Generalized Self-Similarity

We prove the existence of L p functions satisfying a kind of self-similarity condition. This is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space. The solution, a fixed point of the operator, can be obtained by an iterative process, making this model very suitable to use in applications such as fractal image and signal compression. On the other hand, this ‘‘generalized self-similarity equation’’ includes matrix refinement equations of the type f x.s ck f Axyk. which are central in the construction of wavelets and multiwavelets. The results of this paper will therefore yield conditions for the existence of L p-refinable functions in a very general setting.