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Wavelets and Mallat's multiresolution analysis

100%

Wojdyłło P.

Fundamenta Informaticae

1998

tom Vol. 34, nr 4

469-474

We present a simple proof of Mallat's theorem about wavelet and multiresolution analysis. We do not use Fourier transform and the proof is accesible even for younger undergraduate students.

Wavelets for time series analysis - a survey and new results

63%

Mielniczuk J. , Wojdyłło P.

Control and Cybernetics

2005

tom Vol. 34, no 4

1093-1125

In the paper we review stochastic properties of wavelet coefficients for time series indexed by continuous or discrete time. The main emphasis is on decorrelation property and its implications for data analysis. Some new properties are developed as the rates of correlation decay for the wavelet coefficients in the case of long-range dependent processes such as the fractional Gaussian noise and the fractional autoregressive integrated moving average processes. It is proved that for such processes the within-scale covariance of the wavelet coefficients at lag k is O(k^2(H-N)-2), where H is the Hurst exponent and N is the number of vanishing moments of the wavelet employed. Some applications of decorrelation property are briefly discussed.