Fourier--Laplace Injectivity for Oblique and Polygonal Cylinders in Passive Inverse Problems

Abstract

We study Fourier--Laplace transforms on null phases arising in passive inverse problems. Densities with codimension-one dependence in oblique cylinders are first considered. Integration in the distinguished direction reduces the data to an entire transform on a real-analytic maximally totally real submanifold. Holomorphic uniqueness then gives injectivity. A fixed transverse polytope swept along a polygonal curve, forming a polygonal cylinder, is next considered. The density is assumed to be polynomial in the material transverse variables on each swept piece. A weighted Brion--Barvinok expansion separates the vertex terms and the successive pieces. The same representation gives a finite-dimensional Lipschitz estimate when the polynomial degree is fixed. We apply these transform results to passive wave and heat equations in arbitrary dimension. The initial state is zero outside the observation domain. Equality of the Laplace-domain boundary data determines the quantity of the initial data divided by the speed squared in the acoustic model and the quantity of the initial data divided by the coefficient of thermal conductivity in the parabolic model. The question of what remains true when only weighted generalized boundary data on a finite interval \([0,T]\) are available is then studied. These data determine the harmonic moments of the quotient up to the weighted terminal residual. This identity gives a Hölder estimate in an oblique cylinder and a Lipschitz estimate in a polygonal cylinder. Finally, a continuous generating curve is approximated by a polygonal curve and a continuous material density by piecewise transverse polynomials. The resulting reconstruction estimate separates measurement noise, density approximation error, geometric error, and the condition number of the finite sampling matrix.