Abstract
In 1928 Otto Toeplitz constructed a very interesting class of real Bohr almost periodic functions on the line. His construction was based a) on an arbitrary choice of a uniformly convergent sequence f1, f2,... of continuous real functions on the unit interval [O,1] such that fn (O) = O = fn (1) (n = 1,2,...) and b) a binary “filling scheme” which can shortly be described as follows: fill every second unit interval [k - 1, k] with translates of f1 and leave the remaining intervals as “holes”; ect fill every second hole with translates of f2 and leave the remaining unit intervals as “holes” etc. Toeplitz 1 contains formulas for the spectrum and Fourier coefficients of the resulting continuous function f: ℝ → ℝ, which is always Bohr almost periodic.
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References
Besicovitch, A.S.: On generalized almost periodic functions. Proc. London Math. Soc. (2), 25. (1926), 495–512.
Besicovitch, A.S.: Almost periodic functions. Cambridge 1932.
Besicovitch A.S. and H. Bohr: Almost periodicity and general trigonometric series. Acta Math. 52 (1931), 203–292.
Bohr, H.: Zur Theorie der fastperiodischen Funktionen I, Acta Math. 45 (1924), 29–127.
Bohr, H.: Zur Theorie der fastperiodischen Funktionen II, Acta Math. 46 (1925), 101–214.
Bohr, H.: Fastperiodische Funktionen. Ergebnisse der Mathematik 1 Nr. 5, Berlin 1932.
Bohr, H. and E. Føiner: On some types of functional spaces. A. contribution to the theorie of almost periodic functions. Acta Math. 76 (1944), 31–155.
Coven, E. and M. Keane: The structure of substitution minimal sets. Trans.Amer. Math. Soc. 162 (1971), 89–102.
Eberlein, E.: Toeplitz-Folgen und Gruppentranslationen. Arch. Math. 22 (1971), 291–301.
Føiner, E.: On the structure of generalized almost periodic functions. Mat.-Fys. Medd. Danske Vid. Selsk. 21 (1945), 1–30.
Grillenberger, C.: Construction of strictly ergodic systems I. Given entropy, Z. f. Wahrsch. u. verw. Geb. 25 (1973), 323–334; II. K-Systems, 335-342.
Grillenberger, C., and P. Shields: Construction of strictly ergodic systems, III. Bernoulli systems, Z. f. Wahrsch. u. verw. Geb. 33 (1975§76), 215–217
Haller, H.: Kombinatorische Konstruktion fastperiodischer Funktionen, 66S, Dissertation Erlangen 1978.
Jacobs K. and M. Keane: O-I-Sequences of Toeplitz-Type. Z. f. Wahrsch. und verw.Geb. 13 (1969), 123–131.
Loomis,: An introduction to abstract harmonic analysis, Princeton 1953.
Maak, W.: Fastperiodische Funktionen, 2. Auflage, Berlin/Heidelberg/New York 1967.
Stepanoff, W.: Über einige Verallgemeinerungen der fast-periodischen Funktionen. Math. Ann. 95 (1926), 473–498.
Toeplitz, O.: Ein Beispiel zur Theorie der fastperiodischen Funktionen. Math. Ann. 98 (1928), 281–295.
Weyl, H.: Integralgleichungen und fastperiodische Funktionen. Math. Ann. 97 (1926), 338–356.
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© 1982 Springer Basel AG
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Haller, H., Jacobs, K. (1982). Extensions of Otto Toeplitz’ Combinatorial Construction of Almost Periodic Functions on the Real Line. In: Gohberg, I. (eds) Toeplitz Centennial. Operator Theory: Advances and Applications, vol 4. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5183-1_17
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DOI: https://doi.org/10.1007/978-3-0348-5183-1_17
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