##
**Wavelets and operators. I: Wavelets.
(Ondelettes et opérateurs. I: Ondelettes.)**
*(French)*
Zbl 0694.41037

Actualités Mathématiques. Paris: Hermann, Éditeurs des Sciences et des Arts. xii, 215 p. sFr. 186.00 (1990).

This book is concerned with the basic theory of ondelettes \((=wavelets\)) which is useful in locating discontinuities or singularities of a given function and important in mechanics and engineering. The theory is now being developed and the book is an excellent introduction to it.

In Chapter I a theorem on whether a function \(f\) whose Fourier transform has a support contained in \([-T,T]\) can be decided by its sampling \(f(k\delta)\), \(k\in \mathbb Z\), or not is given. Next, the boundedness of a discrete Hilbert transform in \(\ell^ p(\mathbb Z)\) is proved. The final part of this chapter is a brief history of the theory of ondelettes. The study by A. Grossmann and J. Morlet is described concerning the expression of a function in the space \(\mathbb H^ 2(\mathbb R)\) which is the set of all functions \(f\) such that \(f(x+iy)\) is holomorphic in the demiplane \(y>0\) and \(\| f(\cdot +iy)\|_ 2\) is bounded for \(y>0\) by an integral of \(\psi_{(a,b)}(t)=a^{-1/2}\psi ((t-b)/a),\) \(a>0\), \(b\in\mathbb R\), where \(\psi (t)=(t+i)^{-2}\).

In Chapter II there is given the definition of a multi-resolution analysis of \(L^ 2(\mathbb R^ n)\) which is an increasing sequence \(V_ j\), \(j\in\mathbb Z\), of closed subspaces of \(L^ 2(\mathbb R^ n)\) satisfying

(1) \(\cap^{\infty}_{j=-\infty}V_ j=\{0\}\), \(\cup^{\infty}_{j=-\infty}V_ j\) is dense in \(L^ 2(\mathbb R^ n)\),

(2) \(\forall f\in L^ 2(\mathbb R^ n)\), \(\forall j\in\mathbb Z\) \(f(x)\in V_ j\Leftrightarrow f(2x)\in V_{j+1}\),

(3) \(\forall f\in L^ 2(\mathbb R^ n)\), \(\forall k\in\mathbb Z^ n\) \(f(x)\in V_ 0\Leftrightarrow f(x-k)\in V_ 0\),

(4) There exists a function \(g\in V_ 0\) such that \(g(x-k)\), \(k\in\mathbb Z^ n\) is a Riesz base of \(V_ 0\).

A multiresolution analysis is called \(r\)-regular if we can choose the function \(g(x)\) in (4) so that \(| g(x)| \leq C_ m(1+| x|)^{-m}\) for any multiindices \(\alpha\) such that \(| \alpha | \leq r\) and for any \(m\in\mathbb N\). Several examples are given. One of them is the space of splines of order \(r\) for \(n=1\) in which \(V_ 0\) is the set of all functions of class \(C^{r-1}\) whose restrictions to \([k,k+1[\), \(k\in\mathbb Z\), are polynomials of order \(\leq r\). The function \(g\) in (4) is \(\chi*\dots*\chi\) (\(r\)-times) in this example, where \(\chi\) is the characteristic function of the interval \([0,1]\).

It is shown that there exists a function \(\phi\) such that \(\phi(x-k)\), \(k\in\mathbb Z^ n\), forms an orthonormal base of \(V_ 0\). Let \(E_ j\) be the orthogonal projection onto \(V_ j\). \(E_ j\) has a kernel \(E_ j(x,y)=2^{nj}E(2^ jx,2^ jy)\) where \(E(x,y)=\sum \phi (x-k){\bar \phi}(y-k),\) and \(E_ jf(x)\) may be considered as a sampling of \(f\) at the lattice points \(\Gamma_ j=2^{-j}\mathbb Z^ n\). By definition \(E_ jf\to f\) in \(L^ 2(\mathbb R^ n)\) for \(f\in L^ 2(\mathbb R^ n)\). The same holds with \(L^ 2(\mathbb R^ n)\) replaced by the Sobolev space \(H^ s(\mathbb R^ n)\). A multiresolution analysis is also considered in \(L^ p(\mathbb R^ n)\) defining \(V_ j(p)\) appropriately starting from \(V_ j\). The inequality of Bernstein \(\| \partial^{\alpha}f\|_ p\leq C 2^{| \alpha | j}\| f\|_ p\) is established for \(f\in V_ j(p)\), \(| \alpha | \leq r\). The following remarkable inequality is proved: \(\int^{\infty}_{- \infty}E(x,y)y^{\alpha}\,dy=x^{\alpha}\) for \(| \alpha | \leq r\). It is also shown that \(E_ j\) is a pseudodifferential operator whose symbol has an explicit representation with the use of \(\phi\).

Chapter III. A function \(\psi(x)\) of a real variable is called an ondelette of class \(m\) if (a) \(\psi\) and its derivatives of order up to \(m\) belong to \(L^{\infty}(\mathbb R)\), (b) the functions in (a) are rapidly decreasing at infinity, (c) its moment of order up to \(m\) vanish, and (d) \(2^{j/2}\psi (2^ jx-k)\), \(j\in\mathbb Z\), \(k\in\mathbb Z\), is an orthonormal base of \(L^ 2(\mathbb R)\). By virtue of (d) it is possible to expand a function in \(L^ 2(\mathbb R)\) as a series of \(\psi_ I=2^{j/2}\psi (2^ jx-k),\) \(I=[k2^{-j},(k+1)2^{-j}[\). However, this expansion does not function for functions in \(L^ 1(\mathbb R)\) or \(L^{\infty}(\mathbb R)\). In order to avoid this difficulty another function \(\phi\) which is called the father of ondelettes (\(\psi\) is called the mother of ondelettes) is introduced. The function \(\phi\) has the properties (a),(b) above, \(\int^{\infty}_{-\infty}\phi (x)\,dx=1\), and \(\phi(x-k)\), \(k\in \mathbb Z\), \(\psi_ I(x)\), \(| I| \leq 1\), form an orthogonal base of \(L^ 2(\mathbb R)\). The closed subspace spanned by \(2^{j/2}\phi (2^ jx-k)\), \(k\in \mathbb Z\), is \(V_ j\), and that spanned by \(2^{j/2}\psi (2^ jx-k)\), \(k\in\mathbb Z\), is the orthogonal complement of \(V_ j\) in \(V_{j+1}\). This situation is the same in multidimensional cases. In Chapter III a method of constructing \(\psi\) for a given \(\phi\) is stated. Several examples are described in detail.

Chapter IV. Let \(\psi\) be a function in \(L^ 1(\mathbb R)\) whose integral over \(\mathbb R\) vanishes. A sufficient condition in order that \(2^{j/2}\psi (2^ jx-k)\), \(j\in \mathbb Z\), \(k\in\mathbb Z\), is an oblique structure \((=a\) frame) of \(L^ 2(\mathbb R)\) is given. It is stated that for \(\psi (x)=(x+i)^{-2}\) this set of functions constitute a frame of \(\mathbb H^ 2(\mathbb R)\).

In Chapter V the Hardy space \(H^ 1(\mathbb R^ n)\) is defined as the set of functions \(f\in L^ 1(\mathbb R^ n)\) such that the series of ondelettes \(\sum (f,\psi_{\lambda})\psi_{\lambda}\) is convergent unconditionally to \(f\). Ondelettes used here are regular and have compact supports. The definition turns out to be independent of the choice of ondelettes. One of the main theorems of this chapter is that the space of functions of bounded mean oscillation is the dual space of \(H^ 1(\mathbb R^ n).\)

In Chapter VI criteria are stated whether a given function or distribution belongs to various function spaces such as \(L^ p\) spaces, Sobolev spaces, Hardy spaces, the space of Hölder continuous functions, the algebra of Beurling, the algebra of Wiener, Besov spaces, etc.

In Chapter I a theorem on whether a function \(f\) whose Fourier transform has a support contained in \([-T,T]\) can be decided by its sampling \(f(k\delta)\), \(k\in \mathbb Z\), or not is given. Next, the boundedness of a discrete Hilbert transform in \(\ell^ p(\mathbb Z)\) is proved. The final part of this chapter is a brief history of the theory of ondelettes. The study by A. Grossmann and J. Morlet is described concerning the expression of a function in the space \(\mathbb H^ 2(\mathbb R)\) which is the set of all functions \(f\) such that \(f(x+iy)\) is holomorphic in the demiplane \(y>0\) and \(\| f(\cdot +iy)\|_ 2\) is bounded for \(y>0\) by an integral of \(\psi_{(a,b)}(t)=a^{-1/2}\psi ((t-b)/a),\) \(a>0\), \(b\in\mathbb R\), where \(\psi (t)=(t+i)^{-2}\).

In Chapter II there is given the definition of a multi-resolution analysis of \(L^ 2(\mathbb R^ n)\) which is an increasing sequence \(V_ j\), \(j\in\mathbb Z\), of closed subspaces of \(L^ 2(\mathbb R^ n)\) satisfying

(1) \(\cap^{\infty}_{j=-\infty}V_ j=\{0\}\), \(\cup^{\infty}_{j=-\infty}V_ j\) is dense in \(L^ 2(\mathbb R^ n)\),

(2) \(\forall f\in L^ 2(\mathbb R^ n)\), \(\forall j\in\mathbb Z\) \(f(x)\in V_ j\Leftrightarrow f(2x)\in V_{j+1}\),

(3) \(\forall f\in L^ 2(\mathbb R^ n)\), \(\forall k\in\mathbb Z^ n\) \(f(x)\in V_ 0\Leftrightarrow f(x-k)\in V_ 0\),

(4) There exists a function \(g\in V_ 0\) such that \(g(x-k)\), \(k\in\mathbb Z^ n\) is a Riesz base of \(V_ 0\).

A multiresolution analysis is called \(r\)-regular if we can choose the function \(g(x)\) in (4) so that \(| g(x)| \leq C_ m(1+| x|)^{-m}\) for any multiindices \(\alpha\) such that \(| \alpha | \leq r\) and for any \(m\in\mathbb N\). Several examples are given. One of them is the space of splines of order \(r\) for \(n=1\) in which \(V_ 0\) is the set of all functions of class \(C^{r-1}\) whose restrictions to \([k,k+1[\), \(k\in\mathbb Z\), are polynomials of order \(\leq r\). The function \(g\) in (4) is \(\chi*\dots*\chi\) (\(r\)-times) in this example, where \(\chi\) is the characteristic function of the interval \([0,1]\).

It is shown that there exists a function \(\phi\) such that \(\phi(x-k)\), \(k\in\mathbb Z^ n\), forms an orthonormal base of \(V_ 0\). Let \(E_ j\) be the orthogonal projection onto \(V_ j\). \(E_ j\) has a kernel \(E_ j(x,y)=2^{nj}E(2^ jx,2^ jy)\) where \(E(x,y)=\sum \phi (x-k){\bar \phi}(y-k),\) and \(E_ jf(x)\) may be considered as a sampling of \(f\) at the lattice points \(\Gamma_ j=2^{-j}\mathbb Z^ n\). By definition \(E_ jf\to f\) in \(L^ 2(\mathbb R^ n)\) for \(f\in L^ 2(\mathbb R^ n)\). The same holds with \(L^ 2(\mathbb R^ n)\) replaced by the Sobolev space \(H^ s(\mathbb R^ n)\). A multiresolution analysis is also considered in \(L^ p(\mathbb R^ n)\) defining \(V_ j(p)\) appropriately starting from \(V_ j\). The inequality of Bernstein \(\| \partial^{\alpha}f\|_ p\leq C 2^{| \alpha | j}\| f\|_ p\) is established for \(f\in V_ j(p)\), \(| \alpha | \leq r\). The following remarkable inequality is proved: \(\int^{\infty}_{- \infty}E(x,y)y^{\alpha}\,dy=x^{\alpha}\) for \(| \alpha | \leq r\). It is also shown that \(E_ j\) is a pseudodifferential operator whose symbol has an explicit representation with the use of \(\phi\).

Chapter III. A function \(\psi(x)\) of a real variable is called an ondelette of class \(m\) if (a) \(\psi\) and its derivatives of order up to \(m\) belong to \(L^{\infty}(\mathbb R)\), (b) the functions in (a) are rapidly decreasing at infinity, (c) its moment of order up to \(m\) vanish, and (d) \(2^{j/2}\psi (2^ jx-k)\), \(j\in\mathbb Z\), \(k\in\mathbb Z\), is an orthonormal base of \(L^ 2(\mathbb R)\). By virtue of (d) it is possible to expand a function in \(L^ 2(\mathbb R)\) as a series of \(\psi_ I=2^{j/2}\psi (2^ jx-k),\) \(I=[k2^{-j},(k+1)2^{-j}[\). However, this expansion does not function for functions in \(L^ 1(\mathbb R)\) or \(L^{\infty}(\mathbb R)\). In order to avoid this difficulty another function \(\phi\) which is called the father of ondelettes (\(\psi\) is called the mother of ondelettes) is introduced. The function \(\phi\) has the properties (a),(b) above, \(\int^{\infty}_{-\infty}\phi (x)\,dx=1\), and \(\phi(x-k)\), \(k\in \mathbb Z\), \(\psi_ I(x)\), \(| I| \leq 1\), form an orthogonal base of \(L^ 2(\mathbb R)\). The closed subspace spanned by \(2^{j/2}\phi (2^ jx-k)\), \(k\in \mathbb Z\), is \(V_ j\), and that spanned by \(2^{j/2}\psi (2^ jx-k)\), \(k\in\mathbb Z\), is the orthogonal complement of \(V_ j\) in \(V_{j+1}\). This situation is the same in multidimensional cases. In Chapter III a method of constructing \(\psi\) for a given \(\phi\) is stated. Several examples are described in detail.

Chapter IV. Let \(\psi\) be a function in \(L^ 1(\mathbb R)\) whose integral over \(\mathbb R\) vanishes. A sufficient condition in order that \(2^{j/2}\psi (2^ jx-k)\), \(j\in \mathbb Z\), \(k\in\mathbb Z\), is an oblique structure \((=a\) frame) of \(L^ 2(\mathbb R)\) is given. It is stated that for \(\psi (x)=(x+i)^{-2}\) this set of functions constitute a frame of \(\mathbb H^ 2(\mathbb R)\).

In Chapter V the Hardy space \(H^ 1(\mathbb R^ n)\) is defined as the set of functions \(f\in L^ 1(\mathbb R^ n)\) such that the series of ondelettes \(\sum (f,\psi_{\lambda})\psi_{\lambda}\) is convergent unconditionally to \(f\). Ondelettes used here are regular and have compact supports. The definition turns out to be independent of the choice of ondelettes. One of the main theorems of this chapter is that the space of functions of bounded mean oscillation is the dual space of \(H^ 1(\mathbb R^ n).\)

In Chapter VI criteria are stated whether a given function or distribution belongs to various function spaces such as \(L^ p\) spaces, Sobolev spaces, Hardy spaces, the space of Hölder continuous functions, the algebra of Beurling, the algebra of Wiener, Besov spaces, etc.

Reviewer: H. Tanabe (Toyonaka)

### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |