In this paper, we investigate uncertainty principles and stable signal reconstruction in the framework of the generalized Dunkl transform. Extending classical results of Donoho and Stark, we establish quantitative lower bounds linking the concentration of a function and its generalized Dunkl transform on measurable sets. We introduce the concept of strongly annihilating pairs adapted to this transform and prove stability results for the reconstruction of bandlimited functions from incomplete and noisy data. Furthermore, we develop Tikhonov regularization methods and iterative algorithms to address ill-posed recovery problems, providing explicit error bounds and convergence analysis. Our work broadens the scope of harmonic analysis uncertainty principles to singular differential-difference operators and contributes novel tools for signal processing applications within this setting.
Krylov methods have been successfully used to solve large matrix equations with sparse coefficients. In this paper, we are interested on solving the low-rank Lyapunov equation whether in the continuous or discrete case. We propose to modify the block Arnoldi process so that the projection Krylov subspace contains some additional blocks that involves the inverse of the square coefficient of this equation. Our goal with this approach is to reduce the number of iterations and convergence time required by the classical block Arnoldi method. To show the effectiveness of the proposed method, some numerical experiments are carried out.
In this paper, we enhance the oncolytic virotherapy model examined by [23] by incorporating multi-fractional derivatives of the Caputo type, assigning distinct derivative orders to each model variable. We establish the Hadamard well-posedness of the new model and confirm that the solutions remain within the positive half-plane, aligning with the criteria of the modeled problem. Furthermore, we conduct a stability analysis for the equilibrium solutions of these multi-fractional dynamics, demonstrating their dependence on the derivative orders specified in the dynamics. The theoretical insights are supported by numerical simulations.
We obtain new results on the relationship between uniqueness sets and stable sampling sets in function quasinormed spaces. The concept of perturbation of a uniformly discrete set plays an essential role. We also state and prove a result on stability for sampling sets and interpolation sets in a wide class of function quasinormed spaces.
