Abstract
In 1965, Fine & Wilf proved the following theorem: if (f n )n≥0 and (g n ) n ≥0 are periodic sequences of real numbers, of periods h and k respectively, and f n = g n for 0 ≤n < h + k - gcd(h, k), then f n = g n for all n ≥0. Furthermore, the constant h + k - gcd(h, k) is best possible. In this paper we consider some variations on this theorem. In particular, we study the case where f n ≤ g n instead off n =g n . We also obtain a generalization to more than two periods.
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References
J. Berstel. Sturmian words. In M. Lothaire, editor, Algebraic Combinatorics on Words. Cambridge University Press, to appear, 2001. Preliminary version available at http://www-igm.univ-mlv.fr/~berstel/Lothaire/
M. G. Castelli, F. Mignosi, and A. Restivo. Fine and Wilf’s theorem for three periods and a generalization of Sturmian words. Theoret. Comput. Sci. 218 (1999), 83–94.
C. Choffrut and J. Karhumäki. Combinatorics of Words. In G. Rozenberg and A. Salomaa, eds., Handbook of Formal Languages, 1997, pp. 329–438.
M. Crochemore and W. Rytter. Text Algorithms. Oxford University Press, 1994.
L. Fejér. Sur les polynomes harmoniques quelconques. C. R. Acad. Sci. Paris 157 (1913), 506–509.
N. J. Fine and H. S. Wilf. Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965), 109–114.
A. D. Gilbert and C. J. Smyth. Zero-mean cosine polynomials which are non-negative for as long as possible. Preprint, 2000.
J. Justin. On a paper by Castelli, Mignosi, Restivo. Theoret. Inform. Appl. 34 (2000), 373–377.
D. E. Knuth, J. Morris, and V. Pratt. Fast pattern matching in strings. SIAM J. Comput. 6 (1977), 323–350.
G. Pólya and G. Szegö. Problems and Theorems in Analysis II. Springer-Verlag, 1976.
A. Salomaa. Formal Languages. Academic Press, 1973.
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Mignosi, F., Shallit, J., Wang, Mw. (2001). Variations on a Theorem of Fine & Wilf. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_45
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DOI: https://doi.org/10.1007/3-540-44683-4_45
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