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Papers by ubertino battisti

Journal of Pseudo-Differential Operators and Applications

We construct a family of frames describing Sobolev norm and Sobolev seminorm of the space H s (R ... more We construct a family of frames describing Sobolev norm and Sobolev seminorm of the space H s (R n). Our work is inspired by the Discrete Orthonormal Stockwell Transform introduced by R.G. Stockwell, which provides a time-frequency localized version of Fourier basis of L 2 ([0, 1]). This approach is a hybrid between Gabor and Wavelet frames. We construct explicit and computable examples of these frames, discussing their properties.

Journal de Mathématiques Pures et Appliquées

Journal of Noncommutative Geometry

We prove an extension to R^n, endowed with a suitable metric, of the relation between the Einstei... more We prove an extension to R^n, endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. By means of complex powers, we first define the regularised Wodzicki Residue for a class of operators globally defined on R^n. The result is then obtained by using the properties of heat

Given two compact manifolds with boundary X, Y, and a boundary preserving symplectomorphism χ : T... more Given two compact manifolds with boundary X, Y, and a boundary preserving symplectomorphism χ : T * Y \ 0 → T * X \ 0, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with χ. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how-in the spirit of a classical construction by A. Weinstein-a Fredholm operator of this type can be associated with χ and a section of the Maslov bundle. If dim Y > 2 or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.

Journal of Functional Analysis, 2015

Operator Theory: Advances and Applications, 2015

We study a class of Fourier integral operators on compact manifolds with boundary X and Y , assoc... more We study a class of Fourier integral operators on compact manifolds with boundary X and Y , associated with a natural class of symplectomorphisms χ : T * Y \ 0 → T * X \ 0, namely, those which preserve the boundary. A calculus of Boutet de Monvel's type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward way.

2015 8th International Workshop on Advanced Ground Penetrating Radar (IWAGPR), 2015

For most archaeological prospections with GPR, target detection is the most important aspect of t... more For most archaeological prospections with GPR, target detection is the most important aspect of the surveys. The main efforts of GPR data processing are committed to increase the signal to noise ratio in radargrams. The usual processing aims to: filter the radar section from clutters and attenuate the in-band noises; enhance coherent reflections that are likely due to the target response; transform the acquired time section into a depth section; build a 3D image of the reflection targets in the subsoil by correlating reflections from adjacent radargrams. In this paper we show some application of the Stockwell transform, both to synthetic and field data, to enhance the signal to noise ratio in radargrams.

Operator Theory, Pseudo-Differential Equations, and Mathematical Physics, 2012

Annali di Matematica Pura ed Applicata (1923 -), 2015

We study the asymptotic behavior of the counting function of tensor products of operators, in the... more We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds or pseudodifferential operators of Shubin type on R n , respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.

Applied and Computational Harmonic Analysis, 2015

Since its appearing in 1996, the Stockwell transform (S-transform) has been applied to medical im... more Since its appearing in 1996, the Stockwell transform (S-transform) has been applied to medical imaging, geophysics and signal processing in general. In this paper, we prove that the system of functions (so-called DOST basis) is indeed an orthonormal basis of L 2 pr0, 1sq, which is time-frequency localized, in the sense of Donoho-Stark Theorem (1989). Our approach provides a unified setting in which to study the Stockwell transform (associated to different admissible windows) and its orthogonal decomposition. Finally, we introduce a fast-O pN log N q-algorithm to compute the Stockwell coefficients for an admissible window. Our algorithm extends the one proposed by Y. Wang and J. Orchard (2009).

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Journal of Pseudo-Differential Operators and Applications, 2011

We prove an extension to R n , endowed with a suitable metric, of the relation between the Einste... more We prove an extension to R n , endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. By means of complex powers, we first define the regularised Wodzicki Residue for a class of operators globally defined on R n. The result is then obtained by using the properties of heat kernels and generalised Laplacians.

Annals of Global Analysis and Geometry, 2011

We define the Wodzicki Residue TR(A) for A belonging to a space of operators with double order, d... more We define the Wodzicki Residue TR(A) for A belonging to a space of operators with double order, denoted L m 1 ,m 2 cl. Such operators are globally defined initially on R n and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function ζ(A, z). Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator A ∈ L m 1 ,m 2 cl in the case m 1 = m 2. Au(x) = 1 (2π) n e ix•ξ a(x, ξ)û(ξ)dξ, u ∈ S(R n), starting from symbols a(x, ξ) ∈ C ∞ (R n × R n) with the property that, for arbitrary multiindices α, β, there exist constants C αβ ≥ 0 such that the estimates (0.1) |D α ξ D β x a(x, ξ)| ≤ C αβ ξ m1−|α| x m2−|β| hold for fixed m 1 , m 2 ∈ R and all (x, ξ) ∈ R n × R n , where u = 1 + |u| 2 , u ∈ R n. Symbols of this type belong to the class denoted by SG m1,m2 (R n), and the corresponding operators constitute the class L m1,m2 (R n) = Op (SG m1,m2 (R n)). In the sequel we will often simply write SG m1,m2 and L m1,m2 , respectively, fixing the dimension of the (non-compact) base manifold to n. These classes of operators were first introduced on R n by H.O. Cordes [8] and C. Parenti [28], see also R. Melrose [24]. They form a graded algebra, i.e., L r1,r2 • L m1,m2 ⊆ L r1+m1,r2+m2 , whose residual elements are operators with symbols in SG −∞ (R n) = (m1,m2)∈R 2 SG m1,m2 (R n) = S(R 2n), that is, those having kernel in S(R 2n), continuously mapping S ′ (R n) to S(R n). An operator A = Op (a) ∈ L m1,m2

Mathematische Zeitschrift, Feb 16, 2012

We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined ... more We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined on the product of two closed manifolds. We study the asymptotic expansion of the counting function of positive selfadjoint operators in this class. Using a general Theorem of J. Aramaki, we can determine the first term of the asymptotic expansion of the counting function and, in a special case, we are able to find the second term. We give also some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet divisor function.