Dimension And Variation Property Of Recurrent Fractal Interpolation Function

The concept of fractal interpolation functions was firstly proposed by the Ameirica mathematician Bamsley in 1986.It is a new way of interpolating.Its distinctive advantage was showed in many research fields,such as the compression of pictures and the fitting of unsmoothed curves and surfaces.Great successe has been made.The fractal interpolation functions that proposed by Bamsley are continuous functions that are defined in intervals.these functions have no explicit expressions.They are determined by attractors and generated by a set of mappings.Therefore,it needs distinctive ways to research the properties of these functions Generally,some traditional analysis methods can not be used directly and some new methods and theories must be opened up.So,it provides new research fields for theory of functions by interpolation functions.Base on fractal interpolation,the recurrent fractal interpolation is better and more flexible.Structure of recurrent affine fractal interpolation function is got in reference [3],and fractal dimension of graph of this kind of function is studied,the formula of box-counting dimension is provided.What this text involves is more general recurrent fractal interpolation function,studys the structure method of it's IFS,proves the dimensional theorem of graph of recurrent fractal interpolation function.Further, based on studying the variation properties of continuous function,the variation properties of recurrent FIF are studied next.Further,based on the properties,the rank of the general recurrent FIF function variation is estimated.Using this estimation,the dimension theorem can be proved without the condition that the connection matrix is irreducible is given.So Recurrent fractal interpolation function is used more flexibly and practically.So the image is more realistic simulation.