A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology. For this purpose, a binary operation in the space of bounded functions on an interval is established. Two functions give rise to a new one, called in the paper fractal convolution of the originals, whose graph is discontinuous and has a fractal structure in general. The new function approximates one of the chosen pair and, under certain conditions, is continuous. The convolution is used for the definition of discontinuous bases of the space of square integrable functions, whose elements are as close to a classical orthonormal system as desired.
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