Gelfand Triplets, Continuous and Discrete Bases and Legendre Polynomials

We consider a basis of square integrable functions on a rectangle, contained in $R^2$, 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 Gelfand triples appropriate on this context. ``Smoothness'' of functions on space of test functions and some other properties are revisited, as well as te continuity of generators of $su(1,1)$ on this context.