Generalized Self-Similarity

CABRELLI, CARLOS; MOLTER, URSULA

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

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

0022-247X

We prove the existence of \Lp\ functions satisfying a kind of self-similarity condition. This is achieved 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 $f(x) = \sum c_k f(Ax - k)$ which are central in the construction of wavelets and multiwavelets. The results of this paper will therefore yield conditions for the existence of \Lp-refinable functions in a very general setting.