Uncertainty principles and time-invariant filter for the Opdam–Cherednik wavelet transform with its applications

In this paper, we introduce the time-invariant filter for the Opdam–Cherednik wavelet transform and study a few versions of the uncertainty principle for this transform. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho–Stark's uncertainty principle, and Heisenberg-type uncertainty principles for the Opdam–Cherednik wavelet transform. Then, we show that the Opdam–Cherednik convolution operator is a time-invariant filter and give a physical interpretation of a time-invariant filter. As an application, we study a signal recovery problem using uncertainty principles and show that the reconstruction of a transmitted signal from a noisy received signal is possible using the uncertainty principle for the Opdam–Cherednik wavelet transform. Finally, we solve the Fredholm-type integral equation using the time-invariant filter for this transform.