**Ingrid Daubechies**' parents were Marcel and Simone Daubechies. Both are now retired but Marcel was a civil mining engineer while Simone was criminologist with a great interest in history which she went back to university to study as an undergraduate after she retired.

Ingrid's father encouraged her to take an interest in science while she was studying at school. However, even as a child an interest in mathematics was evident. She has written [4]:-

I was always interested in how things worked and how to make things. For instance, I really like weaving and pottery, and I have liked this kind of craft pursuit since my childhood. But I also was interested in seeing how machinery worked, or in why certain mathematical things were true(like the fact that a number is divisible by nine if, when you add all its digits together, you get another number divisible by9- try it with73512and8577, both multiples of9; there is no rule that is quite as simple for divisibility by7, say).

In [4] Daubechies recalls other memories of her childhood:-

When I was eight or nine, the thing I liked best when playing with my dolls was to sew clothes for them. I liked trying to make patterns that would fit them well - it was fascinating to me that by putting together flat pieces of fabric one could make something that was not flat at all, but followed curved surfaces. Around the same time, when I couldn't fall asleep at night, I would compute the powers of2in my head:1, 2, 4, 8, 16, ... (multiplying by2every time). The numbers became very large very quickly but I would keep going quite a while. It was fascinating, again, to see how fast these numbers grew.

After her school education in Belgium, Daubechies entered the Free University Brussels to read for a degree in physics. Many of the mathematicians in this archive have started out as mathematicians and have later come to apply their mathematical skills in other scientific disciplines. Daubechies, however, is one of the few who progressed in the opposite direction, starting off by training in physics. She obtained her Bachelor's degree in Physics in 1975 and continued to undertake research at the Free University Brussels for a doctorate in physics.

From the time that she was awarded a Bachelor's degree, Daubechies became a Research Assistant in the Department for Theoretical Physics at the Free University Brussels. She held this post from 1975 until 1984 and, midway through this period, in 1980 she was awarded her Ph.D. in physics for a thesis entitled *Representation of quantum mechanical operators by kernels on Hilbert spaces of analytic functions*. By this time she had already published quite a number of papers, having written around ten articles. In 1978 *An application of hyperdifferential operators to holomorphic quantization* appeared, then a number of papers written jointly with Dirk Aerts: *A characterization of subsystems in physics; Physical justification for using the tensor product to describe two quantum systems as one joint system; A mathematical condition for a sublattice of a propositional system to represent a physical subsystem, with a physical interpretation*; and *A connection between propositional systems in Hilbert spaces and von Neumann algebras*. In a review of the third of these four papers E G Beltrametti explained the context of both this work and the second of these joint papers:-

One of the basic rules of Hilbert space quantum mechanics is that when two physical systems, sayS_{1}and S, are viewed as the pieces of a compound system S, then the Hilbert space to be associated to S is the tensor product of the Hilbert spaces H_{1}and H associated toS_{1}and S. This rule found little justification in the traditional logico-algebraic approach to quantum mechanics. The authors make a significant contribution to its justification.

In 1981 Daubechies went to the United States, although she continued to hold her Research Assistant position at the Free University Brussels, spending two years there undertaking postdoctoral work. Returning to Belgium she was then appointed to the tenured position of Assistant Professor in the Department for Theoretical Physics at the Free University Brussels in 1984. In fact she received her first major prize in 1984 when she was awarded the Louis Empain Prize for Physics. This prestigious prize is awarded once every five years to a Belgium scientist on the basis of work done before age 29.

In 1987 Daubechies returned to the United States to take up the post of Technical Staff Member at the Mathematics Research Center of AT&T Bell Laboratories, Murray Hill, New Jersey. In the same year she married A Robert Calderbank, who is also a mathematician.

In addition to the events we have mentioned, the year 1987 was an important one for Daubechies from a mathematical point of view. Let us set the scene and put her work in context. In the early 1980s J Morlet had discovered new way to represent geophysical signals. Rather than using Fourier transform methods to analyse signals he had the intuitive idea of using wavelets and later, in collaboration with Alex Grossmann, he put his intuition on a firm mathematical basis by introducing the continuous wavelet transform. In about 1985 Daubechies, in collaboration with Yves Meyer and Alex Grossmann, introduced a discrete approach which enabled functions to be reconstructed from a discrete set of values. It was a breakthrough by Daubechies in 1987, when she constructed compactly supported continuous wavelets, which led to many important applications. Daubechies believes that she is considered a mathematician because these applications are outside physics:-

... even as a physicist, my work was very theoretical, very mathematical. I became interested in applications of mathematics outside physics(especially in engineering), and that is how I am now considered a mathematician.

Although she still remained on the staff at the Bell Laboratories, Daubechies took leave to take university posts. In 1990 she spent six months at the University of Michigan, and in the following year she was appointed Professor in the Mathematics Department at Rutgers University, spending two years there. While at Rutgers, she published *Ten lectures on wavelets* in 1992. This important book led to her being awarded the Steele Prize for mathematical exposition by the American Mathematical Society in 1994. The citation for the award reads:-

The concept of wavelets has its origins in many fields, and part of the accomplishment of Daubechies is finding those places where the concept arose and showing how all the approaches relate to one another. The use of wavelets as an analytical tool is like Fourier analysis - simple and yet very powerful. In fact, wavelets are an extension of Fourier analysis to the case of localization in both frequency and space. And like Fourier analysis, it has both a theoretical side and practical importance....

Ian Stewart has described a number of applications of wavelet analysis. Here are two such descriptions:-

Perhaps the best-known application of wavelet analysis to date derived from the U.S. FBI's decision in1993to use a wavelet transform for encoding digitised fingerprint records. A wavelet transform occupies less computer memory than conventional methods for image storage, and its use was predicted to reduce the amount of computer memory needed for fingerprint records by93 %. ...

In the past two decades, medical centres had come to employ various kinds of scanner-based imaging systems, such as computed tomography and magnetic resonance imaging, that use computers to assemble the digitised data collected by the scanner into two- or three-dimensional pictures of the body's internal structures. ... a poor digitised image can be smoothed and cleaned up by taking a wavelet transform of it, removing unwanted components, and "detransforming" the wavelet representation to yield an image again. The method reduced the time of the patient's exposure to the radiation involved in the scanning process and thus made the imaging technique cheaper, quicker, and safer.

In January 1992 Daubechies gave a lecture *Wavelets making waves in mathematics and engineering* to a joint AMS-MAA meeting in Baltimore, Maryland. The lecture was published on a videocassette by the American Mathematical Society. G Walter's review of the video tells us much about Daubechies' style as a lecturer:-

Most videos of mathematicians giving lectures are not very successful. They are trying to convey ideas but the camera focuses our attention on the appearance of the speaker and his or her idiosyncrasies, for example a bright tie or a habit of pinching the nose. In this video of Ingrid Daubechies' lecture on wavelets, the reverse is true. Her manner of illustrating dilations and translations by stretching her arms contributes to an understanding of the principal concepts of wavelet theory. Furthermore, her obvious enthusiasm for the subject helps focus our attention on it and not on her, and contributes to a very successful video.

Daubechies was a fellow of the John D and Catherine T MacArthur Foundation from 1992 to 1997 and was elected to the American Academy of Arts and Science in 1993. In the following year Daubechies left Bell Laboratories and took up her present position of Professor in the Mathematics Department and the Program in Applied and Computational Mathematics at Princeton University. Between 1997 to 2001 she was Director of the Program in Applied and Computational Mathematics. The American Mathematical Society further honoured Daubechies in 1997 with the award of the Ruth Lyttle Satter Prize in Mathematics for:-

... her deep and beautiful analysis of wavelets and their applications.

The National Academy of Sciences elected Daubechies a Member in 1998, then made their four yearly Award in Mathematics to her in 2000:-

... for fundamental discoveries on wavelets and wavelet expansions and for her role in making wavelets methods a practical basic tool of applied mathematics.

She has received many honours for her remarkable achievements other than those we have mentioned above. For example she was elected a Fellow of The Institute of Electrical and Electronics Engineers in 1998 and, in the same year, was awarded their Information Theory Society Golden Jubilee Award for Technological Innovation. In 1999 she was elected to the Royal Netherlands Academy of Arts and Sciences, and in 2000 she received numerous honours such as an honorary degree from the Université Libre de Bruxelles and the Eduard Rhein Foundation 2000 Basic Research Award.

Daubechies was awarded an honorary degree by the Université Polytechnique Fédérale, Lausanne, Switzerland (2001), the Université Pierre et Marie Curie, Paris, France (2005), the Universita degli Studi di Genova, Genoa, Italy (2006), the Universiteit Hasselt, Belgium (2008) and Oxford University (2013). She was awarded the Gold Medal of the Flemish Royal Academy of Arts and Sciences, Belgium (2005), and the ICIAM Pioneer Prize (2008). In 2004 she was named William R Kenan, Jr. Professor at Princeton University.

**Article by:** *J J O'Connor* and *E F Robertson*