Wavelet&OrthogonalSystems.mws Chapter 1. Orthogonal Systems Figure 1.1 The scaling and mother wavelet for the Haar system Figure 1.2 The scaling function for the Shannon system Figure 1.3 The wavelet for the Shannon system Chapter 2. Tempered Distributions Figure 2.1 A test function in the space S (The Hermite function h4(x)) Figure 2.2 Some approximations to the delta function in S' Chapter 3. Introduction of Wavelets Figure 3.1 Typical functions in the subspaces V0 of the multiresolution analysis for the Haar scaling function Figure 3.2 Typical functions in the subspaces V1 of the multiresolution analysis for the Haar scaling function Figure 3.3 Typical functions in the subspaces V2 of the multiresolution analysis for the Haar scaling function Figure 3.4 The scaling function of the Franklin wavelet araising from the hat function Figure 3.5 The Daubechies scaling function and mother wavelet of example 3 Figure 3.6 The Daubechies mother wavelet of example 3 Figure 3.7 The scaling function and mother wavelet for a Meyer wavelet in frequency domain Figure 3.8 The Scaling function of Figure 3.7 in the time domain Figure 3.9 The mother wavelet of Figure 3.7 in the time domain Figure 3.10 The Reproducing kernel q(x,t) for V0 in the case of Haar wavelets Figure 3.11 The Daubechies scaling function and mother wavelet (N=4) Figure 3.13 The system functions of some continuous filters Low pass High pass Band pass Figure 3.14 The system function of discrete lowpass (halfband) filter Figure 3.15 The decomposition algorithm Figure 3.16 The reconstruction algorithm Chapter 4. Convergence and Summability of Fourier Series Figure 4.1 Dirichlet kernel of Fourier series (n=6) Figure 4.2 Fejer kernel of Fourier series (n=6) Figure 4.3 The saw tooth function Figure 4.4 Gibbs phenomenon for Fourier series; approximation to the saw tooth function using Dirichlet kernel. The approximation to the saw tooth function using Fejer kernel Chapter 5. Wavelets and Tempered Distributions Figure 5.1 A mother wavelet with point support. The vertical bars represent delta functions. Figure 5.2 Continuous function and its impulse train Chapter 6. Orthogonal Polynomials Figure 6.1 Some Legendre Polynomials (n=2, 3, and 6) Figure 6.2 Some Laguerre Polynomials (alpha=1/2, n= 5, 7, and 10) Figure 6.3 Some Hermite Polynomials (modified by constant multiples, n= 4, 5, and 7). Chapter 7. Other Orthogonal Systems Figure 7.1 The Haar mother wavelet Figure 7.2 One of the Rademacher functions Figure 7.3 Two orthogonal Walsh functions Figure 7.4 A bell used for a local cosine basis Figure 7.5 Three elements in the local cosine basis with bell of Figure 7.4 Figure 7.6 Two additional elements of the local cosine basis showing the bell Figure 7.7 A biorthogonal pair of scaling functions with the same MRA The Matlab program Graphs Figure 7.8 An example of biorthogonal pairs of scaling functions and Chapter 8. Pointwise Convergence of Wavelet Expansions Figure 8.1 The delta sequences from Fourier series -- the Drichelet kernel, m=6. Figure 8.2 The delta sequences from Fourier series -- the Fejer kernel, m=6. Figure 8.3 The quasi positive delta sequence qm(x,y) for Daubechies wavelet 2phi(x), m=0, Figure 8.4 the summability function rhor(x) for Daubechies wavelet 2phi(x), m=0, Figure 8.5 The positive delta sequence kr,m(x,y) for Daubechies wavelet 2phi(x), m=0, Chapter 9. A shannon Sampling Theorem in Wavelet Subspaces Figure 9.1 The sampling function for the Daubechies wavelet 2phi(t) with gamma=-1/sqrt(3). Figure 9.2 The function h of Proposition 9.3 Figure 9.3 The partial sum of the Shannon series expansion Chapter 10. Extensions of Wavelet Sampling Theorems Figure 10.1 An example of the scaling function of a Meyer wavelet at scale m=0 and sum of its 5 shifts in next scale m=1. Figure 10.2 The summability function associated with Coiflet of degree 2 with r = 0.22 Figure 10.3 The dual of the positive summability function in Figure 10.2 Figure 10.4 A non negative function f(x)=exp(-x^2/2)Chi[-1/2,1/2](x). Figure 10.5 The positive hybrid sampling series of function in Figure 10.4 (m=4) using Coiflet of degree 2 Figure 10.6 The hybrid series in Figure 10.4 Figure 10.7 An example of cardinal scaling function of Theorem 10.5 -- type two raised cosine wavelet Figure 10.8 The mother wavelet for the scaling function in Figure 10.7 Figure 10.9 The scaling function and wavelet for n=1. Figure 10.10 The two scaling functions and first wavelets for n=2. Chapter 11. Translation and Dilation Invariance in Orthogonal Systems Preliminary Figure 11.1 The approximation of the shifted scaling function by Haar series at scale m=1. Chapter 12. Analytic Representations Via Orthogonal Series Figure 12.1 The kernel Kr(t) given in Lemma 12.1 with m=2 Figure 12.2 The Hermite function h2(x) and the Hermite function of the second kind h~2(x+i*0). Figure 12.3 A Legendre Polynomials (n=3) Figure 12.4 The real and imaginary parts of the analytic representation at y=2 for the Legendre polynomial in Figure 12.3 Chapter 13. Orthogonal Series in Statistics Figure 13.1 Histogram for data in Example 13.1 Figure 13.2 Smooth wavelet estimator for the same data as in Figure 13.1 Figure 13.3 Daubechies wavelet scaling function (N=4). Figuer 13.4 The summability function associated with Daubechies wavelet (N=4). Figuer 13.5 The positive kernel associated with Daubechies wavelet (N=4). Figure 13.6 Density estimate for the Old Faithful geyser data using reproducing kernel associated with Daubechies scaling function (N=4) Figure 13.7 Density estimate for the Old Faithful geyser data using positive kernel in Figure 13.4.