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\(n\)-dimensional chirps and 2-microlocal analysis. (Chirps \(n\)-dimensionnels et analyse 2-microlocale.) (French) Zbl 0883.35005

The aim of the author is to describe, through 2-microlocal spaces and wavelet analysis, equivalent methods for a precise study of chirps, that is, loosely speaking, of functions that rapidly oscillate when \(x\) approaches a given point \(x_0\in\mathbb{R}^n\). In Section 2, he defines a chirp at 0 of type \((\gamma,\beta)\), \(\gamma >-n\), \(\beta>0\), as a function \(f\), integrable around \(0\), with the property that there exists \(r_0>0\) such that for any integer \(N\geq 0\), \(f(x)=\sum_{|\alpha|\leq N}\partial_x^{\alpha}U_{\alpha ,N}(x)\) in \({\mathcal D}'\), for \(x\in\{|x|<r_0\}\), with \(|U_{\alpha,N}(x)|\leq C_N|x|^{\gamma+\beta N+|\alpha|}\). A chirp at \(x_0\) will then be a function \(f\) such that \(f(x+x_0)\) is a chirp at \(0\). Next, he defines, with \(\Omega=\{|x|>R_0\}\), a space of oscillating functions \(g\in L^\infty(\Omega)\) such that, for any integer \(N\geq 1\), \(g(x)=\sum_{|\alpha|=N}\partial^\alpha_xg_\alpha(x)\), where \(g_\alpha\in L^\infty(\Omega)\), and establishes the equivalence: a function \(f\), integrable around \(0\), is a chirp at \(0\) of type \((\gamma,\beta)\) iff \(|x|^{\gamma/\beta}f(x|x|^{-(1+\beta)/\beta})\), \(|x|>R_0\), belongs to the aforementioned space of oscillating functions. In Sections 3 and 4 he defines \(r\)-regular chirps of type \((\gamma,\beta)\) \((r>0)\), characterizes them in terms of Hölder spaces, defines the 2-microlocal spaces \(C_{x_0}^{s,s'}\), \(s,s'\in\mathbb{R}\), \(x_0\in\mathbb{R}^n\), and proceeds in stating the following
Theorem. If \(r,\beta>0\) and \(\gamma> -n\), then \(f\in C_0^{s,s'}\) for any \(s,s'\) such that \(s+s'\leq r\) and \((\beta+1)s+\beta s'\leq\gamma\), iff \(f(x)=u(x)+v(x)\), where \(u\) is an \(r\)-regular chirp of type \((\gamma,\beta)\) at \(0\), and \(v\in C^\infty\) in a neighborhood of \(0\).
In Section 5, he recalls a theorem by S. Jaffard, which relates the 2-microlocal spaces \(C^{s,s'}_{x_0}\) and (local) approximation by polynomials in Hölder norm. In Section 6 he establishes a method for recognizing an \(r\)-regular \((\gamma,\beta)\) chirp through wavelet analysis and finally, in Section 7, he studies the action on chirps of pseudodifferential operators \(T_\zeta\), whose symbols \(\tau_\zeta(\xi)\), independent of \(x\), satisfy the homogeneity condition \(\tau_\zeta(\lambda\xi)=\lambda^\zeta\tau_\zeta(\xi)\), for any \(\lambda>0\) and any \(\xi\in{\mathbb{R}}^n\setminus\{0\}\). One has the following
Theorem. If \(u\) is an \(r\)-regular chirp of type \((\gamma,\beta)\), \(\zeta=\sigma+i\omega\in{\mathbb{C}}\), with \(\sigma<\min\{r,(\gamma+n)/(\beta+1)\}\), then \(T_\zeta u=u_\zeta+v_\zeta\), where \(u_\zeta\) is an \(r-\sigma\)-regular chirp of type \((\gamma-\sigma(\beta+1),\beta)\) and where \(v_\zeta\) is a smooth function.

MSC:

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
42C15 General harmonic expansions, frames
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