Wavelet Theory and Harmonic Analysis in Applied Sciences

The purpose of this note is to describe some results of real analysis related with the Monge-Ampère equation that are proved in [1] and to show its application to the boundedness of certain singular integrals.

The space BMO of those real functions defined on IR ⁿ for which the mean oscillation over cubes is bounded, appears in the pioneer works of J. Moser [13] and John-Nirenberg [9] in the early sixties as a tool for the study of regularity of weak solutions of elliptic and parabolic differential equations. Their main result, known today as John-Nirenberg Theorem, provides a characterization of BMO in terms of the exponential decay of the distribution function on each cube. Although the depth of this result, the space BMO only became well known in harmonic analysis after the celebrated Fefferman-Stein theorem of duality for the Hardy spaces: BMO is the dual of the Hardy space H ¹. Since H ¹ was already known to be the good substitute of L ¹ for many questions in analysis, the space BMO was realized as the natural substitute of L ^∞ in the scale of the Lebesgue spaces. Also due to Fefferman and Stein is the Littlewood-Paley type characterization of BMO in terms of the derivatives of the harmonic extension and Carleson measures (see for example [16]). This result has a discrete version: the Lemarié-Meyer characterization of BMO using wavelets, [10].

The decimated discrete wavelet transform (DWT) gives us a powerful tool in many signal processing applications. It provides stable time—scale representations for any square integrable function as well as a suitable structure of the available information. In connection with this choice, well known families of biorthogonal or orthogonal wavelets are available.

The goal of this chapter is to introduce the new concept of oblique wavelet and multiwavelet bases described in [1]. These wavelet bases contain, as special cases, the orthogonal, semiorthogonal, and biorthogonal theory of wavelets and multiwavelets [6, 9, 12, 13, 22]. The main advantage of oblique wavelets is that they give more flexibility in choosing wavelet bases, without compromising the fast filter bank implementation algorithms.

Orthogonal bases provide a classical method to represent an element of a Hilbert space in terms of simpler ones. A Riesz basis is the image of an orthogonal basis under an invertible continuous linear mapping. In practical applications, when using orthogonal bases one introduces small truncation errors. As we shall see in the sequel, perturbing a Riesz bases (in particular, an orthogonal basis) by a small amount, yields another Riesz basis. This is one of the reasons that motivate the study of such bases.

When solving differential equations by means of a Galerkin approach, the approximating spaces are not only supposed to have good approximation properties, but also they must allow easy and fast computations. In addition, if the goal is the development of a multilevel method to detect and follow local singularities, or a multigrid scheme to solve the resulting linear systems, then hierarchical bases are necessary. As an example, we mention the finite element bases which have been widely used over the last three decades.

Since the two last decades, due to the development of electronic devices and computers, the observation of the fine structure of the electric cardiac signal through non-invasive measurements is more and more a realistic challenge. The technique allowing such measurements is called High Resolution Electrocardiography (HRECG) to make a difference with the classical Electrocardiography (ECG). In classical ECG, the paper records obtained from standardized amplificators, analogic filters and leads configurations enable a rapid observation of the macroscopic behaviour of heart activity: heart rate, rhythm, conduction intervals, morphological aspect of the ECG waves (P, Q, R, S, T) associated to the depolarization and repolarization of the auricles and ventricles. The dynamic of the input signal is a few millivolts.

The study of the rhythmic and nonrhythmic oscillations of the arterial blood pressure (ABP) was first described by Hales [19] two centuries ago. Twenty seven years later, Albrecht von Haller described fluctuations of the cardiac rhythm. In 1847, Carl Ludwig [31], by mean of continuous recordings of physiological events in horses and dogs, was able to graph the rhythmic fluctuations of the ABP. The motivation behind these experiments was to clarify the spontaneous behavior and to overcome the lack of interpretation for these oscillations. It is interesting to remind the first description relating an evident correlation with respiratory fluctuations, mainly because the ease of its visualization, both in laboratory animals and humans beings.

The objective of engineering study is basically, the design and operation of useful systems for humankind development. Historically, this work has been done by trial-and-error procedures. The development of“theory”has always been related to“experiments”that deny or confirm hypothesis. The incorporation of the acquired knowledge to the design of systems, generally using mathematical language, leads to“models”that approach real situations. With these models, a group of equations that relates system variables, it is possible to analyze system behavior and to plan operational changes. In order to conceive a model it is necessary to define its application. In the present work models that emulate a system dynamic behavior that could be used in the design and in the operation of controllers, will be developed. This objective imposes some desirable characteristics to this model, such as: system representation in a wide range of operation, easy implementation and fast prediction of variables of interest.

When modeling steady potential flow problems in polygonal non-convex domains, it is expected to find singularities being develop at the corners. The asymptotic behavior of these singularities in reentering corners depends on the boundary data, on the corner angle and on the permittivity constants associated with the potential equation when modeling inhomogeneous media.

In this work we treat an estimation problem arising from the study of wave propagation in solids. Before dealing with the estimation problem we pay special attention to the physical model, formulated in the space-frequency domain.

The magnetotelluric method consists on inferring the earth’s electric conductivity distribution from measurements of natural electric and magnetic fields on the earth’s surface. Information about characteristics of the earth’s interior may be inferred from the conductivity models by exploiting relationships between electrical conductivity and physical properties of rocks such as composition, texture, temperature, porosity, and fluid content. The magnetotelluric method has been employed in oil exploration in regions where the seismic reflection procedure technique is very expensive or not possible to perform, as in volcanic regions. In the high-frequency range, it has been used to detect groundwater reservoirs and mineral deposits.