Fourier operator

Abstract: We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on R, and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent alpha is an element of]0, 2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators P-t, T-t. The operator P-t is a Fourier operator and is considered here as a perturbation of the Markov operator P = P-0 of the random walk. The operator T-t is related to P-t by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T-0 = T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

Abstract: Summary To accurately image complex structures with strong lateral velocity variations and steep dips, we develop a globally optimized Fourier finite-difference method that uses a rational approximation of the square-root operator in the one-way wave equation. The method uses a split-step Fourier operator coupled with a one-term optimized finite-difference operator. The two coefficients in the rational approximation are obtained by an optimization scheme that maximizes the maximum dip angle of the method for a given model. Our optimized method uses the same coefficients throughout a model in contrast

Abstract: An explicit theory of special functions is developed for the homogeneous space SO(n)/SO(n m) generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on n X m matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of SO(n) and the polynomial representations of GL(m, C).

Abstract: We study the quantitative behavior of the solutions of the one-dimensional Boltzmann equation for hard potential models with Grad’s angular cutoff. Our results generalize those of [5] for hard sphere models. The main difference between hard sphere and hard potential models is in the exponent of the collision frequency \( u(\xi)\approx (1+|\xi|)^\gamma\). This gives rise to new wave phenomena, particularly the sub-exponential behavior of waves. Unlike the hard sphere models, the spectrum of the Fourier operator \(-i\xi^1\eta+L\) is non-analytic in η for hard potential models. Thus the complex analytic methods for inverting the Fourier transform are not applicable and we need to use the real analytic method in the estimates of the fluidlike waves. We devise a new weighted energy function to account for the sub-exponential behavior of waves.