La catástrofe de Asia Menor y sus consecuencias

The central problem of the theory of Fourier series is, how arbitrary periodic func-tions or signals might be written as a series of sine and cosine funcfunc-tions. The sine and cosine functions are also called sinusoidal functions. (See section 1.2.2 for a Sinusoidal function

description of periodic functions or signals and section 2.4.3 for a description of series of functions.) In this section we will first look at the functions that can be constructed if we start from the sine and cosine functions. Next we will examine how, given such a function, one can recover the sinusoidal functions from which it is build up. In the next section this will lead us to the definition of the Fourier coefficients and the Fourier series for arbitrary periodic functions.

The period of periodic functions will always be denoted by T . We would like to approximate arbitrary periodic functions with linear combinations of sine and cosine functions. These sine and cosine functions must then have period T as well. One can easily check that the functions sin(2πt/T ), cos(2πt/T ), sin(4πt/T ), cos(4πt/T ), sin(6πt/T ), cos(6πt/T ) and so on all have period T . The constant function also has period T . Jointly, these functions can be represented by sin(2πnt/T ) and cos(2πnt/T ), where n ∈ N. Instead of 2π/T one often writes ω0, which means that the functions can be denoted by sin nω0t and cos nω0t, where n∈ N. All these functions are periodic with period T . In this context, the constantω0is called the fundamental frequency: sinω0t and cosω0t will complete exactly one cycle on an Fundamental frequency

interval of length T , while all functions sin nω0t and cos nω0t with n> 1 will com-plete several cycles. The frequencies of these functions are thus all integer multiples ofω0. See figure 3.1, where the functions sin nω0t and cos nω0t are sketched for n= 1, 2 and 3. Linear combinations, also called superpositions, of the functions

0

t T/2 1

–T/2

–1

0 t

1

–1 sin3ω0t

sin2ω0t sinω0t

cosω0t cos2ω0t cos3ω0t T/2 –T/2

FIGURE 3.1

The sinusoidal functions sin nω0t and cos nω0t for n= 1, 2 and 3.

sin nω0t and cos nω0t are again periodic with period T . If in such a combination we include a finite number of terms, then the expression is called a trigonometric poly-Trigonometric polynomial

nomial. Besides the sinusoidal terms, a constant term may also occur here. Hence, a trigonometric polynomial f(t) with period T can be written as

f(t) = A + a1cosω0t+ b1sinω0t+ a2cos 2ω0t+ b2sin 2ω0t + · · · + ancos nω0t+ bnsin nω0t withω0= 2π T .

In figure 3.2a some examples of trigonometric polynomials are shown withω0= 1 and so T= 2π. The polynomials shown are

f₁(t) = 2 sin t,

f₂(t) = 2(sin t −¹₂sin 2t),

f₃(t) = 2(sin t −¹₂sin 2t+¹₃sin 3t),

f₄(t) = 2(sin t −¹₂sin 2t+¹₃sin 3t−¹₄sin 4t).

0

t –π

π

–π

0

t π

–π

–2π π 2π

f₁ f₄ f₃ f₂

2π π

–2π –π

a b

FIGURE 3.2

Some trigonometric polynomials (a) and the sawtooth function (b).

In figure 3.2b the sawtooth function is drawn. It is defined as follows. On the interval(−T/2, T/2) = (−π, π) one has f (t) = t, while elsewhere the function is extended periodically, which means that it is defined by f(t + kT ) = f (t) for all k ∈ Z. The function f (t) is then periodic with period T and is called the periodic Periodic extension

extension of the function f(t) = t. The function values at the endpoints of the interval (−T/2, T/2) are not of importance for the time being and are thus not taken into account for the moment. Comparing the figures 3.2a and 3.2b suggests that the sawtooth function, a periodic function not resembling a sinusoidal function at all, can in this case be approximated by a linear combination of sine functions only. The trigonometric polynomials f₁, f₂, f₃and f₄above, are partial sums of the infinite series
_∞

n=1(−1)ⁿ⁻¹(2/n) sin nt. It turns out that as more terms are being included in the partial sums, the approximations improve. When an infinite number of terms is included, one no longer speaks of trigonometric polynomials, but of trigonometric series. The most important aspect of such series is, of course, Trigonometric series

how well they can approximate an arbitrary periodic function. In the next chapter it will be shown that for a piecewise smooth periodic function it is indeed possible to find a trigonometric series whose sum converges at the points of continuity and is equal to the function.

At this point it suffices to observe that in this way a large class of periodic func-tions can be constructed, namely the trigonometric polynomials and series, all based upon the functions sin nω0t and cos nω0t. All functions f which can be obtained

as linear combinations or superpositions of the constant function and the sinusoidal functions with period T can be represented as follows:

f(t) = A +^∞ n=1

(ancos nω0t+ bnsin nω0t) withω0= 2π

T . (3.1)

This, of course, only holds under the assumption that the right-hand side actually exists, that is, converges for all t.

Let us now assume that a function from the previously described class is given, but that the values of the coefficients are unknown. We thus assume that the right-hand side of (3.1) exists for all t. It is then relatively easy to recover these coeffi-cients. In doing so, we will use the trigonometric identities

sinα cos β = ¹₂(sin(α + β) + sin(α − β)) , cosα cos β = ¹₂(cos(α + β) + cos(α − β)) , sinα sin β =¹₂(cos(α − β) − cos(α + β)) .

Using these formulas one can derive the following results for n, m ∈ N with n = 0.

 _T/2

On the basis of the last three equations it is said that the functions from the set {sin nω0t and cos nω0t with n ∈ N} are orthogonal: the integral of a product of Orthogonal

two distinct functions over one period is equal to 0.

After this enumeration of results, we now return to (3.1) and try to determine the unknown coefficients A, a_nand b_n for a given f(t). To this end we multiply the left-hand and right-hand side of (3.1) by cos mω0t and then integrate over the interval(−T/2, T/2). It then follows that

 _T/2

In this calculation we assume, for the sake of convenience, that the integral of the series may be calculated by integrating each term in the series separately. We note here that in general this has to be justified. If we now use the results stated above, then all the terms will equal 0 except for the term with cos nω0t cos mω0t, where n equals m. The integral in this term has value T/2, and so

 _T/2 This means that for a given f(t), it is possible to determine am using (3.2). In an analogous way an expression can be found for bm. Multiplying (3.1) by sin mω0t and again integrating over the interval(−T/2, T/2), one obtains an expression for bm(also see exercise 3.2).

A direct integration of (3.1) over(−T/2, T/2) gives an expression for the con-stant A:

The right-hand side of this equality is, up to a factor 2, equal to the right-hand side of (3.2) for m= 0, because cos 0ω0t = 1. Hence, instead of A one usually takes

All coefficients in (3.1) can thus be determined if f(t) is a given trigonometric poly-nomial or series. The calculations are summarized in the following two expressions, from which the coefficients can be found for all functions in the class of trigonomet-ric polynomials and series, in so far as these coefficients exist and interchanging the order of summation and integration, mentioned above, is allowed:

a_n = 2 T

 _T/2

−T/2f(t) cos nω0t dt for n= 0, 1, 2, . . ., (3.3) b_n = 2

T

 _T/2

−T/2f(t) sin nω0t dt for n= 1, 2, . . .. (3.4) In these equations, the interval of integration is(−T/2, T/2). This interval is pre-cisely of length one period. To determine the coefficients an and bn, one can in general integrate over any other arbitrary interval of length T . Sometimes the inter-val(0, T ) is chosen (also see exercise 3.4).

EXERCISES

Verify that all functions sin nω0t and cos nω0t with n∈ N and ω0 = 2π/T have 3.1

period T .

Prove that if f(t) is a trigonometric polynomial with period T , then bncan indeed