It is known that locally compact groups approximable by finite ones are unimodular, but this condition is not sufficient, for example, the simple Lie groups are not approximable by finite ones as topological groups. In this paper the approximations of locally compact groups by more general finite algebraic systems are investigated. It is proved that the approximation of locally compact groups b...

We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).

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