In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the question of whether the
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
of a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
converges to a given
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is researched by a field known as classical harmonic analysis, a branch of
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.
Determination of convergence requires the comprehension of
pointwise convergence
In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
,
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
,
absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
,
''L''''p'' spaces,
summability method
In mathematics, a divergent series is an infinite series that is not Convergent series, convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit of a sequence, limit.
If a series converges, t ...
s and the
Cesàro mean.
Preliminaries
Consider ''f'' an
integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
function on the interval . For such an ''f'' the Fourier coefficients
are defined by the formula
:
It is common to describe the connection between ''f'' and its Fourier series by
:
The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined:
:
The question of whether a Fourier series converges is: Do the functions
(which are functions of the variable ''t'' we omitted in the notation) converge to ''f'' and in which sense? Are there conditions on ''f'' ensuring this or that type of convergence?
Before continuing, the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any nonneg ...
must be introduced. Taking the formula for
, inserting it into the formula for
and doing some algebra gives that
:
where ∗ stands for the periodic
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
and
is the Dirichlet kernel, which has an explicit formula,
:
The Dirichlet kernel is ''not'' a positive kernel, and in fact, its norm diverges, namely
:
a fact that plays a crucial role in the discussion. The norm of ''D''
''n'' in ''L''
1(T) coincides with the norm of the convolution operator with ''D''
''n'',
acting on the space ''C''(T) of periodic continuous functions, or with the norm of the linear functional ''f'' → (''S''
''n''''f'')(0) on ''C''(T). Hence, this family of linear functionals on ''C''(T) is unbounded, when ''n'' → ∞.
Magnitude of Fourier coefficients
In applications, it is often useful to know the size of the Fourier coefficient.
If
is an
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
function,
:
for
a constant that only depends on
.
If
is a
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
function,
:
If
:
If
and
has
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if
:, f(x)-f ...
,
:
and therefore, if
is in the α-
Hölder class Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
:
Pointwise convergence
There are many known sufficient conditions for the Fourier series of a function to converge at a given point ''x'', for example if the function is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at ''x''. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at ''x'', then the Fourier series converges to the average of the left and right limits (but see
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...
).
The Dirichlet–Dini Criterion states that: if ''ƒ'' is 2–periodic, locally integrable and satisfies
:
then (S
''n''''f'')(''x''
0) converges to ℓ. This implies that for any function ''f'' of any
Hölder class Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
''α'' > 0, the Fourier series converges everywhere to ''f''(''x'').
It is also known that for any periodic function of
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
, the Fourier series converges everywhere. See also
Dini test In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz.
Definition
Let ...
.
In general, the most common criteria for pointwise convergence of a periodic function ''f'' are as follows:
* If ''f'' satisfies a Holder condition, then its Fourier series converges uniformly.
* If ''f'' is of bounded variation, then its Fourier series converges everywhere.
* If ''f'' is continuous and its Fourier coefficients are absolutely summable, then the Fourier series converges uniformly.
There exist continuous functions whose Fourier series converges pointwise but not uniformly; see Antoni Zygmund, ''
Trigonometric Series
In mathematics, a trigonometric series is a infinite series of the form
: \frac+\displaystyle\sum_^(A_ \cos + B_ \sin),
an infinite version of a trigonometric polynomial.
It is called the Fourier series of the integrable function f if the term ...
'', vol. 1, Chapter 8, Theorem 1.13, p. 300.
However, the Fourier series of a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in ''L''
1(T) and the Banach–Steinhaus
uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
. As typical for existence arguments invoking the
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
, this proof is nonconstructive. It shows that the family of continuous functions whose Fourier series converges at a given ''x'' is of
first Baire category, in the
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
of continuous functions on the circle.
So in some sense pointwise convergence is ''atypical'', and for most continuous functions the Fourier series does not converge at a given point. However
Carleson's theorem
Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the res ...
shows that for a given continuous function the Fourier series converges almost everywhere.
It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function ''f'' defined for all ''x'' in
,πby
:
Uniform convergence
Suppose
, and
has
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if
:, f(x)-f ...
; then the partial sums of the Fourier series converge to the function with speed
:
for a constant
that does not depend upon
, nor
, nor
.
This theorem, first proved by D Jackson, tells, for example, that if
satisfies the
-
Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that
: , f(x) - f(y) , \leq C\, ...
, then
:
If
is
periodic and absolutely continuous on