Fractal functions and Schauder bases. (English) Zbl 0838.28009
Summary: The box (entropy) dimension of graphs of real valued functions over \(d\)-dimensional cubes is investigated. Classes of functions with given lower and upper box dimension are described in terms of simple criteria for the coefficients of Schauder and Haar bases (wavelets) expansions.
Keywords:
Haar basis; fractal functions; fractal dimension; box dimension; entropy dimension; Schauder basis; waveletsReferences:
| [1] | Falconner, K., Fractal Geometry, Mathematical Foundations and Applications (1990), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0689.28003 |
| [2] | Ciesielski, Z., On the isomorphism of the space \(H_α\) and \(m\), Bull. Acad. Pol. Serie des Sc. Math., Ast. et Ph., 8, 4, 217-222 (1960) · Zbl 0093.12301 |
| [3] | Deliu, A.; Jawerth, B., Geometric dimension versus smoothness, Constructive Approximation, 8, 211-222 (1992) · Zbl 0782.46031 |
| [4] | Takev, M. D., On the Hausdorff dimension of completed graphs, Comptes rendus de l’Academie bulgare des Sciences, 45, 7, 13-15 (1992) · Zbl 0769.41035 |
| [5] | Ciesielski, Z., Haar orthogonal functions in analysis and probability, (Alfred Haar Memmorial Conference. Alfred Haar Memmorial Conference, Budapest (Hungary). Alfred Haar Memmorial Conference. Alfred Haar Memmorial Conference, Budapest (Hungary), Colloquia Mathematica Janos Bolyai, 49 (1985)), 25-56 · Zbl 0617.42015 |
| [6] | Ryll, J., Schauder bases for the space of continuous functions on an \(n\)-dimensional cube, Commentationes Mathematicae, 27, 1, 201-213 (1973) · Zbl 0265.46026 |
| [7] | Semandeni, Z., Schauder Bases in Banach Spaces of Continuous Functions LNM 918 (1982), Springer-Verlag: Springer-Verlag Berlin · Zbl 0478.46014 |
| [8] | Shvedov, A. S., Construction of functions given their values at dyadic rational points of an \(m\)-dimensional cube, Matematiceskiie Zametki, 44, 2, 250-261 (1988) · Zbl 0706.42020 |
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