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 .

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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.

We classify all functions on a locally compact, abelian group giving equality in an entropy inequality generalizing the Heisenberg Uncertainty Principle. In particular, for functions on a real line, we proof a conjecture of Hirschman published in 1957. r 2004 Elsevier Inc. All rights reserved. MSC: primary 43A25; secondary 94A17

We use a new uncertainty measure, H,,, that predicts the Compactness of digital signal representations to determine a good (non-orthogonal) set of basis vectors. The measure uses the entropy of the signal and its Fourier transform in a manner that is similar to the use of the signal and its Fourier transform in the Heisenberg uncertainty principle. The measure explains why the level of discretization of continuous basis signals can be very important to the compactness of representation. Our use of the measure indicates that a mixture of (non-orthogonal) sinusoidal and impulsive or "blocky" basis functions may be best for compactly representing signals.

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