We consider a basis of square integrable functions on a rectangle, contained in , constructed with Legendre polynomials, suitable, for instance, for the analogical description of images on the plane or in other fields of application of the Legendre polynomials in higher dimensions. After extending the Legendre polynomials to any arbitrary interval of the form [a, b], from its original form on [−1, 1], we generalize the basis of Legendre polynomials to two dimensions. This is the first step to generalize the basis to n-dimensions. We present some mathematical constructions such as Gel’fand triplets appropriate in this context. “Smoothness” of functions on space of test functions and some other properties are revisited, as well as the continuity of generators of su(1, 1) in this context.
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