Generalized Self-Similarity

CABRELLI, CARLOS A.; MOLTER, URSULA M.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Elsevier

Lugar: Amsterdam; Año: 1999 vol. 230 p. 251 - 260

0022-247X

We prove the existence of \Lp\ functions satisfyinga kind of self-similarity condition. This is achievedsolving a functional equation by means of theconstruction of a contractive operator on an appropriate functional space.The solution, a fixed point of the operator, can be obtained byan iterative process, making thismodel very suitable to use in applications suchas fractal image and signal compression.\\On the other hand, this ``generalized self-similarity equation''  includes matrix refinement equations of the type$f(x) = \sum c_k f(Ax - k)$ which are central in the construction of waveletsand multiwavelets. The results of this paper will thereforeyield conditions for the existence of \Lp-refinable functionsin a very general setting.\\*[6mm]{\em Keywords:}  Self-Similarity, Functional Equation, DilationEquation, Refinement Equation, Wavelets,Fixed Points, Fractals, Inverse Problem for Fractals.