Comparison of the values obtained by the Taylor series and by Daubechies wavelets.

## Abstract

This chapter aims to use Daubechies’ wavelets as basis functions to generate analytical functions, thus being able to rewrite the Taylor series using these wavelets. This makes it possible to analyze functions with a high degree of complexity, in problems that require a high degree of precision in their solution. Wavelet analysis can be applied to practical problems that require a high degree of precision, for example, in the study and analysis of electromagnetic propagation in optical fibers, solutions of differential equations involving engineering problems, in the transmission of WiFi signals, in the treatment and analysis of biomedical images, detection of oil sources through the study of seismic signals.

### Keywords

- wavelets
- Daubechies
- analytical functions
- basis functions
- Taylor series

## 1. Introduction

Wavelets [1] were born from the need to generate functions, especially those that present singularities, high gradients, discontinuities both in the time domain and in the frequency domain. Wavelets enable the high-resolution analysis of functions with these characteristics. An example of a problem that occurs when generating functions with a Fourier base is the Gibbs phenomenon. Such a phenomenon occurs because there is no way to represent functions that present discontinuities, even adding more elements in the base that will generate the function. A characteristic of wavelets is that they do not produce such an effect.

Wavelets are widely used in the solution of numerical problems in several areas of knowledge such as image compression, Numerical Harmonic Analysis [2], financial analysis, oil detection, differential Equations [3, 4], biomedical signals, analysis of electromagnetic integral Equations [5], optical fibers [6], among others. Many of these applications use the specific properties of wavelets, such as coefficients that are determined numerically, multi-resolution analysis to decompose a signal, integrals, and derivatives obtained numerically, energy concentrated in its compact and base with orthogonal elements.

## 2. Short introduction to wavelet theory

For the development of topics presented in this chapter, the reader must have as a prerequisite knowledge of functional analysis, linear algebra, measure theory and integration, differential and integral calculus. It is important to note that the wavelet basis is for the wavelet transform as well as the trigonometric basis is for the Fourier transform. Generally, the term wavelet is also used as a wavelet transform. The following subsections present these initial prerequisites to the reader.

### 2.1 Preliminaries on Hilbert spaces

In this subsection, some mathematical concepts necessary for a better formal understanding of the wavelet tool are defined. The definitions, contained in this section, are due to the author [2].

Definition 2.1 The space

Definition 2.2 A set of vectors

Definition 2.3 In Hilbert’s

Definition 2.4 The space

For

In particular

Definition 2.5 Let ^{1}

Definition 2.6 We say that a function

The concepts presented here about orthogonality and support of a

### 2.2 Definition of wavelet

This subsection aims to define wavelet [2], the main mathematical tool used in the development of this chapter. However, it is necessary to define the expansion and translation of mathematical operations beforehand.

Definition 2.7 *Given* *, the expansion operator,* *, defined over a* *function in* *or* *over* *, is given by,*

Definition 2.8 *Given* *, the translation operator,* *, defined over a function* *, in* *or* *over* *, is given by,*

Thus, using the expansion and translation operations defined above, a family of functions

The Definition 2.9, uses the family of functions

Definition 2.9 A function

By varying the values of

The definition 2.10 is another way used to define a wavelet.

Definition 2.10 A wavelet^{2} is a short duration wave, which has an average value equal to zero.

Due to the definition 2.10, wavelets resemble Fourier sine and cosine basis functions. Analogously to what is done in the Fourier transform, which has sine and cosine functions as base functions, in wavelet analysis, a function is decomposed into a base of wavelet functions.

The Fourier transform

The expression (5) means that the Fourier transform is the sum of every

Similarly, the expression of the wavelet transform

Similarly, the expression of the wavelet transform (6) is the internal product of the signal to be transformed by a wavelet function.

In the following subsection, among the most varied types of wavelets, the Daubechies wavelets are highlighted, which are the basis for the development of this chapter.

### 2.3 Daubechies wavelet properties

At

In the Figures 1 and 2 below, we have the graphical representation of the Daubechies wavelet functions

The functions ^{3} is the interval

To determine the filter coefficients

where

## 3. Generating an analytical function of the type x k using wavelets

Analytical functions are those that can be locally around a point

In general according to the author [8], any

The

Since *k* is the degree of the polynomial,

The justification used in the approximation (15) of a polynomial function of type

According to the Eq. (17), the

Definition 3.1 A wavelet has

In general, a Daubechies wavelet of kind

To generate a polynomial with

where

Substituting the Eq. (21) in (20), we have:

where

In the next subsection, the calculation of the moment generating function, which appears in the expression 21 as a coefficient of

## 4. Moment generating function

The calculation of the moment generating function according to the author [11] is of fundamental importance to approximate the functions by wavelets. The deduction of the moment-generating function now begins. For this, the mathematical expression is used

which refers to the moment of the wavelet scale

For

Substituting

Note that the variable

Making the substitution

Using the substitution

Now consider the equations:

Substituting

Now separate the last term of the sum (32),

Using the fact that

Thus, the equations are obtained:

From (35), (34), and (24), we get the moment generating function

The analytical expression for

Similar to what was done with the calculation of the moments for the function

The following is an example of the calculation of the moments for the case of Daubechies wavelets of a kind

Example 4.1 In this example, the Daubechies wavelet of kind

The support of the linear combination (39), represented in Figure 3, is obtained by the intersection of the supports of the translations of the function

Figure 4 shows the graph of translated functions

The calculation using the moment generating function depends on the Daubechies wavelet coefficients of kind 4. These coefficients are obtained by the Eqs. (9)–(12), which gives rise to the following non-linear system.

The solution of this system is the irrational numbers

Using the moment generating function for the case where

Proceeding with the calculations, we obtain:

Replacing the value of

So, the representation for the

In Figure 5, we have the graphical representation of the function obtained of the expression (46). Here the function

The representation for the expression (46) using the summation is given by,

The expression for writing polynomials of degrees

See that to generate the polynomials (48), (49) is necessary to use Daubechies wavelets of kind

### 4.1 Taylor polynomial using Daubechies wavelets

The Taylor polynomial or Taylor series is an expression that allows the calculation of the local value of a function *f* is infinitely derivable in an interval containing a point

The expression (50) developed around

Making use of the expression (21), we have:

The expression (52) is another way of writing Taylor’s polynomial using Daubechies Wavelets.

Example 4.2 Consider the analytical function

Using only two summation terms in the expression (53), we have:

Using the expression (46), we have:

The expression (55) allows us to approximate the exponential function using a base of Daubechies wavelets. This type of approximation, although simple for this case, is very useful in the case of representation for functions other types.

In the following example, the expression (46) is used to approximate Taylor’s series developments for the functions

Example 4.3 For the functions

In order to verify the potentiality of the application of Daubechies wavelets we will calculate the value of the functions in (53), (56), (57) and (58) evaluated at point x = 1. Considering only

Function | Value in ^{4} | Value in ^{5} | Error % |
---|---|---|---|

Table 1 appears here only as a way of showing the quality of the approximations using the Daubechies wavelets of kind

## 5. Conclusions

Daubechies wavelets are quite versatile mathematical tools. They can be used to analyze, generate, decompose a function, or even a signal that is represented by an analytical function. This type of application is widely used, for example, in electrical engineering in studies of magnetic fields and electric fields. The theory exposed in this chapter provides tools to carry out these studies. The use of the Taylor series as a way of approximating analytical functions is a very used technique in applied mathematics. Making use of the Taylor series with wavelets is another option to perform an approximation of analytical functions. In future work, we are researching other wavelets, for example Deslauries-Dubuc interpolets, that have an even better approach quality. As Deslauriers-Dubuc interpolets and others in research.

## Acknowledgments

The author would like to thank UFERSA for support during my doctoral studies.

## Notes

- A set is said to be compact if it is limited and closed
- Anglophone term to designate a small wave, in the sense of having a fast duration.
- We emphasize that the scale function has energy concentrated in its support that is determined by the genus of the wavelet, that is, supp ϕ = 0 N − 1 , and that the total energy of the scale function is unitary, that is, ∫ − ∞ + ∞ ϕdx = 1 .