B-splines of Blaschke product type

In this paper, we construct a class of new splines related to a Blaschke product. They emerge naturally when studying the filter functions of a class of linear time-invariant systems which are related to the boundary values of a Blaschke product for the purpose of sampling non-bandlimited signals using nonlinear Fourier atoms. The new splines generalize the well-known symmetric B-splines. We establish their properties such as integral representation property, a partition of unity property, a recurrence relation and difference property. We also investigate their random behaviour. Finally, our numerical experiments confirm our theories.

[1]  Allan Pinkus,et al.  The B-spline recurrence relations of Chakalov and of Popoviciu , 2003, J. Approx. Theory.

[2]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[3]  Yi Wang,et al.  A sampling theorem for non-bandlimited signals using generalized Sinc functions , 2008, Comput. Math. Appl..

[4]  Khaled H. Hamed,et al.  Time-frequency analysis , 2003 .

[5]  N. Huang,et al.  The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[6]  C. Chui.  Wavelet Analysis and Its Applications , 1992 .

[7]  Bernard C. Picinbono,et al.  On instantaneous amplitude and phase of signals , 1997, IEEE Trans. Signal Process..

[8]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[9]  QIUHUI CHEN,et al.  Functions with spline Spectra and their Applications , 2010, Int. J. Wavelets Multiresolution Inf. Process..

[10]  Michael Unser,et al.  On the asymptotic convergence of B-spline wavelets to Gabor functions , 1992, IEEE Trans. Inf. Theory.

[11]  Qiuhui Chen,et al.  Analytic unit quadrature signals with nonlinear phase , 2005 .

[12]  Qiuhui Chen,et al.  Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms , 2009 .

[13]  E. Bedrosian.  A Product Theorem for Hilbert Transforms , 1963 .

[14]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[15]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals , 1992, Proc. IEEE.

[16]  A. Nuttall,et al.  On the quadrature approximation to the Hilbert transform of modulated signals , 1966 .