Book
9 |
2 Background .1
Cosmology .11 Math .10 Book |
Chapter .2 |

Chapter .1

**Fourier Series and Integrals**

**Jean Baptiste Fourier's Théorie analytique de la chaleur*** is the bible of the
mathematical physicist; it does not only develop the trigonometric series and integrals, named after him, but also executes
admirably general boundary value problems by means of heat conduction.

It is an established custom to emphasize in mathematical lectures about Fourier series the concept of arbitrary functions,
their properties of continuity and singularities (**accumulation of infinitely many maxima and minima!**).
This point of view vanishes in the case of physical applications. In fact, the initial and boundary value problems then under
consideration are always - already due to the actual atomic structure of matter and action - to be viewed as **mean
values** just as the partial differential equations, to which they lead, arise from the **statistical
averaging **of more complicated elementary laws. Thus, we are only concerned with comparatively simple, idealized
functions and their approximations involving smallest possible errors. What has to be understood by this is taught by Gauss in his
**Method of Least Squares**. We will see that it
does not only open up a simple and strict entry into Fourier series, but also into all other expansions in series occurring in
mathematical physics, in spherical, cylindrical and, in general, eigen-functions.

* J.B.Fourier (1768-1830); his book appeared in 1822 in Paris. Fourier also excelled as an algebraist, engineer and historical author about Egypt, to which he accompanied Napoleon.

The response to his book in France and abroad was illuminated by the sentence: "As with Franz Neumann, Fourier's stimulation struck the spark out of the stone in the case of William Thomson." (cf. F. Klein, Lectures on the history of mathematics in the 19-th Century, Vol. I, p. 233.

Let there be given an arbitrary function *f*(*x*) in the interval *p **
x ** ** p , *which, for example, may have
been determined by sufficiently exact and sufficiently many measurements. We want to approximate it by the sum of 2

How are we going to select the available coefficients *A*_{k}*,
B*_{k}? Denote the error *f*(*x*) - S(*x*) by *e*_{n}(*x*),
i.e., set

Following Gauss, we consider the **mean square error**

and make *M *into a minimum by an appropriate choice of *A*_{k}*,B*_{k}.

Note that the corresponding measure of the total error, formed with the linear term *e*_{n}*,
*would be unsuitable, because then errors with different signs would cancel each other and not enter into the overall error.
On the other hand, use of the absolute value |*e*_{n}| would be inconvenient
due to its non-analytic character.

The Russian mathematician Pavputii Ljovovich Tchebychef (1821 - 1894) proposed a quite different
approach. He did not consider the **mean **but the **maximum**
of *|**e*_{n}| in the interval and minimizes this by choice of the
available coefficients.

By (3), one obtains the equations

These are just 2*n* + 1 equations for the 2*n* + 1 unknown coefficients *A*_{k}*,B*_{k}.
It is beautiful here that these equations determine every one of the coefficients directly and do not interlink them. This is due
to the **orthogonality ****conditions
**which hold between the trigonometric functions:

Here and in the sequel, all integrals are to be taken from *-**p* to +*p**.*
In order to justify the use of the word **orthogonality**, we recall that two vectors * u*,

The integrals entering into (5) can obviously be interpreted as sums of infinitely many terms. Cf. the comments regarding the so-called
in **Hilbert
Space.**

In order to prove these formulae, we must now write down the unnecessarily inconvenient addition formulae for the trigonometric
functions; however, it is better to recall their relationship to the exponential functions *e*^{±ikx} and
*e*^{±ilx}*. *The integrands of (5a,b) then only involve 4 terms of the form

all of which vanish during the integration unless *l* = *k*. This proves (5*a*, *b*). However, (5)
is also correct without this restriction because it reduces for *l* = *k *to

In the same way, one finds then also the value of (5*a*, *b*) for *l* = *k* > 0 (when only the
double product of exp(*ikx*) and exp(-*ikx*) contribute): This value is simply *p*
, while for *l = k = *0 the value of (5*a*) is obviously 2*p*. Hence we can
replace (5*a*,*b*) by the double equation, valid for *l = k > *0

For *k *= *l,* (6) is the **normalization condition**.
For the exceptional case *l = k = *0, it must be supplemented by

If we now substitute (5), (6) and (6a) into (4), then among the integrals formed with *S*_{n} all terms
but the *k*-th drop out and we obtain immediately the **Fourier
coefficient formulae**

This determines completely our approximation *S*_{n}. Given an empirical function *f*(*x*),
you have to execute the integrations numerically or by a computer.

In 1947, this meant the use of **harmonic analyzer**.
The most effective one was then the machine of Bush and Caldwell, which can also be used for the integration of arbitrary
simultaneous differential equations (Phys. Rev., **38**, (1931) 1898).

It is immediately clear from (7) that in the case of an even function, i.e., *f*(-*x*) = *f*(+*x*),
all *A*_{k}* *including *A*_{0}* *vanish. Thus, such an even function is
represented by a **pure cosine series**, an odd function by a** ****pure
sine series****.**

Naturally, the quality of the approximation increases with the number of available constants *A*,
*B*, i.e., as *n *increases. In this context, note the fortunate fact that, since the *A*_{k}*,
B*_{k}* *do not depend for *k* < *n *on *n*; in the case of a transition from *n*
to *n* + 1, the *already determined A*_{k}*, B*_{k}* do not change and
one needs only compute A*_{n}*, B*_{n}. Once determined, the *A*_{k}*,
B*_{k}* are **final**!*

There is no reason why one should not let *n *increase, i.e., perform the limit Then
the series, considered hitherto, becomes an **infinite Fourier
series**, the convergence of which will be discussed in 1.2
and 1.3.

The problem of the **completeness** of the basic function
systems is more difficult than that of their convergence. Obviously, if one were to omit in a Fourier series one of the terms, for
example, the *k*-th cosine term, the function *f*(*x*) could not be represented with arbitrary accuracy by
the remaining terms; in other words, the finite error *A*_{k} would remain. For example, to mention the
extremely simple case when cos *nx* is to be represented by the incomplete series of all cosine terms with *k* < *n
*and *k* > *n*, all *A*_{k}* *would vanish due to the orthogonality and the
error would be cos *nx *itself. Obviously, in the case of such a regularly formed system as that of the trigonometric
functions, nobody would have the idea of omitting a single term. However, in more general cases, such a point of view of
mathematical aesthetics would not occur.

What mathematicians tell us by their question regarding **completeness**
is, in fact, nothing else but what the **method of least squares****
**already contains in its formulation. We start here from the observation that a system of functions, say *j*_{0},
*j*_{1}, ··· , *j*_{k}, ··· can only then be complete if for every continuous
function *f*(*x*) the mean error, formed according to (3), goes to zero as
We assume here that the system of the *j*_{k }itself is **orthogonal
and normalized**** **to 1, i.e., that

whence the coefficients *A*_{k} are

If the integration limits in this and the preceding integrals are *a* and *b*, so that the integration interval
becomes *b *- *a*, one has, according to (3),

In the last term, we have used here already (8). The last but one term in (9), apart from its sign, is twice the last term, whence

and one requires, following the preceding remark, that

This is the mathematical formulation of the **completeness relation**,
strongly emphasized in the literature. Obviously, it can hardly be executed as a practical criterion. Since it only concerns the
mean error, it also does not tell anything regarding whether the function *f* is really represented by the Fourier series
at every point (1.3 ).

By following the historical development, we have derived in this introductory section the *final validity of the Fourier
coefficients*** **from the *orthogonality of the trigonometric functions*. In
1.4, we shall readily demonstrate, by the example of the **spherical
functions**, that also, conversely, orthogonality can be derived quite generally from our demand of final validity,
which, from our point of view of approximation, appears to be quite natural. In any case, we emphasize here that both **orthogonality**
and **final validity** are conditioned reciprocally and that they can be replaced by
each other.

Finally, we will give our results so far a mathematically more complete and at the same time physically more useful form. We
perform this immediately in the case of **infinite Fourier series**, but note that what
follows is also valid for the actually more general and more strict form of the truncated series.

We write, transforming and denoting the integration variable by *x*

We can now interpret the last term in the last line so that here instead of the summation being extended over the positive *k
*in exp{-*ik*(*x - **x*)} it is to be taken over the corresponding negative
values of *k *in exp{+*ik*(*x - **x*)}, whence we rewrite this term

This step removes the troublesome exceptional role of the term *k *= 0, which now occurs between the positive and
negative values of *k*, whence

Finally, introducing **complex Fourier coefficients**,
which also may arise for real *f*(*x*),

Obviously, the relationship of the *C *to the *A* and *B*, according to the definition
of the latter in (7), is

Obviously, our complex representation (12) is considerably simpler than the traditional real form; we will find it to be
especially useful when we come to the theory of **Fourier integrals**.

If we extend our representation, initially intended for -*p** *<* x *< +*p**,
*to the regions *x* > *p** *and
*x* < *-**p**, *we arrive at an obviously periodic repetition of the
branch lying between -*p** *and +*p**, *i,e., in
general, not the analytic continuation of our initial function *f*(*x*). In particular, the thus arising periodic
function displays for the odd multiples of the arguments ±*p** ***discontinuities
**unless *f*(-*p*) = *f*(+*p*). We
examine in 1.2 the error which arises at such a location.

**1.2
Example of a discontinuous function. Gibbs' phenomenon and non-uniform convergence****:****
**We consider the function

Fig. 1 displays it with its periodic repetitions, complemented by vertical
links of length 2 at the locations of the discontinuities *x*=0, ±*p*,±2, ···
whereby it becomes a **meander line**. Our function is odd,
whence its Fourier series, as indicated by (1.7), involves only sine terms. It is most
convenient to employ (1.12) for the determination of the coefficients, which yields

Hence, by (1.13),

and we obtain the sine series with odd *k *terms

You can imagine the surprise this series caused when Fourier first obtained
it - a *discontinuous line represented by superposition of an infinite sequence of simplest, continuous functions*! Without
exaggeration, one can say this series made an essential contribution to the concept of real functions. We will see below that it
also strengthened the concept of the convergence of series.

In order to understand how the series approximates the unsteady sequence of
line segments, we draw the functions *S*_{1}, *S*_{2}, *S*_{3} together with *S*_{n}
(Fig. 2 below)

*S*_{1}*
*has its maximum at *x*=*p*/2 with the height

i.e., it exceeds by 27% the line *y *= 1, to be shown. *S*_{3}
has at the same location the minimum

i.e., it drops below the line *y *= 0 by 15%. Moreover, *S*_{3}
has still 2 maxima at *p*/4 and 3*p*/4, 20 % above *y=*1*.
*(Please check these statements!) In contrast, *S*_{5}(*x*) has again a maximum at *x = **p*/2
with the height

which only exceeds *y =* 1 by 10%. Two flat minima are followed by
two steeper maxima which lie already close to *x* = 0 and *x = **p**p. *In
general, the maxima and minima of *S*_{2n+1}* *occur between those of *S*_{2n-1}*
*. (Exercise 1.1)

What has been said here regarding the step by step approximation of the
line *y*=+1 applies, of course, to the line *y*=-1, reflected in the *x*-axis. It is also approximated by **successive
oscillations **in such a way that the approximation *S*_{n}* *with* n *peaks
and *n*+1 valleys oscillates around the line *y*=-1. In the process, the oscillations in the central part of the
line decrease with
increasing *n*; at the points of discontinuity *x = *0, *± **p**, ±*
2*p*, ···, where there does not occur a systematic decrease of the maxima with *n*,
the approximations come closer and closer to the vertical lines of discontinuity, whence the image of an approximation with very
high *n *is shown schematically in Fig.3.below

Let us now inspect more closely the behaviour of *S*_{2n+1}(*x*)
for large *n *near* *a jump, for example, at *x* = 0. For this purpose, we write first of all the sum of *S*_{2n+1}
as integral. (In general, it is easier to discuss an integral instead of a sum with many terms.) We do so as follows:

After extraction of the factor exp (±*x*), the last two sums are
geometric series, which advance with powers of exp (±2*ix*) and are readily summed, whence

Here, the two fractions are reduced, after extraction of suitable factors, to

Thus, (3) becomes

Finally, if *x* is sufficiently small, we can replace sin *x *in
the denominator by *x*, because it has there the large factor *n* + 1. Instead of (3b), we find with a new
integration variable *u* and* a *new argument *v*

We now conclude: If at first we set *x = *0 for finite *n*,
then *v *= 0 and therefore also *S*_{2n+1} = 0. If we then let *u* grow, we obtain *S*_{2n+1}
= 0 also for Hence

However, if *x* is positive, non-zero and one lets first then
and, according to the fundamental formula, derived in Exercise
1.5: *S*_{2n+1}= 1; if one lets now ,
the value of *S*_{2n+1}remains also for *x* = 0 equal to 1, i.e.,

Hence, the two limit processes cannot be inverted. If the function *f*(*x*)
to be represented were continuous at *x* = 0, the result would be indifferent to the order of sequence of the two limit
processes and in contrast to (4a, b)

However,
this does not yet exhaust the strange features of (4); in order to develop them, we consider the tabulated **Integral
sine**

and investigate its general behaviour displayed in Fig. 4. For small values
of *v, *when we can set sin *u* = *u*, it is proportional to *v*, for large values of *v*, it
approaches asymptotically to *p**/*2* *with successively decreasing oscillations;*
*by (5), the maxima and minima occur at *v *=* p*, 2*p**,
*3*p**, *··· ; the ordinate of the first and largest maximum, according to the
tables of the third edition of Jahnke-Emde, Teubner, 1938, is 1.851. There corresponds to the associated abscissa of the *Si*-curve,
in terms of the former variable *x*, because *v *= 2(*n* + 1)*x, *the discontinuous sequence

in which, by (4), the sequence of the approximations *S*_{2n+1},
*S*_{2n+3}, ··· has the fixed value

This value, which exceeds the straight line *y = *1 by 18%, is at
the same time the **upper limit **of the range delivered by the approximations. Its
lower limit *S *= -1.18 is attained when we, coming from the negative side, approach zero in the sequence -*x*_{n},
*x*_{n+1}, ···. Every value between -1.18 and +1.18 can be reached by a special kind of limiting
process, i.e., the locations *S*=0 and *S=*1 in the manner described by (4a) and (4b).

This behaviour of our approximation, especially the presence of values
exceeding the discontinuity ranges ± 1, is referred to as **Gibbs
phenomenon** (William Gibbs, 1844 - 1905, the most outstanding thermo-dynamicist of the USA and, beside
Boltzmann, the founder of Statistical Mechanics). Gibbs' phenomenon occurs whenever one is dealing with a discontinuity. One then
speaks of **non-regular convergence****
**of an approximation.

We still must confirm that we really can reach every value between *S*
= +1.18 and *S* = -1.18 by suitable coupling of the two limit processes. Taking instead the more general value *q*,
we have by (4) *v*=2*q *and by combining (4) and (5)

where now, depending on the choice of* q*, *Si*(2*y*)
can assume all values between 0 and 1.851, as follows directly from Fig. 4
for positive *q*; for negative *q*, one obtains correspondingly all values between 0 and - 1,851. Thus coupled
limiting processes yield not only the approach of our approximating function *S* to the discontinuity -1 to +1, but beyond
also its transgression, i.e., **Gibbs' phenomenon**.

After these fundamental statements, we shall still draw several formal
mathematical conclusions from our Fourier representation (2). If we set there *x* = *p*/2,
we obtain immediately **Leibnitz's sequence**

This series converges slowly; we obtain better convergence for the powers
of *p* by integrating (2)once or several times. Compare what follows with Fig. 5 below.

Restricting consideration
to 0 < *x* < *p*, we rewrite (2)

Integration from 0 to *x *yields

Hence one finds for *x* = *p*/2

Subtraction of (10) from (11) yields

Another integration between 0 and *x* yields

and for *x = **p*/2 as analogue to
Leibnitz's series

Integrating (13) again and setting *x *=* **p*/2
yields

(15) *p*/8(*p**x*²/2
- *x*³/3) = 1 - cos *x + *(1 - cos 3*x*)/3^{4} + (1 - cos 5*x*)/5^{4} + ···

(16) *p*^{
4}/3·32 = 1 + 1/3^{4} + 1/5^{4} + ···.

Finally, subtracting (15) from (16), we obtain

Series (11) and (16) only involve **odd **numbers.
Those involving **even** numbers are 1/4 and 1/16 of the sums involving **all
**integers. If we denote the last series by *S*_{2 }and *S*_{4},
respectively, we have

*p*
²/8 + *S*_{2}/4 =* **S*_{2},
*p*^{ 4}/3·32 + *S*_{4}/16 = *S*_{4},
respectively;

i.e.,

(18) *S*_{2}
= *p* ²/6, *S*_{4} = *p*^{4}/90.

This value of *S*_{4}
was required for the derivation of **Stefan's
radiation law **or **Debye's law** for the energy
content of a solid. The trigonometric series (12), (13), (17) will serve as examples in the following sections. The higher
analogues to Leibnitz's series (8) and (14) as well as to *S*_{2} and *S*_{4}
are given in Exercise 1.2.

**1.3 On the
convergence of Fourier series:**** **We will
prove the **theorem**: *If a function f(x) between -**p**
and +**p **with its first n - 1 derivatives including its bounds is
continuous and differentiable, while its n-th derivative is also in general differentiable, but can have at a finite number of
locations x = x*_{l }*bounded discontinuities, then the coefficients A*_{k}*, B*_{k}*
of its Fourier expansion tend to zero with at least like k*^{ -n-1}.

The addition including its bounds has the
following significance: Every function, represented by a Fourier series, is by its nature periodic. The effective image of its
argument would thus actually not be the range -*p* to +*p*,
but a ring which returns into itself at *x* = ±*p*. Correspondingly, the assumed
continuity extends to the values of *f* and its first *n* - 1 derivatives at the point *x* = ±*p*.
This point does not differ in any way from the internal points of the expansion interval just as it
also is immaterial whether we denote the limits of the interval by -*p*, +*p*
or, for example, by *p*/4, 9*p*/4, etc.

It is convenient to employ for the proof of the theorem the complex version (1.12):

Integration by parts yields

Here, the first term vanishes due to the assumed continuity of *f*(*x*);
the second term can be again transformed by partial integration. After *n* repetitions of the same procedure, one finds

This integral must be decomposed, due to the discontinuities in *f* ^{(n)}(*x*)
at *x = x*_{l}*, *into partial integrals between *x = x*_{l}* *and *x=x*_{l+1};
denote by *D*^{n}_{l}^{ }the jumps of *f*
^{(n) }at the discontinuities. Then, (3) becomes

where in a given case the location *x* = ±*p*
may be contained among the *x* = *x*_{l}. A last integration by parts yields

In view of the fact that the discontinuities *D*^{n}_{l}^{
}were to be bounded and that, in general, *f* ^{(n)} was to be differentiable between the
discontinuities, we conclude from (4) that *C*_{k}* *vanishes at least like *k*^{-n-1}*
*when one lets . Under special
conditions between the *D*^{n}_{l }or special behaviour of *f*
^{(n+1)}(*x*), the order of vanishing could become higher.

This theorem is valid for positive as well as negative *k*, whence
it is also valid for **real Fourier coefficients ***A*_{k}*,
B*_{k}* *(*k *> 0), which according to (1.13) consist of the *C*_{k
}for positive and negative *k*.

A special consequence of the theorem is that an **analytic
function** of period 2*p**, *which is therefore continuous and periodic
with all its derivatives *f* ^{(n)} has Fourier coefficients which with increasing *k* decrease
more than any power of 1/*k*. An example for this would be an arbitrary **polynomial**
in sin *x *and cos* x*. This is represented by a **truncated Fourier series **of
as many terms as are given by the degree of the polynomial, so that all higher order Fourier coefficients vanish. Another example
is the elliptic series, which we will encounter during a heat conduction problem
in 3.15; its Fourier coefficients *C*_{k}* *decrease
like *e*^{-ak²}.

Moreover, the theorem yields that the sum *S**
A*²_{k }converges like *S** k*^{-2 }for every function *f*(*x*)
which otherwise is differentiable, but can still have a finite number of jumps (the case *n *= 0 of the theorem). An
example is our function (2.1) for which, while *S**A*_{k
}diverges, *S**A*²_{k }converges. It also shows that the
completeness relation, as noted already, does not guarantee the complete representability of *f*.
In fact, if we strengthen Definition (2.1) in the sense that *f* = 1 for
and, in contrast, *f*=-1 for *x* < 0. then *f *is not represented at the point *x* = 0 by the
Fourier series (2.2) which yields *f* = 0.

A rich illustration of the theorem is also provided by the sin
and cos series above. We complement these expressions of the function represented, which are given only for the interval 0
< *x* < *p**, *by their expressions for -*p**
*< *x* < 0. The latter just arise simply from the fact that the cos series are **even**,
the sin series **odd** functions of *x*. In the latter, we must therefore invert
for the negative interval, apart from the sign of *x,* also the sign of the entire expression, obtained earlier, in the
former only the sign of *x*. The corresponding expressions are presented in {} after the semicolon. Hence we complete
(2.9), (2.12), (2.15) and (2.17) as follows:

Here, the left hand sides behave increasingly continuously: While the
values of (5) become discontinuous at *x* = 0 and *x*= ±*p*, the values of (6) are continuous and the first
derivative is discontinuous. (7) has also a continuous first derivative and its second derivative has jumps. Only the third
derivative of (8) becomes discontinuous. Every time, the jump present is the same as that of (5) and occurs at the same locations
due to the fact that every *x* = 0 and *x *= ±*p* , corresponding equation
followed from the preceding one by integration.

Fig. 5 illustrates this behaviour. Its curves 0, 1, 2, 3 represent the left
hand sides of (5), (6), (7) and (8). The discontinuity of the tangent of 1 at *x* = 0 is obvious; that of the curvature of
2 at *x*=0 is to be recognized from the behaviour of the two curves meeting here, reflect with respect to each other second
order parabolae. Curve 3 comprises two third order parabolae which meet at *x *= 0 with steady curvature. The scale of the
ordinate used for the different curves was determined by the size of drawing available and can be seen from the ordinates of the
maxima, given on the right hand side.

The **increasing** steadiness of
the curves 0 to 3 is reflected in the **increasing convergence**** **of the
Fourier series on the right hand sides of (5) to (8): In (5), there is a decrease of the coefficients like 1/*k*, in
general, according to the theorem, a decrease with *k*^{-n-1}, where *n *denotes the order of the
firstly arising discontinuous derivative of the respective function.???

The convergence of **Fourier series**
contrasts explicitly that of **Taylor expansions**. The
former depends only on the continuity of the function to be expanded and its differential quotient on the real axis, the latter
also on the position of the singular point in the complex regime. (Indeed, the singular point closest to the starting point of the
power series determines the radius of convergence of the Taylor series.) Correspondingly, the principles of the two expansions
differ fundamentally, as will be discussed further in 1.6: In the case of
Fourier series, one has an **oscillatory approach****
**over the entire expansion interval, in the case of Taylor series, an **osculating
approach **at the origin (Latin: osculum = kiss).

**1.4
Transition to Fourier Integral:**** **The
range of the development interval -*p** *< *x *< *p**
*can be modified in various ways. It cannot only be shifted as has already been noted, but also
its size can be changed, for example, with arbitrary *a* in -*a*<*z<+a. *The last is achieved by the substitution

which changes (1.7) into

In the complex notation (1.12), this becomes

Obviously, one can also employ the interval *b < x < c *with
the substitution

We still refer to certain pure sin and cos series** **which
already were in Fourier's work. In those, one imagine, for example, the function *f*(*x*), restricted to the region
0 < *x *< *p*, extended to the left hand side in an** ****odd
or even manner**, respectively, so that in the case of odd continuation

(Exercise 1.3).

Moreover, if we assume in (3) that *a *is very large, the sequence
of the values

becomes then almost continuous and we will write *w**
*instead of* **w*_{k}*. *Correspondingly, we will write for
the difference of two consecutive* **w*_{k}

Hence, replacing there the symbols *z* and *z**
*by the earlier symbols *x*, *x**, *one has

For the present, we will not denote the limits of this integral by .

Substitution of (6) into the infinite, quasi-continuous series (3) yields
for *f*(*x*), denoting for the time being the upper and lower limits by ± *W**,*

Obviously, the sequence of the limiting processes indicated here is unnecessary: If we were to execute first the transition , we would encounter the totally senseless integral

In order that the first executed limiting process
shall have a defined meaning, *f*(*x*) must vanish for WE
need not investigate how strongly it must vanish, so that also the other limiting process can be executed, since in all meaningful
physical tasks this vanishing will be sufficiently strong.

After these observations, we will abbreviate the more accurate mode of writing (7) to

We now pass on to the in the literature established real form of the **Fourier
integral**** **by setting

Here sin is **odd**** **in
*w**, *i.e., it vanishes therefore during integration with respect to *w*
between ; cos is **even****
**in* **w**, *whence it yields twice the integral from 0 to ,
whence we also have

by the way, without claiming that the real form is better or simpler than the complex form (6). We can also write instead of (9)

with the abbreviations

Especially, if *f*(*x*)
is an even or odd function of *x*, then *b*(*w*) and *a*(*w*),
respectively, vanish; then, according to the **pure cos or sin series** above one has **pure
cos or sin integrals**. We can induce one or the other when (*x*) is only given for *x* > 0 by
continuing it as odd or even function on the negative side. The usefulness of this method will be encountered later on when
dealing with heat conduction problems. Hence, we write for** ****even
continuation**

for **odd
continuation**

We
have chosen the notation for one of the integration variables with a view to what is to come. In general, one uses for vibration
problems the **circular frequency***
w*. Considering temporarily *x* to be the **time co-ordinate**, we
have in (10) a decomposition of an arbitrary process *f*(*x*) into its **harmonic
components**. In the case of the Fourier integral, one is concerned with a **continuous
spectrum** which extends over all frequencies from ,
in the case of the Fourier series with a **discrete spectrum**** **of the
fundamental frequency and overtones. However, one must keep here in mind the following fact: When the physicist designs the
spectrum of a process with a suitable spectral apparatus, he only includes the **amplitude**
belonging to a frequency *w**, *while he does not know the phase of every partial tone.
In our notation, the amplitude corresponds to

the phase *g*(*w*) is given by the
ratio *b*/*a*, best in the combination

The Fourier integral, which describes the process completely, requires for
this both quantities *a* and *b*, i.e., the amplitude and the phase, whence the observable spectrum yields, so to
say, only half of the information contained in the Fourier integral.

This circumstance becomes very noticeable in the nowadays (1947) so
successfully handled **Fourier analysis of crystals****
**. In this analysis, one can only observe the **intensity **of the crystal
reflections, i.e., the squares of the **amplitude**; for a complete knowledge of the
crystal structure, one requires also the **phases**. This methodical defect can only be
partially removed by symmetry considerations.

We will deal in Exercise 1.4 with the spectra of different vibration processes, as examples of the theory of the Fourier integral and at the same time as additions to the spectral statements of Volume IV.

Finally, we return once again to the complex form of Fourier integrals and split them into the two statements which agree with (8)

Apart from the splitting of the denominator and
the notation of the integration variable in the second of these equations, the function *j*(*w*)
is identical with the quantity *a*(*w*) - *ib*(*w*),
whence it yields simultaneously information regarding the amplitude and phase of the vibration process *f*(*x*).

Moreover, (10) shows that the two functions *f *and *j**
*are interrelated reciprocally: The one is determined by the other independently of whether *f *is known and *j**
*is to be found or *vice versa*, and, in fact, both times by **integral
equations**** **with an identical character. One calls the one the **Fourier
transform**** **of the other. Equations (13) are an especially elegant
formulation of the **Fourier integral theorem**.

So far, we have only spoken of functions *f*(*x*) of a single
variable. Obviously, we can also develop functions of several variables in terms of each of them in Fourier series or integrals.
In the space *x, y, z, *we obtain three-fold infinite Fourier series and six-fold Fourier integrals. We will not write down
here the respective long and subscripts heavy formulae, as we will have several times opportunities to explain them during
applications.

**1.5.
Expansion in terms of spherical functions:**** **We
do not claim that our approach here is the most comfortable one, but it follows directly from 1.1,
does not demand preparations from the theory of differential equations and leads to interesting views of far-reaching
generalizations.

Consider the task of approximation of a given function *f*(*x*)
in an interval -1 < *x* < +1 by a sequence of polynomials *P*_{0}, *P*_{1}, *P*_{2},
··· ,* P*_{k}, ··· ,* P*_{n} of degrees 0, 1, 2, ··· ,* k, *···
,* n *by the method of **least squares**, i.e., we
form an *n*-th approximation

and make, as in (1.3), the mean square error

into a minimum by appropriate selection of the *A*_{k}.
As in (1.4), this yields the *n* + 1 equations

We complement this minimum condition by a demand which concerns the
numerical work involved: The coefficients *A*_{k}, computed from (3) for the *n*-th approximation,
are also to be retained in the (*n*+1)-th approximation together with all the succeeding ones; they are to represent for
all the **final ***A*_{k},
which then require for an improved approximation only the determination of the *A*_{k} for *k > n*.
In 1.1, this final relevance of the *A*_{k} arose from the known orthogonality of
the trigonometric functions. Here, conversely, our **final validity
demand** is to serve to justify the necessity of the **orthogonality **of
the *P*_{k}.

The proof is very simple. When expanded, omitting the integration limits ±1, (3) becomes

Since the right hand side does not depend on *n *and the *A*_{n}
are to be final, these equations apply without change to the (*n*+1)-th approximation *S*_{n+1}
except that one has on the left hand side the additional term

However, addition of this term would contradict (4). Hence this term must
disappear and one must have, since *A*_{n+1} (except in the case of a special choice of *f*(*x*))
is not zero, the integral must vanish for all *k* for which (4) was valid, i.e., for all However,
this means that *P*_{n+1} is orthogonal to all *P*_{0}, *P*_{1}, ··· , *P*_{n}
, and that for arbitrary *n*. Hence, having once mutually orthogonalized *P*_{0} and *P*_{1},
our **final validity demand **guarantees the general **orthogonality
condition**

Hence, (4) yields

Thus, the individual *A*_{k} are known (and not
only interlinked by recurrence relations), provided we have an agreement with respect to the **normalization
integral**** **on the left hand side of (6). The simplest such
agreement would appear to be setting it equal to unity - indeed this is what we will do later on in the general theory of **Eigen-functions**.
We prefer here to follow the historical practice and demand instead that

This normalization condition carries the advantage that with it all the
coefficients occurring in the *P*_{k} become rational numbers.

We will now proceed to the step-by-step calculation of the *P*_{0},
*P*_{1}, ··· from (5) and (7)*. P*_{0} is a constant, which, in accordance with (7), we must set
equal to 1. By (5), setting *n = *0, *m* = 1, we obtain from (5) *b* = 0 and from (7) *a* = 1. Setting
*P*_{2} = *ax*² + *bx + c, *we find

whence *P*_{2} = *a*( *x*² - 1/3)* *and,
from (7),

Correspondingly, we find

We see that the *P*_{n}* *are indeed
completely determined by our two conditions, the *P*_{2n}* *as even, the *P*_{2n+1}*
*as odd polynomials with rational coefficients.

The following explicit representation is clearer than this recurrence method:

It is clear that it satisfies (7): For ,
we need only perform the *n*-fold differentiation on the factor (*x - *1)^{n} which yields *n*!;
the factor (*x + *1)^{n} can be set equal to 2^{n}. Thus (8) yields indeed *P*(1)
= 1.

Now, we need only prove that (8) meets also the orthogonality condition (5)
which, as we know, is equivalent to our **final validity condition**.
For this purpose, we introduce the notation

and rewrite the left hand side of (5) with omission of the constant factor, which is here not of interest, as

and demand that *m* > *n*. We now reduce by partial
integration the order of the differential equation of the second factor *D*_{nm}* *and thereby
raise that of the first factor *D*_{nn}. During this process, there arise terms which are free from the
integration vanish for *x* = ±, because during the evaluation of *D*_{n-1,m}*, *following
(9), there remains a factor *x* - 1 and *x *+ 1. Continuing this process, we obtain

Here, according to (9), *D*_{2n,n} is a constant,
in fact, (2*n*)!, whence

This is equal to zero, because the order *m - n - *1 of the required
differentiations is smaller than the number *m* of the factors *x-*1 and *x*+1, to be differentiated. This
also holds for *m = n + *1; it fails for *m = n*, whence the orthogonality has been proved for .

Hence, we now know how to compute the normalization integral in (6):

If we employ here the first line of (11) with *m = n = k*, we find

The factor in front of the last integral is

the integral itself assumes with the substitution the known form

Equation (6) yields now

Inserting in (1) for our *n*-th approximation *S*_{n},
we obtain by the transition (assuming convergence of the
series and completeness of the function system)

The just stated assumptions are justified here as with Fourier series by
the limiting value of the mean square error. Here, as there, the *k*-th expansion function has *k* zeroes within the
expansion interval, except that they are here not like there equidistant. Also, the approximation to the given function *f*
occurs here like there with increasingly narrower and more frequent** ****oscillations**.
Moreover, in this case, there occurs at locations of discontinuities a Gibbs' phenomenon, etc.

**1.6
Generalizations: Oscillating and osculating approximations. An-harmonic analysis. An example of non-final determination of
coefficients****: **There
arises now the question: What is the reason for the difference between the two series despite the equality of the method of
approximation? As we have seen, the form of the *P*_{n}(*x*) is determined completely by our
approximation demands; one might think that, for example, the pure cos series (development of an even function) would become a
series of spherical functions, if one makes in it cos *j** *= *x. *In that case,
cos *kx* like *P*_{n}(*x*) becomes a polynomial of degree *k *in *x*, and the
expansion interval 0 < *j** *<* **p***
**becomes the interval +1 >

While we give all *d**j** *in
the Fourier approximation the same weight, in the *x*-scale the ends ±1 of the interval are preferred due to they
are approximated with greater accuracy than the central part of the interval. Obviously, the same applies to the approximation by
spherical functions which, on transfer to the *j*-scale, treats the ends of the interval with
less accuracy because *g* = sin* **j*. Speaking graphically, one deals in the case
of Fourier series with the uniformly subdivided unit circle between *j** *=* *0
and *p*, which by vertical projection onto the diameter between -1 and +1 yields non-uniform
density, on the other hand, in the case of the series of spherical functions with the uniformly covered diameter, which
corresponds to a non-uniformly covered semi-circle.

**1.6.1
Oscillating and osculating Approximation:** These
different **weight distributions **(**cover
densities**)** ***g, *besides the delimitation of the development interval, cause the
difference between the many series expansions used in mathematical physics. Because of their importance for **wave
mechanics**, we mention here those of the **Hermite and Laguerre polynomials**.
We will not yet deal here with their formal treatment - it could be found as in the case of the spherical functions from the
demand of the best possible and finally valid, step by step determination of the coefficients (Exercise
1.6, where also a normalization rule is indicated); their orthogonality would then be again a necessary consequence of these
conditions. In fact, we will limit consideration to presenting the most important features of the series of polynomials in
a table:

While by these series, just as by Fourier and spherical function series,
the approximation to the image function *f* is achievedby
increasingly approaching oscillations of the successive approximations, we know from the rudiments of differential calculus a
series which does not have an **oscillating**, but an **osculating
character**, namely the **Taylor series**.
With it, the successive approximations *S*_{n}* *approach the task curve at a preferred point of it
in such a way that *S*_{n}* *has in common with the curve *f* all derivatives up to *f*^{(n)}.
Fig. 6 demonstrates the graphics of the power series of sin *x*.

Here, the entire accuracy is concentrated at a single point. With **Dirac**,
we can state briefly that *g*(*x*) is degenerated into a *d*-function. In fact, he
defines as a functional analogue to the algebraic symbol d_{kl}*
*in (1.6) an extreme unsteady function *d*(*x|x*_{0})by
the conditions

for quite arbitrary *e* . We see that in
the case of the Taylor expansion of Fig. 6, in which especially *x*_{0} = 0:

**1.6.2
An-harmonic Fourier analysis:**** **While
we have studied in 1.1 to 1.3 Fourier series which advance by the **harmonic**** **(integer)
**overtones** of a **fundamental tone**, we
now face, with a view to heat conduction problems (3.16), the task of
expanding an arbitrary function *f*(*x*) in the interval 0<*x*<*p* in
the series

where the *l*_{k }are
given by the roots of a **transcendental equation****
**, say,

with *a** *an arbitrary number.
Fig. 7 shows that the equation has infinitely many roots *l*_{k}. We will
encounter another equation of this kind in Exercise 2.1.

First of all, we will show that the sin *l*_{k}*x
*form an* *orthogonal system with weight factor *g*(*x*) = 1, i.e., that

Indeed, one readily obtains by transition from the sin-product to the cos-functions
of the sum and difference for the left hand side of (3)

where now the brackets disappear due to (2a). For *k *= *l*,
one finds in the same way

**(3a)**

This manipulation, which rests on special formulae for the trigonometric functions, will be executed less formally with application of Green's theorem in 3.16.

Equations (3) and (3a) yield for the expansion
coefficients *B*_{k}* *in (1)

This value of *B*_{k}* *is **final
**in the sense of 1.5, because it does not depend on *n *and
makes the mean square error of the approximation

into a minimum, whence the matter of the convergence and completeness is resolved in that the mean square error goes to zero as

*n ***®****
****¥** .

**1.6.3
An example of non-final determination of coefficients****:**
On the other hand, we will consider a much more involved case in which the final validity
demand is not met, as a preparation for an optical (or it is better to say quasi-optical) application. It is a metal mirror which
at first has circular cylindrical shape (Fig. 8) The electric vector of the total oscillation, the direction of which may be
perpendicular to the plane of the drawing, consists of the entering wave *w*, represented
on the mirror by

and the reflected (dispersed ) wave to be denoted by

We must then demand that

firstly, because of the assumed infinitely large conductivity, secondly, because of the required steady link-up of the two fields.

Assuming *w* to be symmetrical with respect to the central axis of
the mirror (for example, as a plane, in this direction advancing wave), we write *

We will see later on in (4.19)
that *g*_{n} and *h*_{n} are Bessel and Hankel functions, respectively; they can
be chosen so that

Then, (5) and the first equation (6) demand

Joining these two equations, one concludes that

independently of whether the preceding sums are extended over all possible
integers *n* or, which is the more general approach, only over the finite number *N , *i.e.,

Hereby Condition (9) is met. Condition (8) yet demands

Moreover, we must still satisfy the second equation, which by (7) becomes

We have added here the factor *a* in front of the bracket in order
to make *g*_{n}* *into a pure number, which we may do due to (11). *C*_{n}*
*must be determined from (10) and (11).

*
It is advisable, with a view to the notation to be used in Chapter 4, to change already here the
summation index *k* into *n*. Instead of *n *employed so far, we will write *N *and use instead of *l*
the subscript *m*.

Once again, this is indicated by the least square method. Thus, we
consider, demanding in the scale of the *j *equal accuracy, the square errors, corresponding to (10) and (11),

The sum of both of them is to be made into a minimum by the choice of *C*_{n}*.
*This yields, by differentiation with respect to *C*_{n}, a system of *N* + 1 linear equations
for *C*_{0}, ··· , *C*_{n}, ··· , *C*_{N}, the (*m *+
1)-th equation being

During the limit , we
would obtain an **infinite system of linear equations for the infinite number of unknowns C**