This letter introduces a new representation of discrete signals based on the mathematical notions of functionals and continuous dual spaces. A new and more general sampling theorem is also suggested. Next, the problems of interpolating and resampling discrete signals are addressed; and a general solution using functional interpolation—which is applicable to many different settings—is proposed. Families of resampling filters dubbed de Boor-Ron filters that use de Boor-Ron interpolation are introduced, and their numerical realization is discussed. Some applications of this research are suggested.