Abstract
The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in \(\mathbb {R}^2\), provides a useful way to classify pointwise singularity. In this paper we characterize all functions whose 2-microlocal frontier at a given point \(x_0\) is a given line. Further, for a general concave downward curve, we obtain a large family of functions (or distributions) for which the 2-microlocal frontier is the given curve. This family contains—as special cases—the constructions given in Meyer (CRM monograph series, 1998), Guiheneuf et al. (ACHA 5(4):487–492, 1998) and Lévy Véhel et al. (Proc Symp Pure Math 72:319–334, 2004). Moreover, following Lévy Véhel et al. (2004), we extend our results to the prescription on a countable dense set.
Similar content being viewed by others
References
Arneodo, A., Bacry, E., Jaffard, S., Muzy, J.F.: Singularity spectrum of multifractal functions involving oscillating singularities. J. Fourier Anal. Appl. 4(2), 159–174 (1998)
Abbott, B.P., et al.: Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016)
Abbott, B.P., et al.: Gw170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017)
Abry, P., Roux, S.G., Jaffard, S.: Detecting oscillating singularities in multifractal analysis: application to hydrodynamic turbulence In: 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4328–4331 (2011)
Balança, P.: Fine regularity of Lévy processes and linear (multi)fractional stable motion. Electron. J. Prob. 19 (2014)
Balança, P., Herbin, E.: 2-Microlocal analysis of martingales and stochastic integrals. Stoch. Process. Appl. 122(6), 2346–2382 (2012)
Bony, J.M.: Second microlocalization and propagation of singularities for semi-linear hyperbolic equations. In: Mizohata, S. (ed.) Hyperbolic Equations and Related Topics, pp. 11–49. Academic Press (1986)
Daoudi, K., Véhel, J.Lévy, Meyer, Y.: Construction of continuous functions with prescribed local regularity. Constr. Approx. 14(3), 349–385 (1998)
Echelard, A.: Analyse 2-microlocale et application au débruitage, Ph.D. thesis, University of Nantes, vol. (197 f.), p. 1 (2007)
Flandrin, P.: Explorations in Time-Frequency Analysis, ch. Small Data are Beautiful, pp. 9–20. Cambridge University Press (2018)
Guiheneuf, B., Jaffard, S., Véhel, J.Lévy: Two results concerning chirps and 2-microlocal exponents prescription. Appl. Comput. Harmon. Anal. 5(4), 487–492 (1998)
Herbin, E., Lévy Véhel, J.: Stochastic 2-microlocal analysis. Stoch. Process. Appl. 119(7), 2277–2311 (2009)
Jaffard, S.: Pointwise smoothness, two microlocalization and wavelet coefficients. Publ. Mat. 35(1), 155–168 (1991)
Jaffard, S.: Functions with prescribed Hölder exponent. Appl. Comput. Harmon. Anal. 2(4), 400–401 (1995)
Jaffard, S.: Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3(1), 1–22 (1997)
Jaffard, S.: Construction of functions with prescribed Hölder and chirp exponents. Revista matemática iberoamericana 16(2), 331–350 (2000)
Jaffard, S., Meyer, Y.: Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions, American Mathematical Society: Memoirs of the American Mathematical Society, vol. 587. American Mathematical Society (1996)
Jaffard, S., Melot, C., Leonarduzzi, R., Wendt, H., Abry, P., Roux, S.G., Torres, M.E.: p-Exponent and p-leaders, part i: negative pointwise regularity. Physica A 448, 300–318 (2016)
Kopsinis, Y., Aboutanios, E., Waters, D.A., McLaughlin, S.: Time-frequency and advanced frequency estimation techniques for the investigation of bat echolocation calls. J. Acoust. Soc. Am. 127(2), 1124–1134 (2010)
Kolwankar, K.M., Véhel, J.Lévy: A time domain characterization of the fine local regularity of functions. J. Fourier Anal. Appl. 8(4), 319–334 (2002)
Lévy Véhel, J., Seuret, S.: 2-microlocal formalism. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proceedings of Symposia in Pure Mathematics, vol. 72, pp. 153–215. AMS (2004)
Meyer, Y.: Wavelets. CRM Monograph Series. American Mathematical Society, Vibrations and Scalings (1998)
Rosenblatt, M.: Un análisis de la regularidad de funciones usando wavelets, Ph.D. thesis, Universidad de Buenos Aires (2019). http://cms.dm.uba.ar/academico/carreras/doctorado/Tesis_MRosenblatt_08_2019.pdf
Seuret, S., Véhel, J.Lévy: The local Hölder function of a continuous function. Appl. Comput. Harmon. Anal. 13(3), 263–276 (2002)
Seuret, S., Véhel, J.Lévy: A time domain characterization of 2-microlocal spaces. J. Fourier Anal. Appl. 9(5), 473–495 (2003)
Acknowledgements
The authors acknowledge support from the Universidad de Buenos Aires UBACyT 20020170100430BA, the CONICET PIP11220150100355, the MinCyT PICT 2014-1480 and from the Universidad Nacional de General Sarmiento PIO CONICET-UNGS 144-20140100011-CO. We thank the anonymous reviewers which helped improve the presentation of this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Stephane Jaffard.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
In this Appendix we complete the technical details required to show that the functions obtained in [22] and [21] can be obtained by appropriately choosing the coefficients as indicated in Sect. 2.3.
1.1 Computation for the Choice of Coefficients of Meyer
Y. Meyer chooses a countable dense set \(\{s'_m\}\) in \(\mathbb {R}\) and defines
He proves that the function f whose wavelet coefficients are given by
with \({\Lambda }_m\) as in (9), and 0 otherwise, has \(s= A(s')\) as its 2-microlocal frontier at \(x_0=0\).
Using Theorem 2.1, we obtain the function f proposed by Y. Meyer, by selecting on one hand \(\mathscr {C}_{j,k}=1\) and on the other, \(\lambda _{j,k}\) such that
To see that for \(j\in {\Lambda }_m\) and \(k=\left[ 2^{jp_m}\right] \),
are the coefficients given in (49), we replace \(\lambda _{j,k}= \frac{1+\left| k\right| }{2^{jp_m}}\) and \(\mathscr {C}_{j,k}=1\). We have
To compute the infimum, recall that \(A(s') \) is downward concave and differentiable. Hence the tangent line at \(s'_m\) is on top of the graph of \(A(s')\), e.g., for all \(s'\),
Hence, using (48) we have that for all \(s'\) \(s+p_ms'\le \tau _m .\) Consequently, since \(S(\sigma )=s\) if and only if \(s=A(\sigma -s)\) we have
Further, the equality \(2^{-j\tau _m}= 2^{-j(S(\sigma )-p_m(S(\sigma )-\sigma ))}\) is true if \(\sigma =\sigma _m=s_m+s'_m\) with \(s_m=A(s'_m)\).
Therefore,
For all indexes j, k not yet considered, we define \(c_{j,k}=0\), and so they clearly satisfy (12) for \(\mathscr {C}_{j,k}=1\) and \(\lambda _{j,k}=1\).
With these choices, the sequences \(\mathscr {C}_{j,k}\) and \(\lambda _{j,k}\) satisfy (i) y (ii) of Theorem 2.1 since,
for j, k such that \(j\ge 0\) and \( | k|<2^j\).
1.2 Computation for the Choice of Coefficients of Lévy Véhel and Seuret
To proof that the function given in [21] is in fact a special case of our formula is rather delicate.
We will exhibit here only the computation for the case that g is a line and strictly increasing. The non-linear case can be found in section 4.3.3 of [23].
In [21], the function whose 2-microlocal frontier at \(x_0=0\) is \(g(s')=Ms' +d\), with \( 0<M \le 1\) and \(d>0\), is given by its wavelet coefficients \(c_{j,k}\). These are defined as 0 if \(k<0\) or \(j=0\) and for \(j>0\) and \(k \ge 0\) as
The Legendre transform of g is
In the \((\sigma ,s)\)-plane, the curve \(s=g(s')\) is
We choose the sequences \(\mathscr {C}_{j,k} \) and \(\lambda _{j,k}\) in (31) and (32) as:
Replacing \( \lambda _{j,k}\) and \(S(\sigma )\) in the equation
we obtain \(|c_{j,k}|=\mathscr {C}_{j,k}\;2^{-jd}. \) Therefore, if in (12) we consider equality and insert \(\mathscr {C}_{j,k}\) given above by (52), we obtain exactly the wavelet coefficients (12) for \(j>0\) and \(k\ge 0\). For \(k<0\) or \(j=0\) we define \(|c_{j,k}|=0\), which also satisfies (12).
It only remains to check that the sequences \( \mathscr {C}_{j,k}\) and \( \lambda _{j,k}\) satisfy (i) and (ii) of Theorem 2.1. In this case, the index set I is
for \(r_m= 1-M\) for all m, where \(1-M\) is the only element in the range of \(\frac{S'(\sigma )}{S'(\sigma )-1}\).
If \((j,k_j)\in I\) we have that \(k_j =[2^{j(1-M)}]\) and hence \(\mathscr {C}_{j,k_j}=1\) or \(\mathscr {C}_{j,k_j}= 2^{-j^2+jd}\) and \(1\le \lambda _{j,k_j}\le 2\), and hence the conditions are satisfied.
If \(k\ne [2^{j(1-M)}]\) we have \(\mathscr {C}_{j,k}=2^{-j^2+jd}\). Further \( \lambda _{j,k}\) is bounded by
and so \(\frac{\log _2(\lambda _{j,k})}{j}\) is bounded.
This proves that for \((j,k_j)\in I\) or \((j,k_j)\notin I\), (i) is satisfied, i.e. for any sequence \((k_j)_j\) such that \( |k_j|<2^j\),
Rights and permissions
About this article
Cite this article
Molter, U., Rosenblatt, M. Generalized 2-Microlocal Frontier Prescription. J Fourier Anal Appl 26, 88 (2020). https://doi.org/10.1007/s00041-020-09791-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00041-020-09791-1