Abstract
In the present paper, we introduce the notion of self-similar function of spectral order zero and study its properties. Such functions have at most countably many discontinuity points, and these points are discontinuity points of the first kind except possibly for a single point, which is a singular point. We derive a formula for calculating the coordinates of this point from the parameters of the self-similar function. We also study the behavior of the self-similar function near the singular point. A nondecreasing function f of spectral order zero belonging to the space L 2[0, 1] generates a self-similar Stieltjes string, namely, a spectral problem of the form
where ρ is a function from the space \( \mathop W\limits^ \circ _{_2^{ - 1} } \left[ {0,1} \right] \) and f′ = ρ. Such a function f that is not of a fixed sign leads to the notion of self-similar indefinite Stieltjes string.
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References
F. Atkinson, “On spectral functions of a string,” in Discrete and Continuous Boundary Problems (Mir, Moscow, 1968), pp. 648–733 [in Russian].
M. Solomyak and E. Verbitsky, “On a spectral problem related to self-similarmeasures,” Bull. London Math. Soc. 27(3), 242–248 (1995).
A. A. Vladimirov and I. A. Sheipak, “Self-similar functions in L 2[0, 1] and the Sturm-Liouville problem with a singular indefinite weight,” Mat. Sb. 197(11), 13–30 (2006) [Russian Acad. Sci. Sb. Math. 197 (11), 1569–1586 (2006)].
A. A. Vladimirov and I. A. Sheipak, “The indefinite Sturm-Liouville problem for some classes of self-similar singular weights,” in Trudy Mat. Inst. Steklov Vol. 109: Functional Spaces, Approximation Theory, Nonlinear Analysis, Collection of papers (Nauka, Moscow, 2006), pp. 88–98 [in Russian] [Proc. Steklov Inst. Math., no. 4 (255), 82–91 (2006)].
A. A. Vladimirov and I. A. Sheipak, Asymptotics of Eigenvalues of Sturm-Liouville Problem with Discrete Self-Similar Weight, arXiv: math. SP/0709.0424 [in Russian].
M. Barnsley, Fractals Everywere (Academic Press, Boston, MA, 1988).
G. de Rham, “Une peu de mathématiques à propos d’une courbe plane,” Elem.Math. 2(4–5), 73–76, 89–97 (1947).
G. de Rham, “Sur une courbe plane,” J. Math. Pures Appl. (9) 35, 25–42 (1956).
G. de Rham, “Sur les courbes limites de polygones obtenus par trisection,” Enseign. Math. (2) 5, 29–43 (1959).
V. Protasov, “Refinement equations with nonnegative coefficients,” J. Fourier Anal. Appl. 6(1), 55–78 (2000).
V. Yu. Protasov, “Fractal curves and wavelets,” Izv. Ross. Akad. Nauk, Ser. Mat. 70(5), 123–162 (2006) [Russian Acad. Sci. Izv. Math. 70 (5), 975–1013 (2006)].
I. A. Sheipak, “On the construction and some properties of self-similar functions in the spaces L p[0, 1],” Mat. Zametki 81(6), 924–938 (2007) [Math. Notes 81 (5–6), 827–839 (2007)].
A. I. Nazarov, “Logarithmic asymptotics of small deviations for some Gaussian processes in the L 2-normwith respect to a self-similar measure,” in Zap. Nauchn. Semin. POMI, Vol. 311: Probability and Statistics. 7 (POMI, St. Petersnurg, 2004), pp. 190–213 [in Russian] [J. Math. Sci., New York 133 (3), 1314–1327 (2006)].
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Original Russian Text © I. A. Sheipak, 2010, published in Matematicheskie Zametki, 2010, Vol. 88, No. 2, pp. 303–316.
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Sheipak, I.A. Singular points of a self-similar function of spectral order zero: Self-similar stieltjes string. Math Notes 88, 275–286 (2010). https://doi.org/10.1134/S0001434610070254
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DOI: https://doi.org/10.1134/S0001434610070254