What Are Wavelets ?

A wavelet is a square integrable function whose translates and dilates form an orthonormal basis for Hilbert space L_2(R^N). To be more precise, let us consider the wavelet functions in Euclidean space R.

A wavelet function f has such a property, letting

f_{m,n}(x)=2^(m/2)f(2^m x-n)

Check here to see how good we can compress images using the wavelets we constructed.

To know more about wavelets, there are several webpages available. Check them out: wavelet digest and wavelet idr

The following is a list of my papers on wavelets [18] M. J. Lai, Construction of multivariate compactly supported prewavelets in L_2 spaces and pre-Riesz basis in Sobolev spaces, accepted for publication in Journal of Approximation Theory, 2006. [17] O. Cho and M. J. Lai, A class of compactly supported orthonormal B-Spline wavelets, to appear in Wavelets and Splines, edited by G. Chen and M. J. Lai, Nashboro Press, 2006. [16] M. J. Lai and J. Stoeckler, Construction of multivariate compactly supported tight wavelet frames, accepted for publication in Applied and Comput. Harmonic Analysis, 2006. [15] M. J. Lai, Construction of multivariate compactly supported orthonormal wavelets , accepted for publication in Advances in Computational Math.. 2004. [14] W. He and M. J. Lai, Construction of trivariate compactly supported biorthogonal box wavelets, J. Approx. Theory, 120(2003), pp. 1--19. [13] M. J. Lai, Methods for Constructing Nonseparable Compactly Supported Orthonormal Wavelets, Wavelet Analysis: Twenty Year's Development, edited by D. X. Zhou, World Scientific, 2002, pp. 231--251. [12] M. J. Lai, and D. W. Roach, The nonexistence of bivariate symmetric wavelets with short support and two vanishing moments, in: {\sl Trends in Approximation Theory}, K. Kopotun, T. Lyche, and M. Neamtu (eds.), Vanderbilt University Press, Nashville, 2001, pp. 213--223. [11] Construction of nonseparable wavelets with short support , (with David Roach), submitted for publication, 2000. [10] Bivariate box spline wavelets in Sobolev spaces, (with Wenjie He) in {\sl Wavelet Applications in Signal and Image Processing VI}, proceedings of SPIE, vol. 3458(1998), pp. 56--66. [9] A new sufficient condition for the orthonormality of refinable functions , (with Wenjie He), in Approximation Theory IX: Computational Aspects Charles K. Chui and Larry L. Schumaker (eds.) Vanderbilt University Press (Nashville), 1998, pp. 121--128. [8] Construction of bivariate nonseparable compactly supported orthonormal multiwavelets with arbitrarily high regularity, (with Wenjie He), 1998, [7] Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets, (with Wenjie He), 1997. Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 3169 (1997), pp. 303--314. Also appeared in IEEE Transactions on Image Processing, 9(2000), 949--953. [6] Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities, (with Wenjie He), Applied Comput. Harmonic Anal., 6(1999), pp. 53--74. [5] On digital filters associated with bivariate box spline wavelets , (with Wenjie He), J. Electronic Imaging, 6(1997), pp. 453--466. (Numerical values associated with these filters in MATLAB format are available here. [4] Bivariate box splines for image processing, Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 2825 (1996), pp. 474--487. [3] Wavelets and Ideal Filters, IEEE Trans. Signal Processing, 43 (1995), pp. 2203--2205. [2] On computation of Battle-Lemarie's wavelets, Mathematics of Computation, 63(1994), pp. 689--699. [1] On Str\"omberg's spline wavelets, Applied and Computational Harmonic Analysis, 1(1994), pp. 188-193.