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In this paper a new class of bivariate refinable functions is constructed by means of the directional convolution product between the tensor product of two univariate refinable functions belonging to a particular class, shortly recalled in this paper, with a third one in the same class. The new functions have many properties useful in applications, such as compact support, positivity, central symmetry, refinability and linear independence of the integer translates. Moreover, the so constructed functions turn out to be suitable for solving the cardinal interpolation problem.