Investigation of smooth functions and analytic sets using fractal dimensions

Emma D'Aniello

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 3, page 637-646
  • ISSN: 0392-4041

Abstract

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We start from the following problem: given a function f : 0 , 1 → 0 , 1 what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of C n functions. We investigate the analogous problem for C n , a functions. These are in a certain way intermediate between C n and C n + 1 functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.

How to cite

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D'Aniello, Emma. "Investigation of smooth functions and analytic sets using fractal dimensions." Bollettino dell'Unione Matematica Italiana 7-B.3 (2004): 637-646. <http://eudml.org/doc/195946>.

@article{DAniello2004,
abstract = {We start from the following problem: given a function $f \colon [0, 1] \to [0, 1]$ what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of $C^\{n\}$ functions. We investigate the analogous problem for $C^\{n, a\}$ functions. These are in a certain way intermediate between $C^\{n\}$ and $C^\{n+1\}$ functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.},
author = {D'Aniello, Emma},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {637-646},
publisher = {Unione Matematica Italiana},
title = {Investigation of smooth functions and analytic sets using fractal dimensions},
url = {http://eudml.org/doc/195946},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - D'Aniello, Emma
TI - Investigation of smooth functions and analytic sets using fractal dimensions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/10//
PB - Unione Matematica Italiana
VL - 7-B
IS - 3
SP - 637
EP - 646
AB - We start from the following problem: given a function $f \colon [0, 1] \to [0, 1]$ what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of $C^{n}$ functions. We investigate the analogous problem for $C^{n, a}$ functions. These are in a certain way intermediate between $C^{n}$ and $C^{n+1}$ functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.
LA - eng
UR - http://eudml.org/doc/195946
ER -

References

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  11. KURATOWSKI, K., Topology, Volume I, Academic Press, INC., New York, 1966. MR217751
  12. LACZKOVICH, M.- PREISS, D., α -variation and transformation into C n functions, Indiana Univ. Math. J., 34, no. 2 (1985), 405-424. Zbl0557.26004MR783923
  13. LEBEDEV, V. V., Homeomorphisms of a segment and smoothness of a function, Mat. Zametki, 40, no. 3 (1986), 364-373. Zbl0637.26006MR869927
  14. MAZURKIEWICZ, S., Sur les fonctions non dérivables, Studia Math., 3 (1931), 92-94. JFM57.0305.04
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