Fractal Interpolation Surface On A Rectangle

This paper discusses the surface fractal interpolation problem (SFIP), it putsforward holographic and intergraphic fractal interpolation iterated function systems(IFS) with vertical contractive ratio varying with point on a rectangular domain. Itproves that holographic IFS and inter-graphic IFS are both contractive, and each hasan unique compact attractor. Interpolation method for the holographic, it gives thefunction space of continuous interpolation and function space of fractal interpolation,gives the holographic interpolation pre-operator and the holographic interpolationoperator. It obtains the necessary and sufficient condition on continuous blending forthe holographic interpolation, proves that holographic interpolation operator iscontractive on the function space of fractal interpolation, and proves there is anunique fixed continuous interpolation function (i.e.: holographic fractal interpolationfunction). It proves that the graph of the holographic fractal interpolation function justis the unique compact attractor of the holographic IFS. Interpolation method for theinter-graphic, it gives the function space of continuous interpolation and functionspace of fractal interpolation, gives the inter-graphic interpolation pre-operator andthe inter-graphic interpolation operator. It obtains the necessary and sufficientcondition on continuous blending for the inter-graphic interpolation, proves thatinter-graphic interpolation operator is contractive on the function space of fractalinterpolation, and proves there is an unique fixed continuous interpolation function(i.e.: inter-graphic fractal interpolation function). It proves that the graph of the inter-graphic fractal interpolation function just is the unique compact attractor of the inter-graphic IFS. The method of this paper provides general and practical approach forconstructiing self-similar continuous surface on a rectangular domain and providessystematic tools for the theoretical study in the fractal interpolation problem on arectangular domain.