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In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation points and vertical scaling factors.2010 Mathematics subject classification: primary 28A80; secondary 41A30.

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similarand proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r m 1 = r n 3 .

In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs.

We introduce a notion of gap sequences for compact sets E ⊂ R d , which is a generalization of the gap sequences of compact sets on the real line. We show that if the gap sequences of two fractal sets are not equivalent, then these two sets cannot be Lipschitz equivalent, where the latter fact is usually very hard to verify. Finally, we show that for some typical fractal sets, the gap sequences characterize the upper box dimension.