| This paper discusses the surface fractal interpolation problem (SFIP), it putsforward a SFIP on a triangular domain (6-SFIP). It gives the interpolation space andthe holographic IFS for-SFIP, and proves that the holographic IFS is contractiveand there is an unique compact attractor. It gives the function space of continuousinterpolation and the function space of fractal interpolation, gives the holographicinterpolation pre-operator and the holographic interpolation operator. It obtains thenecessary and sufficient condition on continuous blending for the holographicinterpolation, proves that the holographic interpolation operator is contractive on thefunction space of fractal interpolation, and proves that there is an unique fixedcontinuous interpolation function (i.e. the holographic fractal interpolation function).It proves that the graph of the holographic fractal interpolation function just is theunique compact attractor of the holographic IFS. It gives four reduced forms for thenecessary and sufficient condition on continuous blending for the holographicinterpolation, and draws the fractal interpolation surface for every forms. At last, itpoints out, the results in this paper can be directly generalized to the case that thetriangular domain with arbitrary subdivision, and also can be directly generalized tothe case that the fractal interpolation method is the intergraphic interpolation. Themethod of this paper provides practical approach and a wealth of examples inconstruction for self-similar continuous surface on a triangular domain, and providessystematic tools for the theoretical study in the fractal interpolation problem on atriangular domain. |
