCINKIEWICZ AND ZYGMUND ON THE CIRCULAR STRUCTURE OF THE SET OF LIMIT POINTS OF THE SEQUENCE OF PARTIAL SUMS OF (C,1) SUMMABLE TAYLOR SERIES.","ΕΣΤΩ Κ Η ΚΛΑΣΗ ΤΩΝ ΤΡΙΓΩΝΟΜΕΤΡΙΚΩΝ ΔΥΝΑΜΟΣΕΙΡΩΝ (ΣΕΙΡΩΝ TAYLOR) Σ CNEINX, N=0 ΠΟΥ ΕΧΟΥΝ ΤΑ ΜΕΡΙΚΑ ΤΟΥΣ ΑΘΡΟΙΣΜΑΤΑ ΠΑΝΩ Σ'ΕΝΑ ΠΕΠΕΡΑΣΜΕΝΟ ΠΛΗΘΟΣ ΚΥΚΛΩΝ ΤΟΥ ΜΙΓΑΔΙΚΟΥ ΕΠΙΠΕΔΟΥ, ΟΤΑΝ ΤΟ Χ ΑΝΗΚΕΙ Σ'ΕΝΑ ΥΠΕΡΑΡΙΘΜΗΣΙΜΟ ΥΠΟΣΥΝΟΛΟ Ε ΤΟΥ (0,2Π). ΤΟ ΚΥΡΙΟ ΑΠΟΤΕΛΕΣΜΑ ΑΥΤΗΣ ΤΗΣ ΕΡΓΑΣΙΑΣ ΕΙΝΑΙ Ο ΧΑΡΑΚΤΗΡΙΣΜΟΣ ΤΩΝ ΣΕΙΡΩΝ ΤΗΣ ΚΛΑΣΗΣ Κ. ΣΥΓΚΕΚΡΙΜΕΝΑ, ΑΠΟΔΕΙΚΝΥΟΥΜΕ ΟΤΙ ΚΑΘΕ ΣΕΙΡΑ ΤΗΣ ΚΛΑΣΗΣ Κ ΕΧΕΙ (ΕΚΤΟΣ ΑΠΟ ΤΕΤΡΙΜΜΕΝΕΣ ΑΛΛΑΓΕΣ) ΜΙΑ ΑΝΑΠΑΡΑΣΤΑΣΗ ΤΗΣ ΜΟΡΦΗΣ: Ρ(Χ) Σ EIKNX, N=0 ΟΠΟΥ Ρ(Χ) ΕΙΝΑΙ ΕΝΑ ΚΑΤΑΛΛΗΛΟ ΤΡΙΓΩΝΟΜΕΤΡΙΚΟ ΠΟΛΥΩΝΥΜΟ ΚΑΙ Κ ΕΝΑΣ ΘΕΤΙΚΟΣ ΑΚΕΡΑΙΟΣ. Η ΑΠΟΔΕΙΞΗ ΕΙΝΑΙ ΣΤΟΙΧΕΙΩΔΗΣ, ΑΛΛΑ ΚΑΠΩΣ ΕΚΤΕΤΑΜΕΝΗ. ΟΙ ΠΑΡΑΠΑΝΩ ΣΕΙΡΕΣ ΣΧΕΤΙΖΟΝΤΑΙ Μ'ΕΝΑ ΘΕΩΡΗΜΑ ΤΩΝ MARCINKIEWICZ ΚΑΙ ZYGMUND, ΠΟΥ ΑΝΑΦΕΡΕΤΑΙ ΣΤΗΝ ΚΥΚΛΙΚΗ ΔΟΜΗ ΤΟΥ ΣΥΝΟΛΟΥ ΤΩΝ ΣΗΜΕΙΩΝ ΣΥΣΣΩΡΕΥΣΗΣ ΤΗΣ ΑΚΟΛΟΥΘΙΑΣ ΤΩΝ ΜΕΡΙΚΩΝ ΑΘΡΟΙΣΜΑΤΩΝ ΤΩΝ ΣΕΙΡΩΝ TAYLOR ΠΟΥ ΕΙΝΑΙ (C,1) ΑΘΡΟΙΣΙΜΕΣ Σ'ΕΝΑ ΥΠΟΣΥΝΟΛΟ Ε ΤΟΥ (0,2Π)."],"dc:identifier":["10.12681/eadd/0691","http://hdl.handle.net/10442/hedi/0691"],"dc:language":["gre"],"dc:publisher":["University of Crete (UOC)","Πανεπιστήμιο Κρήτης"],"dc:subject":["ΑΘΡΟΙΣΙΜΟΤΗΤΑ CESARO","Γωνιακή κατανομή","ΚΥΚΛΙΚΗ ΔΟΜΗ","ΜΕΡΙΚΑΑΘΡΟΙΣΜΑΤΑ","Σειρές Taylor","ΤΡΙΓΩΝΟΜΕΤΡΙΚΕΣ ΔΥΝΑΜΟΣΕΙΡΕΣ","Angular distribution","CESARO SUMMABILITY","CIRCULAR STRUCTURE","PARTIAL SUMS","Taylor series","TRIGONOMETRIC POWER SERIES","Φυσικές Επιστήμες","Μαθηματικά","Natural Sciences","Mathematics"],"dc:title":["ΧΑΡΑΚΤΗΡΙΣΜΟΣ ΔΥΝΑΜΟΣΕΙΡΩΝ ΜΕ ΜΕΡΙΚΑ ΑΘΡΟΙΣΜΑΤΑ ΠΑΝΩ ΣΕ ΠΕΠΕΡΑΣΜΕΝΟ ΠΛΗΘΟΣ ΚΥΚΛΩΝ","CHARACTERIZATION OF POWER SERIES WITH PARTIAL SUMS LYING ON A FINITE NUMBER OF CIRCLES"],"dc:type":["PhD Thesis"]},"updated_at":"2026-07-24T02:25:24Z"}