Abstract. In this paper we establish improved Hardy and Rellich type inequalities on a Riemannian manifold M . Furthermore, we also obtain sharp constants for improved Hardy and Rellich type inequalities on the hyperbolic space H n .
We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold M , started in [17]. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold M . In particular, we obtain sharp constants for these inequalities on the hyperbolic space H n .
We determine all signals giving equality for the discrete Hirschman uncertainty principle. We single out the case where the entropies of the time signal and its Fourier transform are equal. These signals (up to scalar multiples) form an orthonormal basis giving an orthogonal transform that optimally packs a finite-duration discrete-time signal. The transform may be computed via a fast algorithm due to its relationship to the discrete Fourier transform.
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