Wavelet analysis of non-stationary signals with applications

Transcription

1 University of Missouri, St. Louis UMSL Dissertations UMSL Graduate Works Wavelet analysis of non-stationary signals with applications Maria Dorothea Van der Walt University of Missouri-St. Louis, maryke.thom@gmail.com Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Van der Walt, Maria Dorothea, "Wavelet analysis of non-stationary signals with applications" (2015). Dissertations This Dissertation is brought to you for free and open access by the UMSL Graduate Works at UMSL. It has been accepted for inclusion in Dissertations by an authorized administrator of UMSL. For more information, please contact marvinh@umsl.edu.

2 Wavelet analysis of non-stationary signals with applications Maria Dorothea van der Walt M.Sc., Mathematics, Stellenbosch University, South Africa, 2012 B.Sc. Hons., Mathematics, Stellenbosch University, South Africa, 2010 B.Sc., Mathematical Sciences, Stellenbosch University, South Africa, 2009 A dissertation submitted to the Graduate School of the University of Missouri-St. Louis in partial fulllment of the requirements for the degree Doctor of Philosophy in Applied Mathematics May 2015 Advisory Committee: Charles Chui, Ph.D. (chair) Haiyan Cai, Ph.D. (co-chair) Qingtang Jiang, Ph.D. Wenjie He, Ph.D. Yuefeng Wu, Ph.D.

3 Van der Walt, Maria, 2015, UMSL, p.i Abstract The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang et al in 1998, is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF's), after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu and H.-T. Wu in 2011, is rst applied to estimate the IF's of all signal components of the given signal, based on one single frequency reassignment rule, under the assumption that the signal components satisfy certain strict properties of the so-called adaptive harmonic model, before the signal components of the model are recovered, based on the estimated IF's. The objective of this dissertation is to develop a hybrid EMD- SST computational scheme by applying a modied SST to each IMF produced by a modied EMD, as an alternative approach to the original EMD-HSA method. While our modied SST assures nonnegative instantaneous frequencies of the IMF's, application of the EMD scheme eliminates the dependence on a single frequency reassignment rule as well as the guessing work of the number of signal components in the original SST approach. Our modication of the SST consists of applying analytic vanishing moment wavelets (introduced in a recent paper by C.K. Chui, Y.-T. Lin and H.-T. Wu) with stacked knots to process signals on bounded or half-innite time intervals, and spline curve tting with optimal smoothing parameter selection through generalized cross-validation. In addition, we modify EMD by formulating a local spline interpolation scheme for bounded intervals, for real-time realization of the EMD sifting process. This scheme improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-innite time intervals.

4 Van der Walt, Maria, 2015, UMSL, p.ii Acknowledgments I am deeply indebted to my advisor, Professor Charles Chui, for his guidance and vision throughout my studies at UMSL. His broad knowledge, sound instincts and ingenuity in the eld of mathematics have been and will continue to be an inspiration for me. A word of thanks to the other four members of my advisory committee, Professors Cai, Jiang, He and Wu, for their interest, time and eort. My time at the math department at UMSL was made much easier and less stressful (and therefore more enjoyable) by the excellent direction and help, in all administrative matters, of Kimberly Stanger and Raina Traore- Gress; thank you very much. I am also grateful for the nancial support of the Graduate School at UMSL during the course of this study. My time at UMSL was truly enriched by having the opportunity to teach undergraduate mathematics courses I will always remember fondly the hours I spent in Clark Hall 409 with my trigonometry students. The code to execute the EMD algorithm in the numerical experiments of Chapter 9 of this dissertation was initially programmed by my fellow student, Joe Koester. I am grateful for his enthusiasm for EMD and for just about any topic in mathematics, really. I would also like to thank Professor Hau-tieng Wu of the Department of Mathematics at the University of Toronto, for graciously providing me with his code to execute the SST. I feel that these acknowledgments would not be complete without thanking Professor Johan de Villiers of the Department of Mathematical Sciences

5 Van der Walt, Maria, 2015, UMSL, p.iii at Stellenbosch University, for instilling in me a passion for conducting honest research and solving problems in mathematics. He is truly a remarkable person with an unquenchable zest for life. A special thank you to the Kruger and Miller families from Harvester Christian Church and the Stulacs from Memorial Presbyterian Church, for being our family in a foreign country. I want to thank my family for their love and encouragement. First, thank you to my loving parents, Jaco and Elmien Thom, who raised me to love learning new things, to work hard and to persevere, and who believed in me and supported me in all my pursuits. Second, I want to thank my lovely sisters, Matildie and Elmientjie, for their support, inspiration and camaraderie. This last word of acknowledgment I have saved for my wonderful husband, Tjaart. Thank you for leaving everything we had in South Africa so we could embrace this adventure together; for encouraging me and helping me keep a clear perspective; for your common sense and steadfastness; for believing in me and supporting me throughout this study. I could not have done it without you. Soli Deo Gloria.

6 Van der Walt, Maria, 2015, UMSL, p.iv Contents Acknowledgments Contents List of Figures List of Tables Abbreviations ii iv vi ix x 1 Introduction 1 2 Preliminaries Fourier series Fourier transform B-splines Spline interpolation Continuous wavelet transform Hilbert spectral analysis Quasi-interpolation Quasi-interpolation: Scheme E Quasi-interpolation: Scheme H Approximation order Blending interpolation Local interpolation: Scheme E Local interpolation: Scheme H Blending interpolation

7 Van der Walt, Maria, 2015, UMSL, p.v 4.4 Approximation order Stationary and non-stationary signals Stationary signals Non-stationary signals Overview of time-frequency methods Empirical mode decomposition EMD algorithm Variations on EMD Limitations of EMD Synchrosqueezed wavelet transform SST Limitations of SST Analytic vanishing moment wavelets VM wavelets Analytic VM wavelets Hybrid EMD-SST scheme Implementation Numerical experiments Final remarks 145 Bibliography 147

8 Van der Walt, Maria, 2015, UMSL, p.vi List of Figures 6.1 Illustration of sifting. (a) Original signal f(t) = cos 2π(8t) cos 2π(4t) cos 2πt. (b) Construction of upper envelope (in red) and lower envelope (in blue). (c) Calculation of mean envelope (in purple). (d) Result of subtracting mean envelope from input signal Illustration of IMF-expansion obtained through EMD. (a) C 1 (t) (b) C 2 (t) (c) C 3 (t) (d) R 3 (t) End result of EMD and HSA. (a)-(c) IMF's C 1 (t), C 2 (t), C 3 (t). (d)-(f) IF's θ 1(t), θ 2(t), θ 3(t) Illustration of SST. (a) Original signal f(t) = cos 2π(8t)cos 2π(4t) cos 2πt. (b) SST output Interior wavelet ψ x,4,1,0 on the interval [ 5, 5] Boundary wavelets on the interval [ 5, 5]. (a)-(c) ψ x,4,1, 3, ψ x,4,1, 2 and ψ x,4,1, 1. (d)-(f) ψ x,4,1,1, ψ x,4,1,2 and ψ x,4,1, Ex. 1: graphs of originals. (a) Original signal f(t). (b)-(d) Components f 1 (t), f 2 (t), f 3 (t) Ex. 1: IMF's constructed through EMD with our real-time cubic spline interpolation scheme. (a)-(c) C 1 (t), C 2 (t), C 3 (t) Ex. 1: digital image output of SST (in grayscale) with IF's estimated through curve tting (in red). (a)-(c) φ 1(t), φ 2(t), φ 3(t)

9 Van der Walt, Maria, 2015, UMSL, p.vii 9.4 Ex. 1: comparison of IMF's. (a),(d),(g) True components f 1 (t), f 2 (t), f 3 (t). (b),(e),(h) IMF's C1 O (t), C2 O (t), C3 O (t), obtained from the original EMD with standard cubic spline interpolation. (c),(f),(i) IMF's C1 S (t), C2 S (t), C3 S (t), obtained through applying EMD with our real-time cubic spline interpolation scheme Ex. 1: comparison of IF estimation. (a),(d),(g) True IF's φ 1(t), φ 2(t), φ 3(t). (b),(e),(h) Estimated IF's φ O 1 (t), φ O 2 (t), φ O 3 (t), obtained by applying HSA to the original EMD. (c),(f),(i) Estimated IF's φ S 1 (t), φ S 2 (t), φ S 3 (t), obtained through smoothing spline curve tting of our modied SST applied to each IMF Cj S, j = 1, 2, 3 separately Ex. 2: graphs of originals. (a) Original signal g(t). (b)-(d) Components g 1 (t), g 2 (t), g 3 (t) Ex. 2: IMF's constructed through EMD with our real-time cubic spline interpolation scheme. (a)-(c) C 1 (t), C 2 (t), C 3 (t) Ex. 2: digital image output of SST (in grayscale) with IF's estimated through curve tting (in red). (a)-(c) φ 1(t), φ 2(t), φ 3(t) Ex. 2: comparison of IMF's. (a),(d),(g) True components g 1 (t), g 2 (t), g 3 (t). (b),(e),(h) IMF's C1 O (t), C2 O (t), C3 O (t), obtained from the original EMD with standard cubic spline interpolation. (c),(f),(i) IMF's C1 S (t), C2 S (t), C3 S (t), obtained through applying EMD with our real-time cubic spline interpolation scheme Ex. 2: comparison of IF estimation. (a),(d),(g) True IF's φ 1(t), φ 2(t), φ 3(t). (b),(e),(h) Estimated IF's φ O 1 (t), φ O 2 (t), φ O 3 (t), obtained by applying HSA to the original EMD. (c),(f),(i) Estimated IF's φ S 1 (t), φ S 2 (t), φ S 3 (t), obtained through smoothing spline curve tting of our modied SST applied to each IMF Cj S, j = 1, 2, 3 separately Ex. 3: graphs of originals. (a) Original signal h(t). (b)-(c) Components h 1 (t), h 2 (t) Ex. 3: IMF's constructed through EMD with our real-time cubic spline interpolation scheme. (a)-(b) C 1 (t), C 2 (t) Ex. 3: digital image output of SST (in grayscale) with IF's estimated through curve tting (in red). (a)-(b) φ 1(t), φ 2(t)

10 Van der Walt, Maria, 2015, UMSL, p.viii 9.14 Ex. 3: comparison of IMF's. (a),(d) True components h 1 (t), h 2 (t). (b),(e) IMF's C1 O (t), C2 O (t), obtained from the original EMD with standard cubic spline interpolation. (c),(f) IMF's C1 S (t), C2 S (t), obtained through applying EMD with our real-time cubic spline interpolation scheme Ex. 3: comparison of IF estimation. (a),(d) True IF's φ 1(t), φ 2(t). (b),(e) Estimated IF's φ O 1 (t), φ O 2 (t), obtained by applying HSA to the original EMD. (c),(f) Estimated IF's φ S 1 (t), φ S 2 (t), obtained through smoothing spline curve tting of our modied SST applied to each IMF Cj S, j = 1, 2 separately Ex. 3: comparison of our hybrid EMD-SST scheme and original SST approach. (a) The output from the (improved) SST (with our analytic VM wavelet and boundary considerations) applied to the mixed input signal. (b)-(c) The output from the hybrid EMD-SST approach

11 Van der Walt, Maria, 2015, UMSL, p.ix List of Tables 9.1 Ex. 1: comparison of maximum errors produced by EMD-HSA and EMD-SST approaches Ex. 1: comparison of mean errors produced by EMD-HSA and EMD-SST approaches Ex. 1: comparison of standard deviation of errors produced by EMD-HSA and EMD-SST approaches Ex. 2: comparison of maximum errors produced by EMD-HSA and EMD-SST approaches Ex. 2: comparison of mean errors produced by EMD-HSA and EMD-SST approaches Ex. 2: comparison of standard deviation of errors produced by EMD-HSA and EMD-SST approaches Ex. 3: comparison of maximum errors produced by EMD-HSA and EMD-SST approaches Ex. 3: comparison of mean errors produced by EMD-HSA and EMD-SST approaches Ex. 3: comparison of standard deviation of errors produced by EMD-HSA and EMD-SST approaches

12 Van der Walt, Maria, 2015, UMSL, p.x Abbreviations AHM: adaptive harmonic model AM: amplitude-modulated CWT: continuous wavelet transform EEMD: ensemble empirical mode decomposition EMD: empirical mode decomposition FM: frequency-modulated FRA: frequency reassignment GCV: generalized cross-validation HSA: Hilbert spectral analysis IA: instantaneous amplitude IF: instantaneous frequency IMF: intrinsic mode function NHT: normalized Hilbert transform SST: synchrosqueezed wavelet transform VM: vanishing moment

13 Van der Walt, Maria, 2015, UMSL, p.1 Chapter 1 Introduction Time-frequency analysis is one of the most important and powerful tools in signal processing for understanding the oscillatory features of signals. Let us consider the signal (or function) f(t) = a 0 N a j cos(2πω j t), (1.0.1) j=1 for arbitrary real values ω j > 0 and a j R. It is clear from (1.0.1) that f is a superposition of the signal components f j (t) = a j cos(2πω j t), j = 1,..., N, each with a frequency of ω j Hz. Since each ω j is a constant, independent of the time variable t, such a signal is classied as stationary. However, most real-world signals (for example, biological signals, speech signals and music signals) are non-stationary, meaning that their frequencies may change with time. In the literature, these types of signals are represented by a generalization of the model in (1.0.1), namely f(t) = A 0 (t) N A j (t) cos 2πφ j (t), (1.0.2) j=1 where A j (t) 0 and each φ j (t) is a general C 1 function such that φ j(t) > 0 (where C 1 denotes the space of all functions with continuous rst derivatives). The derivative φ j(t) is a natural extension of the frequency ω j in (1.0.1), and is called the instantaneous frequency (IF) of the component

14 CHAPTER 1 Van der Walt, Maria, 2015, UMSL, p.2 f j (t) = A j (t) cos 2πφ j (t), j = 1,..., N. While the mathematical theory to analyze stationary signals is well developed in the literature (and is mainly founded on Fourier analysis), the study of non-stationary signals is still a relatively new eld, only developing over the last thirty years [31, 17, 23, 2]. The contributions of this dissertation can be grouped into two parts: rst, the development of a spline interpolation scheme, which is then applied in the second part, which concerns the non-stationary signal analysis problem described above. In the rst part, we formulate a new spline interpolation scheme for a bounded interval [a, b], in terms of the m th order B-splines. We start by developing a quasi-interpolation operator Q m with a local formulation (in the sense that the value of Q m applied to a given function f at any x [a, b] only depends on the values of f in a small neighborhood of x ), which preserves polynomials of degree m 1. This quasi-interpolation operator is based on a scheme introduced in [9]; however, the method in [9] is formulated for an unbounded interval, and is adapted here (in a non-trivial way) for a bounded interval. The next step is to develop a local spline interpolation operator R m, such that R m f interpolates the function f at a given sequence of discrete data points in [a, b] and satises certain Hermite interpolation conditions at the endpoints of the interval x = a and x = b. We base our local spline interpolation operator on an idea described in [15]; it is adapted here to include the Hermite interpolation conditions at the endpoints. The quasi-interpolation operator Q m and interpolation operator R m are then combined in a smart way to form a so-called blending operator P m (rst considered in [13]), such that P m meets all the requirements met by Q m and R m. Corresponding error bounds for both Q m and P m are also derived rigorously. The local formulation and boundary considerations make this spline interpolation method particularly useful. A specic application for this scheme is presented in our approach to instantaneous frequency estimation of non-stationary signal components, considered in the second part of this dissertation.

15 CHAPTER 1 Van der Walt, Maria, 2015, UMSL, p.3 The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang and others in 1998 [36], is currently one of the most popular mathematical schemes for non-stationary signal decomposition and timefrequency analysis. The objective of EMD is to decompose a given (nonstationary) signal into a number of oscillating components, called intrinsic mode functions (IMF's), and a monotone or slowly oscillating remainder. This is done through an algorithm that is based on standard cubic spline interpolation. Each IMF is then extended to an amplitude-frequency modulated (AM-FM) signal through Hilbert spectral analysis (HSA), based on the Hilbert transform, in order to compute its instantaneous frequency and amplitude. However, the EMD scheme, and the current modications and improvements of it [53, 55], have several limitations. Firstly, there is no guarantee that the AM-FM extension of an IMF will yield a non-negative IF. This is a serious defect, since negative frequency is meaningless for signals and limits the application of EMD. Moreover, since the Hilbert transform is dened for functions on an unbounded interval, while real-life signals are typically dened on bounded or half-innite intervals, articial extension of an IMF to the real line is necessary in order to apply the Hilbert transform, often yielding unreliable results. Instead of computing the IF's after the signal is decomposed as is done when applying EMD, the approach that I. Daubechies and others [20, 21] proposed is to rst estimate the IF's of the signal components, under the assumption that the signal satises certain strict properties of the model in (1.0.2), before recovering the signal components of the model. For this purpose, the notion of the synchrosqueezed wavelet transform (SST), based on the continuous wavelet transform, was introduced to compute a single reassignment rule, or IF reference function, through which the IF's of all the signal components are squeezed out from the input signal in the form of a digital image displaying a set of IF curves, allowing the estimation of the individual IF functions and the signal components themselves. Again, there are a few limitations to the SST. First, to estimate the instantaneous frequencies of signal components, the IF curves represented in the digital image output of the SST must be extracted, one by one, through

16 CHAPTER 1 Van der Walt, Maria, 2015, UMSL, p.4 a suitable curve tting scheme. In general, this can be quite complicated, particularly for over four of ve IF curves, and therefore, the process must be supervised. Moreover, the SST's original formulation is not suited to real-time implementation. Our approach to instantaneous frequency estimation of non-stationary signal components consists of combining the best parts of EMD and SST to form a hybrid EMD-SST scheme. In a nutshell, we apply a modied SST to each IMF produced by a modied EMD. With this approach, we are assured of non-negative instantaneous frequencies of the IMF's through the SST, while the EMD eliminates the need to extract multiple IF curves from the digital image output of the original SST approach. In addition, since the Hilbert transform of the original EMD approach is replaced by our modied SST, articial extension of the IMF's to the real line is avoided, solving many computational issues. The modication of the SST consists in applying so-called analytic vanishing moment wavelets with stacked knots (rst considered in [15]) to allow processing of signals on bounded or half-innite time intervals, as well as applying spline curve tting with optimal smoothing parameter selection through generalized cross-validation to identify the IF curve displayed in the digital image output of the SST. The modication of EMD consists in replacing the standard cubic spline interpolation in the original algorithm with our real-time spline interpolation scheme for bounded intervals. According to our results, the instantaneous frequency estimation of signal components is remarkably more accurate when using our approach, compared to the original EMD-HSA approach. The construction of IMF's through EMD is also improved by our real-time spline interpolation scheme. The next chapter is devoted to the study of the preliminary results that we will rely on in this dissertation. Specically, we will study the theory of Fourier series and the Fourier transform, B-splines and spline interpolation, the continuous wavelet transform, and the Hilbert transform. In Chapter 3, we introduce our quasi-interpolation scheme for a bounded interval in terms of the m th order B-splines. We consider two variants of this scheme, developed for dierent types of knot sequences and sampling points. A corresponding approximation order analysis is also derived. We

17 CHAPTER 1 Van der Walt, Maria, 2015, UMSL, p.5 formulate our local spline interpolation operator in Chapter 4, again considering two variants for dierent knot sequences and sampling points. The blending operator and corresponding error bounds are also derived in Chapter 4. Next, we move on to the second part of the dissertation, namely the study of instantaneous frequency estimation of non-stationary signal components. To this end, we start, in Chapter 5, by considering the notions of stationary and non-stationary signals in greater detail, as well as dierent time-frequency methods in the literature. The EMD algorithm, some modications and improvements of it and its limitations are studied in detail in Chapter 6, while the SST and its limitations are discussed in Chapter 7. The essential properties and computational algorithms of the analytic vanishing moment wavelets are described in Chapter 8. In Chapter 9, we describe our hybrid EMD-SST scheme in greater detail and provide graphical results, numerical experiments and comparisons, based on a number of representative test signals. Final remarks and conclusions follow in Chapter 10.

18 Van der Walt, Maria, 2015, UMSL, p.6 Chapter 2 Preliminaries This chapter is devoted to the development of preliminary results that will be needed later in this dissertation. In Section 2.1, we discuss Fourier series, with specic reference to the Fourier cosine series (which occur often in signal processing applications). The Fourier transform and its properties are studied in Section 2.2. Section 2.3 is devoted to B-splines and its properties, and in Section 2.4, we consider the basic theory of spline interpolation, including quasi-interpolation and Hermite interpolation. Next, in Section 2.5, we dene the notions of wavelets and the continuous wavelet transform. Lastly, in Section 2.6, we consider the Hilbert spectral analysis technique, a method in the signal analysis literature to estimate a signal's instantaneous frequency, which is based on the Hilbert transform. 2.1 Fourier series Traditionally, signal analysis has been based on nding a Fourier representation of a signal. In this section, we examine the notions of Fourier series and Fourier cosine series. To this end, let L 2 [ L, L ] denote the set of square-integrable functions 2 2 on [ L, L] (for some L > 0). Any function f 2 2 L2 [ L, L ] can be extended 2 2

19 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.7 to an L-periodic function F (x) on R through the denition F (x) = f(x), x ( L, L); 2 2 F ( L 2 ) = F ( L) = ( f( L ) f( L)) ; 2 2 F (x kl) = F (x), k Z. For convenience, we will rename F (x) as f(x) also. With this denition, we denote the inner product space of all L-periodic square-integrable functions by L 2 [ L, L ], with inner product 2 2 j=1 f, g = L 2 L 2 f(x)g(x)dx. Denition (Fourier series) If f L 2, its Fourier series Sf is dened by (Sf)(x) = a 0 2 ( ( ) ( )) 2πjx 2πjx a j cos b j sin, (2.1.1) L L where a j = 2 L b j = 2 L ( ) 2πjx f(x) cos dx, j = 0, 1, 2,... ; L ( ) 2πjx f(x) sin dx, j = 1, 2,.... L (2.1.2) The signicance of the Fourier series in Denition is the following (see [14, Theorem 2, p.282]): Theorem (Fourier series) For L > 0, the family { ( ) ( ) } 1 2 2πjx 2 2πjx, L L cos, L L sin : j = 1, 2,... L is an orthonormal basis of L 2 [ L, L 2 2 ]. Therefore, any f L2 [ L, L ] can be 2 2 represented by its Fourier series, namely f(x) = (Sf)(x) = a 0 2 j=1 ( a j cos ( ) ( 2πjx 2πjx b j sin L L )),

20 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.8 which converges to f in L 2 [ L 2, L 2 ], where a j, j = 0, 1, 2,... and b j, j = 1, 2,... are dened in (2.1.2). The convergence of the Fourier series in Theorem rests upon the fact that the partial sums of the Fourier series of f are best L 2 -approximations of f from the space of all trigonometric polynomials, and is proved in [14, Chapter 6]. For signal analysis, to reduce computational complexity, it is customary in the literature to nd a Fourier cosine series representation of a signal instead of the Fourier series representation in terms of both cosines and sines. To this end, we consider a function f L 2 [0, L ], and extend f to an 2 even function f e on [ L, L ] through the denition 2 2 { f(x), 0 x L 2 f e (x) :=, f( x), L x < 0. 2 This implies that the coecient b j of the Fourier series of f e (dened in (2.1.2)) is 0, since sin ( ) 2πjx L in (2.1.2) is an odd function. Therefore, the Fourier series in (2.1.1) of f e reduces to the Fourier cosine series (Sf e )(x) = a 0 2 ( ) 2πjx a j cos, L j=1 with a j = 2 L L 2 L 2 ( 2πjx f e (x) cos L ) dx = 4 L 2 f(x) cos L 0 ( 2πjx L ) dx, for j = 0, 1, 2,.... From Theorem 2.1.1, we have that (S n f e )(x) = a n ( ) 0 2πjx 2 a j cos L converges to f e in L 2 [ L 2, L 2 ] as n, so that S nf e restricted to [0, L 2 ] converges to f in L 2 [0, L 2 ]. We therefore have the following (see [14, Theorem 3, p.284]): j=1

21 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.9 Theorem (Fourier cosine series) For L > 0, the family { ( ) } 1 2 2πjx, L L cos : j = 1, 2,... L Therefore, any f L2 [0, L ] can be 2 is an orthonormal basis of L 2 [0, L 2 ]. represented by its Fourier cosine series, namely f(x) = (S e f)(x) = a 0 2 ( 2πjx a j cos L which converges to f in L 2 [0, L 2 ], where a j, j = 0, 1, 2,... is given by a j = 4 L L 2 0 ( 2πjx f(x) cos L j=1 ), ) dx, j = 0, 1, 2,.... We note that a Fourier sine representation may be found in a similar way (by extending the function f L 2 [0, L ] to an odd function instead of an 2 even one), but this is not conventional for signal analysis in the literature. 2.2 Fourier transform The Fourier series, considered in Section 2.1, provides us with a method to study the frequency contents of periodic functions (as we shall see in Chapter 5). In this section, we consider the Fourier transform, which may be used instead to study the frequency contents of stationary signals. Denition (Fourier transform) Let f be a function in L 1 (R), where L 1 denotes the space of all integrable functions. Then the Fourier transform of f, which we denote by ˆf or Ff, is dened by ˆf(ω) = (Ff)(ω) = f(x)e i2πωx dx, ω R. (2.2.1) In Theorem 2.2.1, we list some important properties of the Fourier transform of a function f in L 1 (R) or L 1 [0, ). We will make use of the following function operations:

22 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.10 ˆ Even extension: For f L 1 [0, ), the even extension f e of f is dened by { f(x), x 0; f e (x) := f( x), x < 0. (2.2.2) ˆ Translation: For f L 1 (R) and b R, the translation operator T b is given by (T b f)(x) := f(x b). (2.2.3) ˆ Dilation: For f L 1 (R) and a > 0, the dilation operator D a is dened by (D a f)(x) := f(ax). (2.2.4) ˆ Frequency modulation: For f L 1 (R) and c R, c 0, the (frequency) modulation operator M c is dened by (M c f)(x) := f(x)e i2πcx. (2.2.5) ˆ Convolution: Let f, h be functions on R. dened by (f h) (x) := Then the convolution of f with h is f(t)h(x t)dt. (2.2.6) We note that each of the function operations in (2.2.2)-(2.2.5) are in L 1 (R). If f, h L 1 (R), then the convolution f h in (2.2.6) is also in L 1 (R); that is, (f h) (x) dx f(t)h(x t) dtdx <. (2.2.7) This follows from Fubini's Theorem (see, for example, [14, Theorem 1, p.376]), by which we may interchange the order of integration in (2.2.7). Theorem (Properties of Fourier transform) The Fourier transform satises the following properties:

23 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.11 (i) Let f e be the even extension of f L 1 [0, ). Then ˆf e (ω) = 2 (ii) For f L 1 (R) and b R, 0 f(x) cos 2πωxdx; (T b f)(ω) = e i2πωb ˆf(ω); (2.2.8) (iii) For f L 1 (R) and a > 0, (D a f)(ω) = 1 a ˆf ( ω a ) ; (2.2.9) (iv) For f L 1 (R) and c R, c 0, (M c f)(ω) = ˆf (ω c) ; (2.2.10) (v) Let f, h L 1 (R). Then (f h)(ω) = ˆf(ω)ĥ(ω). Proof: (i) From (2.2.1) and (2.2.2), we have ˆf e (ω) = = 0 = = 0 f e (x)e i2πωx dx f( x)e i2πωx dx f(x)e i2πωx dx 0 0 f(x)e i2πωx dx f(x)e i2πωx dx f(x) [ e i2πωx e i2πωx] dx = from Euler's formula e iθ = cos θ i sin θ for θ R. f(x) cos 2πωxdx,

24 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.12 (ii) From (2.2.1) and (2.2.3), it follows that (T b f)(ω) = = f(x b)e i2πωx dx f(x)e i2πω(xb) dx = e i2πωb f(x)e i2πωx dx = e i2πωb ˆf(ω). (iii) From (2.2.1) and (2.2.4), it follows that (D a f)(ω) = = 1 a = 1 a f(ax)e i2πωx dx f(x)e i2πω(x/a) dx (iv) From (2.2.1) and (2.2.5), it follows that (M c f)(ω) = = f(x)e i2π(ω/a)x dx = 1 a ˆf f(x)e i2πcx e i2πωx dx ( ω a ). f(x)e i2π(ω c)x dx = ˆf(ω c). (v) From (2.2.1), (2.2.6) and Fubini's theorem, we have (f h)(ω) = = = = = (f h)(x)e i2πωx dx [ ] f(t)h(x t)dt e i2πωx dx [ ] f(t) h(x t)e i2πωx dx dt [ ] f(t) h(x)e i2πω(xt) dx dt f(t)ĥ(ω)e i2πωt dt = ˆf(ω)ĥ(ω).

25 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.13 Example (a) For f(x) = cos 2πcx with c > 0, ˆf(ω) = 1 (δ(ω c) δ(ω c)). (2.2.11) 2 (b) For f(x) = sin 2πcx with c > 0, ˆf(ω) = 1 (δ(ω c) δ(ω c)). (2.2.12) 2i Solution: (a) From the denition of the Fourier transform in (2.2.1), we have ˆf(ω) = = 1 2 = 1 2 cos(2πcx)e i2πωx dx ( e i2πcx e i2πcx) e i2πωx dx ( e i2π(ω c)x e i2π(ωc)x) dx = 1 (δ (ω c) δ (ω c)), 2 where δ denotes the Dirac delta distribution. (b) The solution is obtained in a similar way to the solution of (2.2.11) above. If f L 2 (R) such that its Fourier transform ˆf is in L 1 (R), f may be recovered from its Fourier transform (as shown in [14, Theorem 4, p.335]): Theorem (Inverse Fourier transform) For a function f L 2 (R), let g(ω) = ˆf(ω) L 1 (R). Then f(x) = ǧ(x) := g(ω)e i2πωx dω. The function ǧ is called the inverse Fourier transform of f.

26 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.14 In Section 5.1 in Chapter 5, we will see how the Fourier transform may be applied to reveal the frequency content of stationary signals. 2.3 B-splines We start this section by dening the m th order spline space S x,m [a, b] with knot sequence x, for m 1. Denition (Spline space) For positive integers m and N and a, b R, let x : x m1 = = a = x 0 < x 1 < < x N1 = b = = x Nm. (2.3.1) We denote by S x,m [a, b] the linear space of m th order polynomial splines on [a, b] with knots in x, namely { } S x,m [a, b] = f(x) C m 2 [a, b] : f [xi,x i1 ] π m 1, i = 0,..., N, (2.3.2) where C m 2 [a, b] denotes the space of all functions on [a, b] with m 2 continuous derivatives, and π m 1 denotes the space of all polynomials of degree m 1. As shown in [19],[6, Theorem IX.1], a locally supported basis for S x,m [a, b] is given by the set of normalized m th order B-splines {N x,m,j : j = m 1,..., N}, where each N x,m,j is dened in terms of divided dierences of truncated powers (to be made precise in Denition below). Divided dierences are dened by [u,..., u]g := gl (u) (2.3.3) l! if there are l 1 entries in [u..., u], and [u 0,..., u n ]g := [u 1,..., u n ]g [u 0,..., u n 1 ]g u n u 0 (2.3.4) if u 0 u 1 u n with u n > u 0, where [u i ]g := g(u i ). The truncated powers are given by x n := (max {0, x}) n. (2.3.5)

27 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.15 Denition (B-splines) For the sequence x in (2.3.1), the normalized m th order B-splines N x,m,j, j = m 1,..., N, are dened by N x,m,j (x) := (x jm x j )[x j,..., x jm ]( x) m 1, j = m 1,..., N. (2.3.6) For the knot sequence x in (2.3.1), the B-splines N x,m,j, j = 0,..., N m 1, are called interior B-splines, while N x,m,j, j = m 1,..., 1; N m 2,..., N, are called left hand side and right hand side boundary B- splines, respectively. We note that N x,m,j 0, j / { m 1,..., N}. (2.3.7) By expanding the divided dierences in the denition of B-splines (2.3.6), we obtain the following special formulations for the boundary B-splines N x,m, m1 and N x,m,n. Theorem (Boundary B-splines) For the knot sequence x in (2.3.1), the boundary B-splines N x,m, m1 and N x,m,n satisfy the formulations ( m 1 x 1 x N x,m, m1 (x) = x 1 a), a x x1 ; (2.3.8) 0, otherwise, and ( ) m 1 x x N N x,m,n (x) = b x N, xn x b; 0, otherwise. (2.3.9) Proof: We proceed to prove (2.3.9); the proof of (2.3.8) is similar. From (2.3.6) and the denition of divided dierences in (2.3.3)-(2.3.4), we have N x,m,n (x)

28 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.16 = (x Nm x N ) [x N,..., x Nm ] ( x) m 1 = (x Nm x N ) [x N1,..., x Nm ] ( x) m 1 [x N,..., x Nm 1 ] ( x) m 1 x Nm x N = (m 1)! (b x)0 (m 1)! = (m 1)! (b x)0 (m 1)! = (m 1)! (b x)0 (m 1)! i=0 [x N1,..., x Nm 1 ] ( x) m 1 [x N,..., x Nm 2 ] ( x) m 1 x Nm 1 x N (m 1)! (b x)1 (m 2)!1!(b x N ) [x N1,..., x Nm 2 ] ( x) m 1 [x N,..., x Nm 3 ] ( x) m 1 (b x)(x Nm 2 x N ) (m 1)! (b x)1 (m 2)!1!(b x N ) (m 1)! (b x)2 (m 3)!2!(b x N ) 2 [x N1,..., x Nm 3 ] ( x) m 1 [x N,..., x Nm 4 ] ( x) m 1 (b x N ) 2 (x Nm 3 x N ) = m 1 ( ) m 1 (b x) = ( 1) i i i (b x N ) ( 1)m (x N x) m 1. i (b x N ) m 1 From the denition of truncated powers (2.3.5), we have Therefore, let x b, so that N x,m,n (x) = 0, x > b. N x,m,n (x) m 1 ( ) ( ) i m 1 b x = ( 1) i ( 1)m (x N x) m 1 i b x i=0 N (b x N ) m 1 m 1 ( ) ( ) i m 1 x b = ( 1)m (x N x) m 1 i b x i=0 N (b x N ) m 1 ( ) m 1 x b = 1 ( 1)m (x N x) m 1 b x N (b x N ) m 1 ( ) m 1 x xn = ( 1)m (x N x) m 1, b x N (b x N ) m 1

29 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.17 from the binomial theorem. If x < x N, it follows from the denition of truncated powers (2.3.5) that ( ) m 1 x xn N x,m,n (x) = ( 1)m (x N x) m 1 b x N (b x N ) m 1 ( ) m 1 ( ) m 1 x xn x xn = = 0, b x N b x N while, if x N x b, we have ( ) m 1 x xn N x,m,n (x) =, b x N again from (2.3.5), completing our proof of (2.3.9). The normalized m th order B-splines N x,m,j, j = m1,..., N, in (2.3.6) satisfy the following properties (see, for example, [12, Theorem 6.4] and [22, Theorems , , ] and [5]). Theorem (Properties of B-splines) The B-splines {N x,m,j : j = m 1,..., N} with knot sequence x, as dened in (2.3.6) and (2.3.1), respectively, satisfy the following properties: (i) suppn xm,j = [a, b] [x j, x jm ], j = m 1,..., N; (2.3.10) (ii) N x,m,j, j = m 1,..., N, may be computed recursively through N x,m,j (x) = where x x j N x,m 1,j (x) x jm x N x,m 1,j1 (x), x jm 1 x j x jm x j1 (2.3.11) N x,1,j (x) = χ [xj,x j1 )(x), j = 0,..., N, (2.3.12) the characteristic function on the interval [x j, x j1 ); (iii) N x,m,j (x) > 0, x (x j, x jm ); (2.3.13)

30 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.18 (iv) The B-splines {N x,m,j : j = m 1,..., N} form a partition of unity; that is, N N x,m,j (x) = 1, x [a, b]; (2.3.14) j= m1 (v) The derivative of N x,m,j, j = m 1,..., N, may be computed recursively through N x,m,j(x) = for m 2. m 1 m 1 N x,m 1,j (x) N x,m 1,j1 (x), x jm 1 x j x jm x j1 (2.3.15) Proof: (i) First, let x > x jm for some xed j. Then N x,m,j (x) = (x jm x j )[x j,..., x jm ]( x) m 1 = 0, from the denition of truncated powers. On the other hand, if we let x < x j, we have N x,m,j (x) = (x jm x j )[x j,..., x jm ]( x) m 1 = (x jm x j )[x j,..., x jm ]( x) m 1. (2.3.16) We now use the fact that, if g is a function with n continuous derivatives in the smallest interval containing the points {u 0,..., u n }, then [u 0,..., u n ]g = g(n) (ξ) n! for some point ξ in the smallest interval containing the points {u 0,..., u n } (see [22, Theorem 2.1.2]). With g := ( x) m 1, (2.3.16) therefore becomes N x,m,j (x) = (x jm x j )[x j,..., x jm ]g = (x jm x j ) g(m) (ξ) m! = 0, where ξ is some point in the smallest interval containing {x j,..., x jm }, since g π m 1.

31 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.19 (ii) The formulation (2.3.12) for N x,1,j, j = 0,..., N, follows directly from the denition (2.3.6) with m = 1 and the denition of divided dierences (2.3.4). For j = m 1 and j = N, the formulation (2.3.11) follows from Theorem Indeed, if j = N and x R is xed, we have x x N N x,m 1,N (x) x Nm x N x,m 1,N1 (x) x Nm 1 x N x Nm x N1 = x x N b x N ( ) m 2 x xn 0 b x N ( ) m 1 x xn = = N x,m,n(x), b x N using also (2.3.7). The result follows similarly if j = m 1. Hence, let j { m 2,..., N 1}, so that x jm 1 > x j and x j > x j1 both hold, and let x R be xed. To prove (2.3.11), we recall the Leibniz rule for the divided dierence of a product, namely, for two functions f and g and any sequence of points {u 0,..., u n }, [u 0,..., u n ](fg) = n [u 0,..., u j ]f[u j,..., u n ]g. j=0 (see [45, Theorem 2.52]). Using also (2.3.6) and (2.3.4), we have N x,m,j (x) = (x jm x j )[x j,..., x jm ]( x) m 1 = (x jm x j )[x j,..., x jm ]( x)( x) m 2 jm = (x jm x j ) k=j [x j,..., x k ]( x)[x k,..., x jm ]( x) m 2 = (x jm x j ) [ (x j x)[x j,..., x jm ]( x) m 2 ] [x j1,..., x jm ]( x) m 2 = (x jm x j ) [(x j x) [x j1,..., x jm ]( x) m 2 [x j x jm 1 ]( x) m 2 x jm x j

32 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.20 ] [x j1,..., x jm ]( x) m 2 = (x j x)[x j1,..., x jm ]( x) m 2 (x j x)[x j,..., x jm 1 ]( x) m 2 (x jm x j )[x j1,..., x jm ]( x) m 2 = [x j1,..., x jm ]( x) m 2 (x j x x jm x j ) (x x j )[x j,..., x jm 1 ]( x) m 2 = x jm x (x jm x j1 ) [x j1,..., x jm ]( x) m 2 x jm x j1 x x j (x jm 1 x j ) [x j,..., x jm 1 ]( x) m 2 x jm 1 x j = x jm x x jm x j1 N x,m 1,j1 (x) x x j x jm 1 x j N x,m 1,j (x). (iii) We know that suppn x,m,j = [x j, x jm ], j = m 1,..., N, so that N x,m,j (x) = 0, x R \ [x j, x jm ]. We now proceed to show by induction that (2.3.13) holds. Since N x,1,j (x) = χ [xj,x j1 )(x) (as follows from (ii)), it is clear that the result holds for m = 1. Next, we assume that (2.3.13) holds for a xed non-negative integer m 1, and let x (x j, x jm ). From the recursion formula (2.3.11), we have N x,m,j (x) = x x j N x,m 1,j (x) x jm x N x,m 1,j1 (x). x jm 1 x j x jm x j1 Next, we note that, from the support property (2.3.10), { N x,m 1,j (x) = 0, x [x jm 1, x jm ); N x,m 1,j1 (x) = 0, x (x j, x j1 ], for all j = m 1,..., N. Therefore, using also the induction hypothesis, we deduce that N x,m,j (x) > 0, x (x j, x jm ), for all j = m 1,..., N, which completes our inductive proof of (2.3.13). (iv) Our proof is once again by induction on m. Since N x,1,j (x) = χ [xj,x j1 )(x), it is clear that (2.3.14) holds for m = 1. Next, we assume the result

33 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.21 holds for a xed non-negative integer m 1 and let x [a, b]. From the recursive formulation in (2.3.11), it follows that N j= m1 = = = N j= m1 N j= m1 N j= m2 N x,m,j (x) x x j x jm 1 x j N x,m 1,j (x) x x j x jm 1 x j N x,m 1,j (x) x x j x jm 1 x j N x,m 1,j (x) N j= m1 N1 j= m2 N j= m2 x jm x x jm x j1 N x,m 1,j1 (x) x jm 1 x x jm 1 x j N x,m 1,j (x) x jm 1 x x jm 1 x j N x,m 1,j (x), from the support property (2.3.10). It therefore follows that N j= m1 N x,m,j (x) = = = N j= m2 N j= m2 N j= m2 x x j x jm 1 x N x,m 1,j (x) x jm 1 x j x jm 1 x j x jm 1 x j N x,m 1,j (x) N x,m 1,j (x) = 1, from the induction hypothesis, completing our proof of (2.3.14). (v) Lastly, to verify (2.3.15), we dierentiate (2.3.6) and use (2.3.4) to obtain N x,m,j(x) = (m 1) (x jm x j ) [x j,..., x jm ] ( x) m 2 = (m 1) (x jm x j ) [x j1,..., x jm ] ( x) m 2 [x j,..., x jm 1 ] ( x) m 2 x jm x j = (m 1) [x j1,..., x jm ] ( x) m 2 (m 1) [x j,..., x jm 1 ] ( x) m 2

34 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.22 from (2.3.6). = m 1 (x jm x j1 ) [x j1,..., x jm ] ( x) m 2 x jm x j1 m 1 (x jm 1 x j ) [x j,..., x jm 1 ] ( x) m 2 x jm 1 x j m 1 m 1 = N x,m 1,j (x) N x,m 1,j1 (x), x jm 1 x j x jm x j1 2.4 Spline interpolation B-splines are useful in many applications, including spline interpolation and approximations problems (the applications that we will be studying in this dissertation). Before dening a spline interpolation operator in terms of B-splines of any arbitrary order, we start this section by reviewing the standard cubic spline interpolation scheme in [6, Chapter IV], since it is one of the most popular interpolation methods in current applications (due to the simplicity to implement it). For a function f : [a, b] R and a sequence y : a = y 1 < y 2 < < y n = b, we proceed to investigate the construction of a cubic spline interpolant S, with knot sequence y, that satises the interpolation conditions S(y i ) = f(y i ), i = 1,..., n. Since S is a cubic polynomial spline, we know that (as in (2.3.2)). Therefore, we have S C 2 and S [yi,y i1 ] π 3 S(x) = p i (x), x [y i, y i1 ], for all i = 1,..., n 1, for some cubic polynomials p i (x) = a i b i (x y i )c i (x y i ) 2 d i (x y i ) 3, i = 1,..., n 1, (2.4.1)

35 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.23 such that { pi (y i ) = f(y i ); p i (y i1 ) = f(y i1 ); p i(y i ) = s i ; p i(y i1 ) = s i1, (2.4.2) for i = 1,..., n 1, where s 1,..., s n are free parameters determining the slope of each polynomial p i, and such that p i 1(y i ) = p i (y i ), i = 2,..., n 1. (2.4.3) By substituting the conditions (2.4.2) in the polynomial formulations (2.4.1), we may solve for a i, b i, c i, d i, i = 1,..., n 1, in terms of the parameters s 1,..., s n. Assuming that the parameters s 1 and s n are chosen by the user in some way, the remaining n 2 free parameters s 2,..., s n 1 are then determined uniquely by the n 2 conditions in (2.4.3). There exist dierent approaches in the literature to choosing the boundary slopes s 1 and s n. The most popular techniques include the so-called natural spline interpolation, where s 1 and s n are chosen to satisfy the free-end condition S (y 1 ) = S (y n ) = 0, and the spline resulting from the not-a-knot condition p 1 = p 2 and p n 2 = p n 1 (so that the knots y 2 and y n 1 are not active). We now turn our attention to spline interpolation schemes in terms of B-splines with arbitrary order m (the type of spline interpolation methods that we will be considering in this dissertation), and where the spline knot sequence x need not coincide with the interpolation points y. For a function f : [a, b] R, we will be interested in nding a spline interpolation operator S m : C[a, b] S x,m, where C[a, b] denotes the space of continuous functions on the interval [a, b] and with the spline space S x,m dened in (2.3.2) for the knot sequence x in (2.3.1), such that the spline S m f interpolates the function f at a given sequence of discrete data points on the interval [a, b]. Since the set of normalized B-splines {N x,m,j : j = m 1,..., N} forms a basis for the spline space S x,m, the dimension of S x,m is mn. This means that we can accommodate m N interpolation conditions on the interval [a, b].

36 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.24 Denition (Spline interpolation operator) Given a function f : [a, b] R, let y = {y 1,..., y mn } denote a sequence of m N distinct points in the interval [a, b], with y : a y 1 < y 2 < y mn b. (2.4.4) Then the spline interpolation operator S m : C[a, b] S x,m, with knot sequence x given in (2.3.1), satises the m N interpolation conditions (S m f)(y i ) = f(y i ), i = 1,..., m N. (2.4.5) Spline interpolation has the advantage over traditional polynomial interpolation (for example, the Lagrange and Newton interpolation formulas [22, Sections 1.2, 1.3]) that the approximation accuracy may be improved by decreasing the distance between consecutive knots in x while keeping the polynomial degree m 1 relatively low. Now, since {N x,m,j : j = m 1,..., N} forms a basis for the spline space S x,m, we know that there exists a spline S m f satisfying (2.4.5) if and only if N (S m f)(x) = c f j N x,m,j(x), x [a, b] for certain coecients N j= m1 j= m1 { } c f m1,..., c f N In other words, the vector c f := R, such that c f j N x,m,j(y i ) = f(y i ), i = 1,..., m N. (2.4.6) ( c f m1,..., c f N) T R mn, where R mn denotes the (m N)-dimensional real space, is a solution to the matrix equation A m,n c f = f mn,

37 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.25 with f mn := (f(y 1 ),..., f(y mn )) T R mn, and where A m,n is the (m N) (m N) coecient matrix N x,m, m1 (y 1 ) N x,m, m2 (y 1 ) N x,m,n (y 1 ) N x,m, m1 (y 2 ) N x,m, m2 (y 2 ) N x,m,n (y 2 ) A m,n =.... N x,m, m1 (y mn ) N x,m, m2 (y mn ) N x,m,n (y mn ) (2.4.7) A necessary and sucient condition for the matrix A m,n to be invertible is given by the Schoenberg-Whitney theorem [44]: Theorem (Schoenberg-Whitney) The matrix A m,n in (2.4.7) of the linear system (2.4.6) is invertible if and only if N x,m, mi (y i ) 0, i = 1,..., m N. In other words, from the B-spline support property (2.3.10), the linear system (2.4.6) has a unique solution if and only if x mi y i x i, i = 1,..., m N, in which case the spline interpolant is given by (S m f)(x) = N j= m1 ( A 1 m,n f ) mn N j x,m,j(x), x [a, b]. (2.4.8) This gives an existence and uniqueness result for spline interpolation in terms of the normalized m th order B-splines, for any arbitrary order m. In general, the inverse matrix A 1 m,n is not banded, but a full matrix. This means that the value of (S m f)(x) in (2.4.8) at any x [a, b] depends on most, if not all, of the function values {f(y 1 ),..., f(y mn )}. The spline interpolant in (2.4.8) is therefore not a local interpolant. In Chapter 4, we will investigate the construction of local interpolation schemes in terms of B-splines. In some applications, it might be advantageous to not only interpolate function values f(y i ), i = 1,..., m N, as in (2.4.5) in Denition 2.4.1,

38 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p.26 but derivative values of f as well. If {r 1,..., r mn } is a sequence of m N non-negative integers such that r 1 r mn = m N, the m N interpolation conditions in (2.4.5) in Denition become (S m f) (l) (y i ) = f (l) (y i ), l = 1,..., r i ; i = 1,..., m N, called Hermite interpolation conditions. We will investigate spline interpolation operators with this type of interpolation conditions in Chapter 4. We note that, in practice, the l th order derivative values of the function f may also be approximated with the l th order divided dierences of f, with the divided dierences dened in (2.3.3)-(2.3.4). In this dissertation, we will also be interested in certain spline approximation operators, called quasi-interpolation operators, rst introduced by De Boor and Fix in [7]. Denition (Spline quasi-interpolation operator) The spline approximation operator Q m : C[a, b] S x,m with knot sequence x in (2.3.1) is called a quasi-interpolation operator if it reproduces polynomials of degree n for any non-negative integer n; that is, for polynomials p π n. (Q m p)(x) = p(x), x [a, b], (2.4.9) Again, since {N x,m,j : j = m 1,..., N} forms a basis for the spline space S x,m, we know that there exists a spline Q m f satisfying (2.4.9) if N (Q m f)(x) = c f j N x,m,j(x), x [a, b] for certain coecients j= m1 { } c f m1,..., c f N N j= m1 R, such that c p j N x,m,j(x) = p(x), p π n. We will investigate the construction of quasi-interpolation operators, with a local formulation in terms of the normalized m th order B-splines, in Chapter 3.

39 CHAPTER 2 Van der Walt, Maria, 2015, UMSL, p Continuous wavelet transform Let us consider a function ψ L 2 (R). If ψ satises the conditions ψ(x) 0, x ±, (2.5.1) and A p.v. ψ(x)dx = lim ψ(x)dx = 0, (2.5.2) A A where p.v. denotes the Cauchy principal value, then ψ is called a wavelet. From (2.5.2), we see that ψ oscillates (ψ has a wavy shape), while the condition (2.5.1) ensures that the function ψ dies down as x ±, so that the graph of ψ looks like a short wave or wavelet. Given a wavelet ψ, we can generate a whole family of wavelets through ψ b,a (x) := 1 ( ) x b a ψ, (2.5.3) a where b R and a > 0. The factor a is used to adjust the scale and length of the wavelet, while the support interval of ψ b,a can be shifted over the entire real axis by changing the value of b. There are many applications of wavelets in the literature. In this dissertation, we will mainly be interested in the continuous wavelet transform (CWT), where the wavelets ψ b,a (2.5.3) generated by ψ are used as integration kernel. Denition (Continuous wavelet transform) For a function f L 2 (R), the CWT W ψ f of f at the time-scale point (b, a) is dened as the inner product of f with the wavelet ψ b,a ; that is, (W ψ f)(b, a) = f, ψ b,a = 1 ( x b f(x)ψ a a ) dx. (2.5.4) The CWT is a time-frequency method used to analyze the time and frequency contents of a function f. In this regard, the wavelet ψ b,a is called a window function, and it is used to localize f in order to examine its time and frequency contents. This localization depends on the width of the window function. In general, the window width of a time-frequency window function is calculated as follows.