Jean-Baptiste Fourier (French, 1:02:36)
by Jean-Pierre Kahane, 1926-2017 (Institut Fourier, 2011年05月18日).
Séries trigonométriques lacunaires et
prolongement minimal (French, 59:03)
by Jean-Pierre Kahane, 1926-2017 (CIRM, 2015年12月02日).
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.
Consider the function f defined by the following sum.
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 :
This is true because...
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年12月21日) 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).
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:
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).
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 :
Wikipedia : Dirichlet_conditions
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:
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 :
tan (p/2 - e) sin 2n (p/2 - e) = - (cos np) (sin 2ne) / (tan e) ~ 2n (-1)n+1
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:
Fourier Expansion of tan(x)
by Jeremy Orloff (MIT).
Numericana :
Divergent Series Redux
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Cauchy principal values
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).
Wikipedia : D. E. Menshov (1892-1988)
This can be obtained by integrating a rectangular wave.
