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Preface xi
Chapter 1. Inner Product Spaces (Pre-Hilbert) 1
1.1. Real and complex inner products 1
1.2. The norm associated with an inner product and normed vector spaces 6
1.2.1. The parallelogram law and the polarization formula 9
1.3. Orthogonal and orthonormal families in inner product spaces 11
1.4. Generalized Pythagorean theorem 11
1.5. Orthogonality and linear independence 13
1.6. Orthogonal projection in inner product spaces 15
1.7. Existence of an orthonormal basis: the Gram-Schmidt process 19
1.8. Fundamental properties of orthonormal and orthogonal bases 20
1.9. Summary 28
Chapter 2. The Discrete Fourier Transform and its Applications to Signal and Image Processing 31
2.1. The space l2(ZN) and its canonical basis 31
2.1.1. The orthogonal basis of complex exponentials in l2(ZN) 34
2.2. The orthonormal Fourier basis of l2(ZN) 38
2.3. The orthogonal Fourier basis of l2(ZN) 40
2.4. Fourier coefficients and the discrete Fourier transform 41
2.4.1. The inverse discrete Fourier transform 44
2.4.2. Definition of the DFT and the IDFT with the orthonormal Fourier basis 46
2.4.3. The real (orthonormal) Fourier basis 47
2.5. Matrix interpretation of the DFT and the IDFT 48
2.5.1. The fast Fourier transform 51
2.6. The Fourier transform in signal processing 51
2.6.1. Synthesis formula for 1D signals: decomposition on the harmonic basis 51
2.6.2. Signification of Fourier coefficients and spectrums of a 1D signal 53
2.6.3. The synthesis formula and Fourier coefficients of the unit pulse 54
2.6.4. High and low frequencies in the synthesis formula 55
2.6.5. Signal filtering in frequency representation 59
2.6.6. The multiplication operator and its diagonal matrix representation 60
2.6.7. The Fourier multiplier operator 60
2.7. Properties of the DFT 61
2.7.1. Periodicity of ¿ and ? 62
2.7.2. DFT and shift 63
2.7.3. DFT and conjugation 67
2.7.4. DFT and convolution 68
2.8. The DFT and stationary operators 73
2.8.1. The DFT and the diagonalization of stationary operators 75
2.8.2. Circulant matrices 77
2.8.3. Exhaustive characterization of stationary operators 78
2.8.4. High-pass, low-pass and band-pass filters 82
2.8.5. Characterizing stationary operators using shift operators 83
2.8.6. Frequency analysis of first and second derivation operators (discrete case) 84
2.9. The two-dimensional discrete Fourier transform (2D DFT) 88
2.9.1. Matrix representation of the 2D DFT: Kronecker product versus iteration of two 1D DFTs 91
2.9.2. Properties of the 2D DFT 93
2.9.3. 2D DFT and stationary operators 95
2.9.4. Gradient and Laplace operators and their action on digital images 97
2.9.5. Visualization of the amplitude spectrum in 2D 97
2.9.6. Filtering: an example of digital image filtering in a Fourier space 100
2.10. Summary 102
Chapter 3. Lebesgue's Measure and Integration Theory 105
3.1. Riemann versus Lebesgue 105
3.2. s-algebra, measurable space, measures and measured spaces 106
3.3. Measurable functions and almost-everywhere properties (a.e) 108
3.4. Integrable functions and Lebesgue integrals 109
3.5. Characterization of the Lebesgue measure on R and sets with a null Lebesgue measure 111
3.6. Three theorems for limit operations in integration theory 113
3.7. Summary 114
Chapter 4. Banach Spaces and Hilbert Spaces 115
4.1. Metric topology of inner product spaces 116
4.2. Continuity of fundamental operations in inner product spaces 120
4.2.1. Equivalence of separated topologies in finite-dimension vector spaces 128
4.3. Cauchy sequences and completeness: Banach and Hilbert 129
4.3.1. Completeness of vector spaces 133
4.3.2. Characterizing the completeness of normed vector spaces using series 135
4.3.3. Banach fixed-point theorem 139
4.4. Remarkable examples of Banach and Hilbert spaces 145
4.4.2. L¿ and l¿ spaces 156
4.4.3. Inclusion relationships between lp spaces 161
4.4.4. Inclusion relationships between Lp spaces 163
4.4.5. Density theorems in Lp(X,A,µ) 165
4.5. Summary 169
Chapter 5. The Geometric Structure of Hilbert Spaces 171
5.1. The orthogonal complement in a Hilbert space and its properties 171
5.2. Projection onto closed convex sets: theorem and consequences 174
5.2.1. Characterization of closed vector subspaces in Hilbert spaces 180
5.3. Polar and bipolar subsets of a Hilbert space 182
5.4. The (orthogonal) projection theorem in a Hilbert space 185
5.5. Orthonormal systems and Hilbert bases 188
5.5.1. Bessel's inequality and Fourier coefficients 189
5.5.2. The Fischer-Riesz theorem 192
5.5.3. Characterizations of a Hilbert basis (or complete orthonormal system) 194
5.5.4. Isomorphisms between Hilbert spaces 199
5.5.5. l2(N,K) as the prototype of separable Hilbert spaces of infinite dimension 201
5.6. The Fourier Hilbert basis in L2 202
5.6.1. L2[-p, p] or L2[0, 2p] 202
5.6.2. L2(T) 204
5.6.3. L2[a, b] 205
5.6.4. Real Fourier series 206
5.6.5. Pointwise convergence of the real Fourier series: Dirichlet's theorem 212
5.6.6. The Gibbs phenomenon and Cesàro summation 214
5.6.7. Speed of convergence to 0 of Fourier coefficients 214
5.6.8. Fourier transform in L2(T) and shift 218
5.7. Summary 219
Chapter 6. Bounded Linear Operators in Hilbert Spaces 221
6.1. Fundamental properties of bounded linear operators between normed vector spaces 223
6.1.1. Continuity of linear operators defined on a finite-dimensional normed vector space 226
6.2. The operator norm, convergence of operator sequences and Banach algebras 227
6.2.1. A classical example of a non-bounded linear operator on a vector space of infinite dimension 238
6.3. Invertibility of linear operators 239
6.4. The dual of a Hilbert space and the Riesz representation theorem 244
6.4.1. The scalar product induced on the dual of a Hilbert space 249
6.5. Bilinear forms, sesquilinear forms and associated quadratic forms 249
6.5.1. The Lax-Milgram theorem and its consequences 257
6.6. The adjoint operator: presentation and properties 261
6.7. Orthogonal projection operators in a Hilbert space 269
6.7.1. Bounded multiplication operators and their relation to orthogonal projectors 278
6.7.2. Geometric realization of orthogonal projection operators via orthonormal systems 280
6.8. Isometric and unitary operators 286
6.8.1. Characterizations of isometric and unitary operators 288
6.8.2. Relationship between isometric and unitary operators and orthonormal systems 293
6.9. The Fourier transform on S(Rn), L1(Rn) and L2(Rn) 296
6.9.1. The invariance of the Schwartz space with respect to the Fourier transform 296
6.9.2. Extension of the Fourier transform of S(Rn) to L1(Rn): the Riemann-Lebesgue theorem 301
6.9.3. Extension of the Fourier transform to a unitary operator on L2(Rn): the Fourier-Plancherel transform 302
6.9.4. Relationship between the Fourier-Plancherel transform and the Hermitian Hilbert basis 305
6.9.5. The Fourier transform and convolution 306
6.9.6. Convolution and Fourier transforms in L2: localization of the Fourier transform 309
6.10. The Nyquist-Shannon sampling theorem 310
6.10.1. The Nyquist frequency: aliasing and oversampling 312
6.11. Application of the Fourier transform to solve ordinary and partial differential equations 313
6.11.1. Solving an ordinary differential equation using the Fourier transform 313
6.11.2. The Fourier transform and partial differential equations 315
6.11.3. Solving the partial differential equation for heat propagation using the Fourier transform 316
6.12. Summary 319
Appendix 1: Quotient Space 323
Appendix 2: The Transpose (or Dual) of a Linear Operator 329
Appendix 3: Uniform, Strong and Weak Convergence 331
References 335
Index 337
This chapter will focus on inner product spaces, that is, vector spaces with a scalar product, specifically those of finite dimension.
In real Euclidean spaces R2 and R3, the inner product of two vectors v, w is defined as the real number:
where ? is the smallest angle between v and w and ? ? represents the norm (or the magnitude) of the vectors.
Using the inner product, it is possible to define the orthogonal projection of vector v in the direction defined by vector w. A distinction must be made between:
where is the unit vector in the direction of w. Evidently, the roles of v and w can be reversed.
The absolute value of the scalar projection measures the "similarity" of the directions of two vectors. To understand this concept, consider two remarkable relative positions between v and w:
When the position of v relative to w falls somewhere in the interval between the two vectors described above, the absolute value of the scalar projection of v in the direction of w falls between 0 and ?v?; this explains its use to measure the similarity of the direction of vectors.
In this book, we shall consider vector spaces which are far more complex than R2 and R3, and the measure of vector similarity obtained through projection supplies crucial information concerning the coherence of directions.
Before we can obtain this information, we must begin by moving from Euclidean spaces R2 and R3 to abstract vector spaces. The general definition of an inner product and an orthogonal projection in these spaces may be seen as an extension of the previous definitions, permitting their application to spaces in which our representation of vectors is no longer applicable.
Geometric properties, which can only be apprehended and, notably, visualized in two or three dimensions, must be replaced by a set of algebraic properties which can be used in any dimension.
Evidently, these algebraic properties must be necessary and sufficient to characterize the inner product of vectors in a plane or in real space. This approach, in which we generalize concepts which are "intuitive" in two or three dimensions, is a classic approach in mathematics.
In this chapter, the symbol V will be used to describe a vector space defined over the field , where is either R or and is of finite dimension n < +8. field contains the scalars used to construct linear combinations between vectors in V . Note that two finite dimensional vector spaces are isomorphic if and only if they are of the same dimension. Furthermore, if we establish a basis B = (b1, . . . , bn) for V , an isomorphism between V and n can be constructed as follows:
that is, I associates each v ? V with the vector of n given by the scalar components of v in relation to the established basis B. Since I is an isomorphism, it follows that n is the prototype of all vector spaces of dimension n over a field .
DEFINITION 1.1.- Let V be a vector space defined over a field .
A -form over V is an application defined over V × V with values in , that is:
DEFINITION 1.2.- Let V be a real vector space. A couple (V, <, >) is said to be a real inner product space (or a real pre-Hilbert space) if the form <, > is:
1) bilinear, i.e.1 linear in relation to each argument (the other being fixed):
and:
2) symmetrical: <v, w> = <w, v>, ?v, w ? V ;
3) defined: <v, v> = 0 v = 0V , the null vector of the vector space V ;
4) positive: <v, v> > 0 ?v ? V , v ? 0V .
Upon reflection, we see that, for a real form over V , the symmetry and bilinearity requirements are equivalent to requiring symmetry and linearity on the left-hand side, that is:
The simplest and most important example of a real inner product is the canonical inner product, defined as follows: let v = (v1, v2, . . . , vn), w = (w1, w2, . . . , wn) be two vectors in Rn written with their components in relation to any given, but fixed, basis in Rn. The canonical inner product of v and w is:
where vt and wt in the final equations are the transposed vectors of v and w, giving us the matrix product of a line vector (treated as a 1 × n matrix) and a column vector (treated as an n × 1 matrix).
The extension of these definitions to complex vector spaces is not particularly straightforward. First, note that if V is a complex vector space, then there is no bilinear and definite-positive transformation over V × V . In this case, any vector v ? V would give the following:
As we shall see, the property of positivity is essential in order to define a norm (and thus a distance, and by extension, a topology) from a complex inner product. To obtain an algebraic structure for complex scalar products which remains compatible with a topological structure, we are therefore forced to abandon the notion of bilinearity, and to search for an alternative.
We could consider antilinearity2, i.e.
But it has the same problem as bilinearity, <iv, iv> = (-i)(-i)<v, v> = i2<v, v> = -<v, v>2 ? 0.
A simple analysis shows that, in order to avoid losing the positivity, it is sufficient to request the linearity with respect to one variable and the antilinearity with respect to the other. This property is called sesquilinearity3.
The choice of the linear and antilinear variable is entirely arbitrary.
By convention, the antilinear component is placed on the right-hand side in mathematics, but on the left-hand side in physics.
We have chosen to adopt the mathematical convention here, i.e. <av, ßw> = a߯<v, w>.
Next, it is important to note that sesquilinearity and symmetry are incompatible: if both properties were verified, then <v, aw> = <v, w>, and also <v, aw> = <aw, v> = a<w, v> = a<v, w>. Thus, <v, aw> = <v, w> = a<v, w> which can only be verified if a ? R.
Thus <, > cannot be both sesquilinear and symmetrical when working with vectors belonging to a complex vector space.
The example shown above demonstrates that, instead of symmetry, the property which must be verified for every vector pair v, w is , that is, changing the order of the vectors in <, > must be equivalent to complex conjugation.
A transform which verifies this property is said to be Hermitian4.
These observations provide full justification for Definition 1.3.
DEFINITION 1.3.- Let V be a complex vector space. The pair (V, <, >) is said to be a complex inner product space (or a complex pre-Hilbert space) if <, > is a complex form which is:
1) sesquilinear:
? v1, v2, w1, w2 ? V , and:
? a, ß ? , ? v, w ? V ;
2) Hermitian: , ?v, w ? V ;
3) definite: <v, v> = 0 v = 0V , the null vector of the vector space V;
As in the case of the canonical inner product, for a complex form over V , the symmetry and sesquilinearity requirement is equivalent to requiring the Hermitian property and linearity on the left-hand side; if these properties are verified, then:
Considering the sum of n, rather than two, vectors, sesquilinearity is represented by the following...
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