Unlike most Numericana pages,
which generally starts with the easiest aspects of a topic,
this page starts with a fairly complete historical discussion about the convergence and unicity of Fourier series
(with tough counterexamples and the introduction of summable divergent series) before presenting a tame
practical catalog of noteworthy Fourier series.
(2018-05-17) Euler-Fourier formulas (Euler, 1777)
Term-by-term integration retrieves a harmonic component from a sum.
Consider the function f defined by the following sum.
f (x)
ao
¥
å
n=1
[ an cos(nx) + bn sin(nx) ]
2
For introductory pursoses,
we could have presented the sum as finite,
but there's no great difficulty in generalizing to a
convergent infinite series.
(We'll even consider one example of a
divergent series shortly.)
The form of the constant term is just for future convenience
(it make the formulas below valid for n=0 and would also a seamless
complex expression therefof).
Now, if f is assumed to be of the above
form but not directly given as such, we can retrieve
the coefficients of the triginometric series with the following equations,
known as Euler's formulas :
an
1
ò
2p
0
f (x) cos(nx) dx
p
bn
1
ò
2p
0
f (x) sin(nx) dx
p
This is true because...
(2018-05-17) Fourier expansions (Fourier, 1807)
Could any function be just the sum of all its harmonic components?
Well, the answer turns out to be no,
in particular because the only type of discontinuity
which a convergent infinite sum of harmonic functions can possess is a
jump discontinuity where the value at the point of discontinuity is
the half-sum of its left-limit and right-limit.
That's no big deal in most practical applications where
we may as well consider only normalized functions whose values at
a few jump discontinuities have been redefined as needed to make this true:
f (x) = ½
[ f (x-) + f (x+) ]
Indeed, functions which have a left-limit and a right-limit at every point
can be so normalized.
This is clearly no all there is to it, but this was good enough at first for
Joseph Fourier (1768-1830)
whose primary concern was to have a shot at solving differential equations
whose restrictions to harmonic functions were easily dealt with
(especially the equation of heat).
He left open for future generationss the intricate theoretical considerations so raised.
Another key contribution of Fourier was to realize that,
over any bounded interval, any function is equal to the restriction to
that interval of a periodic function with a period at least equal
to the length of the interval. This paved the way
for the revolutionary notion of a Fourier transforms.
When he submitted his work to the French Academy of Sciences (1807-12-21)
Fourier's mathematics wasn't yet airtight and he did get some
flak for it from
Lagrange and
Laplace after he submitted his work, in 1808.
Nevertheles, Fourier's momentous ideas got him a well-deserved prize in 1811 for
solving the equation of heat (the aforementioned objections still held back publication).
(2018-05-14) The Dirichlet conditions
(Dirichlet, 1828)
Sufficient conditions for a function to be equal to its Fourier expansion.
A real-valued periodic function of a real variable
is a function f for which there's
a number T such that f (x+T)
and f (x) are equal for any value of x.
Such a number T is then called a period of f.
When f is constant, any number will do.
Otherwise, there's a smallest positive period, called
the period of f and any period is an integral
multiple of that simplest one.
Without loss of generality, we'll consider only functions
for which 2p is a period
(any other periodic function of x with period T
is of the form f (2px/T),
where f is a function of period 2p).
Such a periodic function is said to satisfy the Dirichlet conditions when:
It's of bounded variation (i.e., [0,2p]
contains only finitely many extrema).
There are only finitely many points of discontinuity in [0,2p]
and all of them are just jump discontinuities (i.e., the left-limit and right-limit
are both well-defined and finite).
The value at every point is equal the half-sum of the left-limit and right-limit.
The last condition is often not listed among Dirichlet's conditions (for historical reasons)
but I beg to differ. It's the only viewpoint which make the following theorem easy to state and,
more importantly, easy to use. Every function which doesn't verify it
is best thrown out in favor of a normalized function which does
(lest the tools of harmonic analysis become unwieldy). In particular,
I argue against maintaining two versions of the
prime-counting function p
(just drop the nought subscript historically given to the normalized version).
In 18??, Dirichlet
proved that this is a sufficient condition for a function to
have a Fourier expansion whose coefficients are given by Euler's formula.
This is, by no means, a necessary
condition since much weirder functions do.
In fact, Georg Cantor
was originally led to the study of sets of real when he
wondered about the possible collections of dicontinuities a
Dourier sum could have. (At first Cantor defined sets
only over the real line).
Dirichlet's theorem :
If a (normalized) periodic function f
verifies the above Dirichlet conditions,
then it is equal to the convergent sum of a trigonometric series
(called its Fourier expansion) whose coefficients are given by
the Euler formulas below.
Or, more formally:
If a function f (x) = ½
[ f (x-) + f (x+)]
has period 2p and verifies
Dirichlet's conditions then it's equal to the sum of the following
convergent Fourier expansion :
f (x)
ao
¥
å
n=1
[ an cos(nx) + bn sin(nx) ]
2
The coefficients an and bn are equal to
twice the average values of
cos(nx) f (x) and sin(nx) f (x)
as given by Euler's formulas, namely:
(2018-05-15)
The Fourier expansion of the tangent function:
It's a divergent series everywhere
(except at multiples of p/2).
Most functions we discuss satisfy the above Dirichlet conditions
but the tangent function doesn't
(its singularities aren't jump discontinuities).
tan x = tg x = sin x / cos x
Dirichlet's theorem doesn't apply in this case.
Actually, that function has a Fourier expansion
which doesn't converge at any point besides its zeroes and singularities
(where all terms of the series are trivially zero). Namely:
2 sin 2x - 2 sin 4x +
2 sin 6x - 2 sin 8x +
2 sin 10x ...
In this, the coefficient cn = 2 (-1)n+1
of sin 2nx was obtained thusly:
cn =
2
ó
p
tan x sin 2nx dx
p
õ
0
This is just Euler's formula as an
integral over an interval containing just one singularity (at p/2).
We don't even have to worry about
Cauchy principal values since the integrand itself has a
finite limit at p/2 :
Manipulating this monstrosity will invariably result in divergent series which necessitate
summation methods beyond regular convergence.
For example, its derivative at 0 is given by:
2 (2 - 4 + 6
- 8 + 10
- 12 + ... ) = 1
More interestingly, integration yields a
convergent Fourier expansion for a periodic function
which doesn't satisfy Dirichlet's conditions:
-Log | cos x | = Log 2 -
cos 2x + (cos 4x)/2 - (cos 6x)/3 ...
The constant term is a constant of integration obtained by equating the two sides
for x = 0,
knowing one way Log 2 can be expressed as a series.
This gives the (otherwise nontrivial) value of one improper integral:
(2018-05-18) Riemann-Cantor theorem and the birth of Set Theory.
The zero series is the only trigonometric series vanishing everywhere.
This fundamental result was famously postulated by
Bernhard Riemann (1826-1866)
in his Habilitationsschrift
(1854)
which was only published posthumously (1867)
by Richard Dedekind (1831-1916).
The theorem itself was proved by
Georg Cantor (1845-1918)
when he was 25 (1870).
That establishes the uniqueness of convergent Fourier expansions
(since the difference of two Fourier series of equal sums is a trigonometric
series which converges to zero everywhere).
The next year (1871) Cantor defined a
set of uniqueness
(U-set) as a set of points
such that the only trigonometric series which converges to zero
everywhere outside of it is the trivial zero series
(all coefficients must be 0).
In 1872, Cantor inroduced limit-points
and properly defined real numbers as equivalence classes
of Cauchy sequences.
He defined the derived set
E' of a set E as the set of the limit-points of E
(which made the newly-defined real numbers form the derived set of the rationals).
This is arguably what launched Set Theory,
originally limited to subsets of the real numbers.
At that time, Cantor proved that the derived set of a U-set is a U-set.
He went on to show that a set with countably many
limit-points is a U-set.
That establishes the unicity of the Fourier expansions of functions well beyond the
limited scope of Dirichlet's conditions.
The subsets of the trigonometric circle (i.e.,
the reals modulo 2p)
which are not U-sets are known as M-sets or
multiplicity sets.
They're also called Menshov sets,
in honor of Dmitrii Menshov
(1892-1988) who established in 1916 that there are some M-sets of measure zero
(thereby disproving an earlier conjecture).
In 1958,
U-sets were the subject of the doctoral dissertation of
Paul Cohen (1934-2008)
who went on to earn a Fields Medal in 1966, for his
proof of the undecidability
(in the sense of Gödel)
of Cantor's Continuum Hypothesis (1963).
