Table of contents
Introduction p. 1
I. Modulus of continuity and two-microlocalization p. 8
1.1 Modulus of continuity p. 8
1.2 Pointwise smoothness and two-microlocalization p. 14
1.3 Pseudodifferential operators and two-microlocalization p. 22
1.4 Characterizations of two-microlocal spaces p. 25
II Singularities of functions in Sobolev spaces p. 30
II. 1 Hausdorff dimension of the singularities of functions in Lp s p. 32
11.2 Packing dimension of strong ^-singularities p. 37
11.3 Singularities of BV functions p. 41
11.4 Sobolev regularity of trace functions p. 44
III Wavelets and lacunary trigonometric series p. 48
111.1 Pointwise regularity of Hardy functions p. 48
111.2 Convergence and derivability of wavelet series p. 52
IV Properties of chirp expansions p. 56
IV. 1 Indefinitely oscillating functions and chirps p. 56
IV.2 Characterization of chirps: statement of results p. 60
IV.3 Proof of Theorem 5.2: (I) (II) p. 65
IV.4 Proof of Theorem 5.2: (II) = (I) p. 66
IV.5 Proof of Theorem 5.2: (I) & (III) p. 70
V Trigonometric chirps p. 72
V.l Definition and statement of the main theorem p. 72
V.2 Some asymptotic expansions p. 75
V.3 Proof of Theorem 5.1 p. 81
VI Logarithmic chirps p. 89
VII The Riemann series p. 96
VII. 1 Trigonometric chirps at rationals p. 96
VII.2 Logarithmic chirps at quadratic irrationals p. 102
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