MRA of processes synthesized by fractional integration

1203 of the standard FFT (for N < 16, the Dirichlet kernel interpolation is always more efficient). It is also worth noting that for these values, and small N , the CP is more attractive than the Yaroslavsky; otherwise, there is little computational difference between the two FFT-based methods. In addition, as N decreases, the computational saving introduced by using the Dirichlet kernel, instead of an FFT-based method, increases. However, it should be noted that the complexity of the FFT-based algorithms may be improved by " pruning " [7], [14] (the elimination of operations on zeros) but requires further careful programming. In light of this, the Dirichlet kernel interpolation method appears to be the most straightforward to implement and the least computationally complex, numerically stable method. The equivalence of the Schanze and Cavicchi interpolation has been demonstrated. The interpolation formulas are shown to be equivalent to Dirichlet kernels, which are directly numerically implementable in " cosine " form. Relationships with Dirichlet kernels and fractional delay filters are outlined. As a result, the roles of these methods in bandlim-ited interpolation has been clarified. In addition, the Dirichlet kernels are shown to be computationally simplest to generate, out of the numerically stable interpolation kernels examined, for most data lengths. REFERENCES [1] S. R. Dooley and A. K. Nandi, " Adaptive subsample time delay estimation using Lagrange interpolators, " Evaluation of several variable FIR fractional-sample delay filters, " in Proc. IEEE Int. Conf. Abstract—In this correspondence, we show that the multiresolution analysis of the reproducing kernel Hilbert space, associated with processes synthesized by fractional integration of white process, leads to an orthogonal, multiresolution decomposition of the process. We show that the decomposition converges almost surely and uniformly on finite intervals. We also show that the random coefficients in the decomposition can be evaluated almost surely.