Diophantine approximations and almost periodic functions

• Autorzy: Nawrocki A.

Identyfikatory

DOI

10.1515/dema-2017-0011

• Języki publikacji: EN

Abstrakty

EN

In this paper we investigate the asymptotic behaviour of the classical continuous and unbounded almost periodic function in the Lebesgue measure. Using diophantine approximations we show that this function can be estimated by functions of polynomial type and we give the best polynomial estimation.

Słowa kluczowe

EN

almost periodic function in the Lebesgue measure; continued fraction; Stepanov almost periodic function

PL

funkcja okresowa; miara Lebesgue'a; ułamek łańcuchowy; ułamek ciągły

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2017
• Tom: Vol. 50, nr 1
• Strony: 100--104
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Nawrocki A.

• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland

Bibliografia

• [1] Bohr H., Zur Theorie der fastperiodischen Funktionen, I. Teil Eine Verallgemeinerung der Theorie der Fourierreihen, Acta Math., 1925, 45, 29–127
• [2] Bohr H., Zur Theorie der fastperiodischen Funktionen, II. Teil Zusammenhang der fastperiodischen Functionen mit Funktionen von unendkich vielen Variablen; gleichmassige Approximation durch trigonometrische Summen, Acta Math., 1925, 46, 101–214
• [3] Bohr H., Zur Theorie der fastperiodischen Funktionen, III. Teil Dirichletentwicklung analytischer Functionen, Acta Math., 1926, 47, 237–281
• [4] Besicovitch A.S., On generalized almost periodic functions, Proc. London Math. Soc., 1926, 25, no. 2, 495–512
• [5] Bochner S., Properties of Fourier series of almost periodic functions, Proc. London Math. Soc., 1926, 26, no. 2, 433–452
• [6] Levitan B. M., Pochti-periodicheskie Funkcii, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953, (in Russian)
• [7] Neumann J., Almost periodic functions in a group, Trans. Amer. Math. Soc., 1934, 36, 445–492
• [8] Stepanov V. V., Uber einige Verallgemeinerungen der fastperiodischen Funktionen, Ann. Math., 1926, 95, 437–498
• [9] Weyl H., Integralgleichungen und fastperiodische Funktionen, Ann. Math., 1926, 97, 338–356
• [10] Andres J., Bersani A. M., Grande R. F., Hierarchy of almost-periodic function spaces, Rend. Mat. Appl.(7), 2006, 26, 121–188
• [11] Franklin P., Approximation theorems for generalized almost periodic functions, Math. Z., 1929, 29, no. 1, 70–86
• [12] Stoiński S., Almost periodic functions in the Lebesgue measure, Comment. Math. Prace Mat., 1994, 34, 189–198
• [13] Stoiński S., On compactness of almost periodic functions in the Lebesgue measure, Fasc. Math., 1999, 30, 171–175
• [14] Bugajewski D., Nawrocki A., Some remarks on almost periodic functions in view of the Lebesgue measure with applications to linear differential equations, Ann. Acad. Sci. Fenn. Math., (in press)
• [15] Hardy G. H., Wright E. M., An introduction to the number theory, 4th ed., Clarendon Press, 1971

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).