Beginnings

Joseph Fourier wrote^[13]

.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}

φ
(
y
)
=

a

0

cos
⁡

π
y

2

+

a

1

cos
⁡
3

π
y

2

+

a

2

cos
⁡
5

π
y

2

+
⋯
.

{displaystyle varphi (y)=a_{0}cos {frac {pi y}{2}}+a_{1}cos 3{frac {pi y}{2}}+a_{2}cos 5{frac {pi y}{2}}+cdots .}

Multiplying both sides by

cos
⁡
(
2
k
+
1
)

π
y

2

{displaystyle cos(2k+1){frac {pi y}{2}}}

, and then integrating from

y
=
−
1

{displaystyle y=-1}

to

y
=
+
1

{displaystyle y=+1}

yields:

a

k

=

∫

−
1

1

φ
(
y
)
cos
⁡
(
2
k
+
1
)

π
y

2

d
y
.

{displaystyle a_{k}=int _{-1}^{1}varphi (y)cos(2k+1){frac {pi y}{2}},dy.}

This immediately gives any coefficient a_k of the trigonometric series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral

∫

−
1

1

φ
(
y
)
cos
⁡
(
2
k
+
1
)

π
y

2

d
y

=

∫

−
1

1

(

a
cos
⁡

π
y

2

cos
⁡
(
2
k
+
1
)

π
y

2

+

a
′

cos
⁡
3

π
y

2

cos
⁡
(
2
k
+
1
)

π
y

2

+
⋯

)

d
y

{displaystyle {begin{aligned}&int _{-1}^{1}varphi (y)cos(2k+1){frac {pi y}{2}},dy\&=int _{-1}^{1}left(acos {frac {pi y}{2}}cos(2k+1){frac {pi y}{2}}+a’cos 3{frac {pi y}{2}}cos(2k+1){frac {pi y}{2}}+cdots right),dyend{aligned}}}

can be carried out term-by-term. But all terms involving

cos
⁡
(
2
j
+
1
)

π
y

2

cos
⁡
(
2
k
+
1
)

π
y

2

{displaystyle cos(2j+1){frac {pi y}{2}}cos(2k+1){frac {pi y}{2}}}

for j ≠ k vanish when integrated from −1 to 1, leaving only the

k

th

{displaystyle k^{text{th}}}

term, which is 1.

In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d’Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.

When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: “…the manner in which the author arrives at these equations is not exempt of difficulties and…his analysis to integrate them still leaves something to be desired on the score of generality and even rigour“.^[14]

Fourier’s motivation

The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula

s
(
x
)
=

x
π

{displaystyle s(x)={tfrac {x}{pi }}}

, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier’s motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose sides measure

π

{displaystyle pi }

meters, with coordinates

(
x
,
y
)
∈
[
0
,
π
]
×
[
0
,
π
]

{displaystyle (x,y)in [0,pi ]times [0,pi ]}

. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by

y
=
π

{displaystyle y=pi }

, is maintained at the temperature gradient

T
(
x
,
π
)
=
x

{displaystyle T(x,pi )=x}

degrees Celsius, for

x

{displaystyle x}

in

(
0
,
π
)

{displaystyle (0,pi )}

, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by

T
(
x
,
y
)
=
2

∑

n
=
1

∞

(
−
1

)

n
+
1

n

sin
⁡
(
n
x
)

sinh
⁡
(
n
y
)

sinh
⁡
(
n
π
)

.

{displaystyle T(x,y)=2sum _{n=1}^{infty }{frac {(-1)^{n+1}}{n}}sin(nx){sinh(ny) over sinh(npi )}.}

Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of the equation from Analysis § Example by

sinh
⁡
(
n
y
)

/

sinh
⁡
(
n
π
)

{displaystyle sinh(ny)/sinh(npi )}

. While our example function

s
(
x
)

{displaystyle s(x)}

seems to have a needlessly complicated Fourier series, the heat distribution

T
(
x
,
y
)

{displaystyle T(x,y)}

is nontrivial. The function

T

{displaystyle T}

cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier’s work.

Other applications

Another application is to solve the Basel problem by using Parseval’s theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.

Synthesis

A Fourier series can be written in several equivalent forms, shown here as the

N

th

{displaystyle N^{text{th}}}

partial sums

s

N

(
x
)

{displaystyle s_{N}(x)}

of the Fourier series of

s
(
x
)

{displaystyle s(x)}

:^[21]

The harmonics are indexed by an integer,

n
,

{displaystyle n,}

which is also the number of cycles the corresponding sinusoids make in interval

P

{displaystyle P}

. Therefore, the sinusoids have:

• a wavelength equal to

  P
  n

  {displaystyle {tfrac {P}{n}}}

  in the same units as

  x

  {displaystyle x}

  .

• a frequency equal to

  n
  P

  {displaystyle {tfrac {n}{P}}}

  in the reciprocal units of

  x

  {displaystyle x}

  .

These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum. In the limit

N
→
∞

{displaystyle Nto infty }

, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.

Analysis

The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform. In the latter case, the exponential form of Fourier series synthesizes a discrete-time Fourier transform where variable

x

{displaystyle x}

represents frequency instead of time. In general, the coefficients are determined by analysis of a given function

s
(
x
)

{displaystyle s(x)}

whose domain of definition is an interval of length

P

{displaystyle P}

.^[B][22]

The

2
P

{displaystyle {tfrac {2}{P}}}

scale factor follows from substituting Eq.1 into Eq.3 and utilizing the orthogonality of the trigonometric system.^[23] The equivalence of Eq.1 and Eq.2 follows from Euler’s formula

cos
⁡
x
=

e

i
x

+

e

−
i
x

2

,

sin
⁡
x
=

e

i
x

−

e

−
i
x

2
i

,

{displaystyle cos x={frac {e^{ix}+e^{-ix}}{2}},quad sin x={frac {e^{ix}-e^{-ix}}{2i}},}

resulting in:

with

c

0

{displaystyle c_{0}}

being the mean value of

s

{displaystyle s}

on the interval

P

{displaystyle P}

.^[24] Conversely:

Amplitude-phase form

If the function

s
(
x
)

{displaystyle s(x)}

is real-valued then the Fourier series can also be represented as^[25][26]

where

A

n

{displaystyle A_{n}}

is the amplitude and

φ

n

{displaystyle varphi _{n}}

is the phase shift of the

n

t
h

{displaystyle n^{th}}

harmonic.

The equivalence of Eq.4 and Eq.1 follows from the trigonometric identity:

cos
⁡

(

2
π

n
P

x
−

φ

n

)

=
cos
⁡
(

φ

n

)
cos
⁡

(

2
π

n
P

x

)

+
sin
⁡
(

φ

n

)
sin
⁡

(

2
π

n
P

x

)

,

{displaystyle cos left(2pi {tfrac {n}{P}}x-varphi _{n}right)=cos(varphi _{n})cos left(2pi {tfrac {n}{P}}xright)+sin(varphi _{n})sin left(2pi {tfrac {n}{P}}xright),}

which implies^[27]

a

n

=

A

n

cos
⁡
(

φ

n

)

and

b

n

=

A

n

sin
⁡
(

φ

n

)

{displaystyle a_{n}=A_{n}cos(varphi _{n})quad {text{and}}quad b_{n}=A_{n}sin(varphi _{n})}

are the rectangular coordinates of a vector written in polar coordinates as

A

n

∠

φ

n

=

a

n

+
i

b

n

{displaystyle A_{n}angle varphi _{n}=a_{n}+ib_{n}}

where

A

n

=

a

n

2

+

b

n

2

and

φ

n

=
atan2
⁡
(

b

n

,

a

n

)
=
−
Arg
⁡
(

c

n

)

{displaystyle A_{n}={sqrt {a_{n}^{2}+b_{n}^{2}}}quad {text{and}}quad varphi _{n}=operatorname {atan2} (b_{n},a_{n})=-operatorname {Arg} (c_{n})}

An example of determining the parameter

φ

n

{displaystyle varphi _{n}}

for one value of

n

{displaystyle n}

is shown in Figure 2. It is the value of

φ

{displaystyle varphi }

at the maximum correlation between

s
(
x
)

{displaystyle s(x)}

and a cosine template,

cos
⁡
(
2
π

n
P

x
−
φ
)

{displaystyle cos(2pi {tfrac {n}{P}}x-varphi )}

. The blue graph is the cross-correlation function, also known as a matched filter:

X

(
φ
)

=

∫

P

s
(
x
)
⋅
cos
⁡

(

2
π

n
P

x
−
φ

)

d
x

φ
∈

[

0
,
2
π

]

=
cos
⁡
(
φ
)

∫

P

s
(
x
)
⋅
cos
⁡

(

2
π

n
P

x

)

d
x

⏟

X
(
0
)

+
sin
⁡
(
φ
)

∫

P

s
(
x
)
⋅
sin
⁡

(

2
π

n
P

x

)

d
x

⏟

X
(
π

/

2
)

{displaystyle {begin{aligned}mathrm {X} (varphi )&=int _{P}s(x)cdot cos left(2pi {tfrac {n}{P}}x-varphi right),dxquad varphi in left[0,2pi right]\&=cos(varphi )underbrace {int _{P}s(x)cdot cos left(2pi {tfrac {n}{P}}xright)dx} _{X(0)}+sin(varphi )underbrace {int _{P}s(x)cdot sin left(2pi {tfrac {n}{P}}xright)dx} _{X(pi /2)}end{aligned}}}

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum:

X
′

(
φ
)
=
sin
⁡
(
φ
)
⋅
X
(
0
)
−
cos
⁡
(
φ
)
⋅
X
(
π

/

2
)
=
0
,

at

 
φ
=

φ

n

.

{displaystyle X'(varphi )=sin(varphi )cdot X(0)-cos(varphi )cdot X(pi /2)=0,quad {textrm {at}} varphi =varphi _{n}.}

Hence

φ

n

≡
arctan
⁡
(

b

n

/

a

n

)
=
arctan
⁡
(
X
(
π

/

2
)

/

X
(
0
)
)
.

{displaystyle varphi _{n}equiv arctan(b_{n}/a_{n})=arctan(X(pi /2)/X(0)).}

Common notations

The notation

c

n

{displaystyle c_{n}}

is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (

s
,

{displaystyle s,}

in this case), such as

s
^

(
n
)

{displaystyle {widehat {s}}(n)}

or

S
[
n
]
,

{displaystyle S[n],}

and functional notation often replaces subscripting:

s
(
x
)

=

∑

n
=
−
∞

∞

s
^

(
n
)
⋅

e

i
2
π

n
P

x

common mathematics notation

=

∑

n
=
−
∞

∞

S
[
n
]
⋅

e

i
2
π

n
P

x

common engineering notation

{displaystyle {begin{aligned}s(x)&=sum _{n=-infty }^{infty }{widehat {s}}(n)cdot e^{i2pi {tfrac {n}{P}}x}&&scriptstyle {text{common mathematics notation}}\&=sum _{n=-infty }^{infty }S[n]cdot e^{i2pi {tfrac {n}{P}}x}&&scriptstyle {text{common engineering notation}}end{aligned}}}

In engineering, particularly when the variable

x

{displaystyle x}

represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:

S
(
f
)
 
≜
 

∑

n
=
−
∞

∞

S
[
n
]
⋅
δ

(

f
−

n
P

)

,

{displaystyle S(f) triangleq sum _{n=-infty }^{infty }S[n]cdot delta left(f-{frac {n}{P}}right),}

where

f

{displaystyle f}

represents a continuous frequency domain. When variable

x

{displaystyle x}

has units of seconds,

f

{displaystyle f}

has units of hertz. The “teeth” of the comb are spaced at multiples (i.e. harmonics) of

1
P

{displaystyle {tfrac {1}{P}}}

, which is called the fundamental frequency.

s
(
x
)

{displaystyle s(x)}

can be recovered from this representation by an inverse Fourier transform:

F

−
1

{
S
(
f
)
}

=

∫

−
∞

∞

(

∑

n
=
−
∞

∞

S
[
n
]
⋅
δ

(

f
−

n
P

)

)

e

i
2
π
f
x

d
f
,

=

∑

n
=
−
∞

∞

S
[
n
]
⋅

∫

−
∞

∞

δ

(

f
−

n
P

)

e

i
2
π
f
x

d
f
,

=

∑

n
=
−
∞

∞

S
[
n
]
⋅

e

i
2
π

n
P

x

 
 
≜
 
s
(
x
)
.

{displaystyle {begin{aligned}{mathcal {F}}^{-1}{S(f)}&=int _{-infty }^{infty }left(sum _{n=-infty }^{infty }S[n]cdot delta left(f-{frac {n}{P}}right)right)e^{i2pi fx},df,\[6pt]&=sum _{n=-infty }^{infty }S[n]cdot int _{-infty }^{infty }delta left(f-{frac {n}{P}}right)e^{i2pi fx},df,\[6pt]&=sum _{n=-infty }^{infty }S[n]cdot e^{i2pi {tfrac {n}{P}}x} triangleq s(x).end{aligned}}}

The constructed function

S
(
f
)

{displaystyle S(f)}

is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.^[C]

Symmetry relations

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:^[30][31]

T
i
m
e
 
d
o
m
a
i
n

s

=

s

R
E

+

s

R
O

+

i
 

s

I
E

+

i
 

s

I
O

⇕

F

⇕

F

 
 

⇕

F

 
 

⇕

F

 
 

⇕

F

F
r
e
q
u
e
n
c
y
 
d
o
m
a
i
n

S

=

S

R
E

+

i
 

S

I
O

+

i
 

S

I
E

+

S

R
O

{displaystyle {begin{array}{rlcccccccc}{mathsf {Time domain}}&s&=&s_{mathrm {RE} }&+&s_{mathrm {RO} }&+&i s_{mathrm {IE} }&+&i s_{mathrm {IO} }\&{Bigg Updownarrow }{mathcal {F}}&&{Bigg Updownarrow }{mathcal {F}}&& {Bigg Updownarrow }{mathcal {F}}&& {Bigg Updownarrow }{mathcal {F}}&& {Bigg Updownarrow }{mathcal {F}}\{mathsf {Frequency domain}}&S&=&S_{mathrm {RE} }&+&i S_{mathrm {IO} },&+&i S_{mathrm {IE} }&+&S_{mathrm {RO} }end{array}}}

From this, various relationships are apparent, for example:

• The transform of a real-valued function

  (

  s

  R
  E

  +

  s

  R
  O

  )

  {displaystyle (s_{mathrm {RE} }+s_{mathrm {RO} })}

  is the conjugate symmetric function

  S

  R
  E

  +
  i
   

  S

  I
  O

  .

  {displaystyle S_{mathrm {RE} }+i S_{mathrm {IO} }.}

  Conversely, a conjugate symmetric transform implies a real-valued time-domain.

• The transform of an imaginary-valued function

  (
  i
   

  s

  I
  E

  +
  i
   

  s

  I
  O

  )

  {displaystyle (i s_{mathrm {IE} }+i s_{mathrm {IO} })}

  is the conjugate antisymmetric function

  S

  R
  O

  +
  i
   

  S

  I
  E

  ,

  {displaystyle S_{mathrm {RO} }+i S_{mathrm {IE} },}

  and the converse is true.

• The transform of a conjugate symmetric function

  (

  s

  R
  E

  +
  i
   

  s

  I
  O

  )

  {displaystyle (s_{mathrm {RE} }+i s_{mathrm {IO} })}

  is the real-valued function

  S

  R
  E

  +

  S

  R
  O

  ,

  {displaystyle S_{mathrm {RE} }+S_{mathrm {RO} },}

  and the converse is true.

• The transform of a conjugate antisymmetric function

  (

  s

  R
  O

  +
  i
   

  s

  I
  E

  )

  {displaystyle (s_{mathrm {RO} }+i s_{mathrm {IE} })}

  is the imaginary-valued function

  i
   

  S

  I
  E

  +
  i
   

  S

  I
  O

  ,

  {displaystyle i S_{mathrm {IE} }+i S_{mathrm {IO} },}

  and the converse is true.

Riemann–Lebesgue lemma

If

S

{displaystyle S}

is integrable,

lim

|

n

|

→
∞

S
[
n
]
=
0

{textstyle lim _{|n|to infty }S[n]=0}

,

lim

n
→
+
∞

a

n

=
0

{textstyle lim _{nto +infty }a_{n}=0}

and

lim

n
→
+
∞

b

n

=
0.

{textstyle lim _{nto +infty }b_{n}=0.}

Parseval’s theorem

If

s

{displaystyle s}

belongs to

L

2

(
P
)

{displaystyle L^{2}(P)}

(periodic over an interval of length

P

{displaystyle P}

) then:

1
P

∫

P

|

s
(
x
)

|

2

d
x
=

∑

n
=
−
∞

∞

|

S
[
n
]

|

2

.

{displaystyle {frac {1}{P}}int _{P}|s(x)|^{2},dx=sum _{n=-infty }^{infty }{Bigl |}S[n]{Bigr |}^{2}.}

Plancherel’s theorem

If

c

0

,

c

±
1

,

c

±
2

,
…

{displaystyle c_{0},,c_{pm 1},,c_{pm 2},ldots }

are coefficients and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{textstyle sum _{n=-infty }^{infty }|c_{n}|^{2}

∑

n
=
−
∞

∞

|

c

n

|

2

<
∞

{textstyle sum _{n=-infty }^{infty }|c_{n}|^{2}<infty }

<img src="//wikimedia.org/api/rest_v1/media/math/render/svg/9a28fe1237efdca7ee9ae68f6351757eb7aa1022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.255ex; height:3.509ex;" alt="{textstyle sum _{n=-infty }^{infty }|c_{n}|^{2} then there is a unique function

s
∈

L

2

(
P
)

{displaystyle sin L^{2}(P)}

such that

S
[
n
]
=

c

n

{displaystyle S[n]=c_{n}}

for every

n

{displaystyle n}

.

Convolution theorems

Given

P

{displaystyle P}

-periodic functions,

s

P

{displaystyle s_{P}}

and

r

P

{displaystyle r_{P}}

with Fourier series coefficients

S
[
n
]

{displaystyle S[n]}

and

R
[
n
]
,

{displaystyle R[n],}

n
∈

Z

,

{displaystyle nin mathbb {Z} ,}

• The pointwise product:

  h

  P

  (
  x
  )
  ≜

  s

  P

  (
  x
  )
  ⋅

  r

  P

  (
  x
  )

  {displaystyle h_{P}(x)triangleq s_{P}(x)cdot r_{P}(x)}

  is also

  P

  {displaystyle P}

  -periodic, and its Fourier series coefficients are given by the discrete convolution of the

  S

  {displaystyle S}

  and

  R

  {displaystyle R}

  sequences:

  H
  [
  n
  ]
  =
  {
  S
  ∗
  R
  }
  [
  n
  ]
  .

  {displaystyle H[n]={S*R}[n].}

  

• The periodic convolution:

  h

  P

  (
  x
  )
  ≜

  ∫

  P

  s

  P

  (
  τ
  )
  ⋅

  r

  P

  (
  x
  −
  τ
  )

  d
  τ

  {displaystyle h_{P}(x)triangleq int _{P}s_{P}(tau )cdot r_{P}(x-tau ),dtau }

  is also

  P

  {displaystyle P}

  -periodic, with Fourier series coefficients:

  H
  [
  n
  ]
  =
  P
  ⋅
  S
  [
  n
  ]
  ⋅
  R
  [
  n
  ]
  .

  {displaystyle H[n]=Pcdot S[n]cdot R[n].}

  

• A doubly infinite sequence

  {

  c

  n

  }

  n
  ∈
  Z

  {displaystyle left{c_{n}right}_{nin Z}}

  in

  c

  0

  (

  Z

  )

  {displaystyle c_{0}(mathbb {Z} )}

  is the sequence of Fourier coefficients of a function in

  L

  1

  (
  [
  0
  ,
  2
  π
  ]
  )

  {displaystyle L^{1}([0,2pi ])}

  if and only if it is a convolution of two sequences in

  ℓ

  2

  (

  Z

  )

  {displaystyle ell ^{2}(mathbb {Z} )}

  . See ^[32]

Derivative property

If

s

{displaystyle s}

is a

P

{displaystyle P}

-periodic function on

R

{displaystyle mathbb {R} }

which is

k

{displaystyle k}

times differentiable, and its

k

th

{displaystyle k^{text{th}}}

derivative is continuous, then

s

{displaystyle s}

belongs to the function space

C

k

(

R

)

{displaystyle C^{k}(mathbb {R} )}

.

• If

  s
  ∈

  C

  k

  (

  R

  )

  {displaystyle sin C^{k}(mathbb {R} )}

  , then the Fourier coefficients of the

  k

  th

  {displaystyle k^{text{th}}}

  derivative of

  s

  {displaystyle s}

  can be expressed in terms of the Fourier coefficients

  s
  ^

  [
  n
  ]

  {displaystyle {widehat {s}}[n]}

  of

  s

  {displaystyle s}

  , via the formula

  s

  (
  k
  )

  ^

  [
  n
  ]
  =
  (
  i

  2
  π
  n

  P

  )

  k

  s
  ^

  [
  n
  ]
  .

  {displaystyle {widehat {s^{(k)}}}[n]=(i{frac {2pi n}{P}})^{k}{widehat {s}}[n].}

  In particular, since for any fixed

  k
  ≥
  1

  {displaystyle kgeq 1}

  we have

  s

  (
  k
  )

  ^

  [
  n
  ]
  →
  0

  {displaystyle {widehat {s^{(k)}}}[n]to 0}

  as

  n
  →
  ∞

  {displaystyle nto infty }

  , it follows that

  |

  n

  |

  k

  s
  ^

  [
  n
  ]

  {displaystyle |n|^{k}{widehat {s}}[n]}

  tends to zero, i.e., the Fourier coefficients converge to zero faster than the

  k

  th

  {displaystyle k^{text{th}}}

  power of

  |

  n

  |

  {displaystyle |n|}

  .

Compact groups

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L²(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the [−π,π] case.

An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups.

Riemannian manifolds

If the domain is not a group, then there is no intrinsically defined convolution. However, if

X

{displaystyle X}

is a compact Riemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold

X

{displaystyle X}

. Then, by analogy, one can consider heat equations on

X

{displaystyle X}

. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type

L

2

(
X
)

{displaystyle L^{2}(X)}

, where

X

{displaystyle X}

is a Riemannian manifold. The Fourier series converges in ways similar to the

[
−
π
,
π
]

{displaystyle [-pi ,pi ]}

case. A typical example is to take

X

{displaystyle X}

to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.

Locally compact Abelian groups

The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.

This generalizes the Fourier transform to

L

1

(
G
)

{displaystyle L^{1}(G)}

or

L

2

(
G
)

{displaystyle L^{2}(G)}

, where

G

{displaystyle G}

is an LCA group. If

G

{displaystyle G}

is compact, one also obtains a Fourier series, which converges similarly to the

[
−
π
,
π
]

{displaystyle [-pi ,pi ]}

case, but if

G

{displaystyle G}

is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is

R

{displaystyle mathbb {R} }

.

Fourier-Stieltjes series

Formally, the Fourier-Stieltjes series can be defined as the Fourier series whose coefficients are given by

c

n

=

μ
^

(
n
)
=

1
P

∫

0

P

e

−
i
2
π

n
P

x

d
μ
(
x
)
,

∀
n
∈

Z

,

{displaystyle c_{n}={hat {mu }}(n)={frac {1}{P}}int _{0}^{P} e^{-i2pi {tfrac {n}{P}}x},dmu (x),quad forall nin mathbb {Z} ,}

for any

μ
∈
M

{displaystyle mu in M}

, where

M

{displaystyle M}

is the space finite Borel measures on the interval

[
0
,
P
]

{displaystyle [0,P]}

. As such, when

μ
∈
M

{displaystyle mu in M}

, the function

μ
^

(
n
)

{displaystyle {hat {mu }}(n)}

is also referred to as a Fourier-Stieltjes transform.^[33][34]

This follows from an earlier and more concrete representation of a Radon measure (i.e. a locally finite Borel measure) on

R

{displaystyle mathbb {R} }

, given by F. Riesz. That is, if

F

{displaystyle F}

is function of bounded variation on the interval

[
0
,
P
]

{displaystyle [0,P]}

then the Fourier coefficients can be expressed by the Riemann-Stieltjes integral

c

n

=

1
P

∫

0

P

e

−
i
2
π

n
P

x

d
F
(
x
)
,

∀
n
∈

Z

,

{displaystyle c_{n}={frac {1}{P}}int _{0}^{P} e^{-i2pi {tfrac {n}{P}}x},dF(x),quad forall nin mathbb {Z} ,}

called the Fourier-Stieltjes coefficients of

F

{displaystyle F}

.^[35] As the distributional derivative of

F

{displaystyle F}

is a Radon measure, it is subject to the Lebesgue decomposition and can be expressed as

d
F
=

F
′

d
x
+
d

F

s

{displaystyle dF=F’dx+dF_{s}}

.^[36][37] If

d

F

s

=
0

{displaystyle dF_{s}=0}

the expression reduces to the original definition of the Fourier coefficients, hence a Fourier series is a Fourier-Stieltjes series.

The question whether or not

μ

{displaystyle mu }

exists for a given sequence of

c

n

{displaystyle c_{n}}

forms the basis of the trigonometric moment problem.^[38]

The Fourier series can be generalized still further from measures to distributions. If the Fourier coefficients are determined by a distribution

F
∈

D

′

{displaystyle Fin {mathcal {D}}’}

then the series is sometimes described as a Fourier-Schwartz series.^[39]

While it is often extremely difficult to decide whether a given series is a Fourier or a Fourier-Stieltjes series, deciding whether or not it is a Fourier-Schwartz series is relatively trivial.^[40]

Fourier series on a square

We can also define the Fourier series for functions of two variables

x

{displaystyle x}

and

y

{displaystyle y}

in the square

[
−
π
,
π
]
×
[
−
π
,
π
]

{displaystyle [-pi ,pi ]times [-pi ,pi ]}

:

f
(
x
,
y
)

=

∑

j
,
k
∈

Z

c

j
,
k

e

i
j
x

e

i
k
y

,

c

j
,
k

=

1

4

π

2

∫

−
π

π

∫

−
π

π

f
(
x
,
y
)

e

−
i
j
x

e

−
i
k
y

d
x

d
y
.

{displaystyle {begin{aligned}f(x,y)&=sum _{j,kin mathbb {Z} }c_{j,k}e^{ijx}e^{iky},\[5pt]c_{j,k}&={frac {1}{4pi ^{2}}}int _{-pi }^{pi }int _{-pi }^{pi }f(x,y)e^{-ijx}e^{-iky},dx,dy.end{aligned}}}

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the JPEG image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.

For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.^[41]

Fourier series of a Bravais-lattice-periodic function

A three-dimensional Bravais lattice is defined as the set of vectors of the form

R

=

n

1

a

1

+

n

2

a

2

+

n

3

a

3

{displaystyle mathbf {R} =n_{1}mathbf {a} _{1}+n_{2}mathbf {a} _{2}+n_{3}mathbf {a} _{3}}

where

n

i

{displaystyle n_{i}}

are integers and

a

i

{displaystyle mathbf {a} _{i}}

are three linearly independent but not necessarily orthogonal vectors. Let us consider some function

f
(

r

)

{displaystyle f(mathbf {r} )}

with the same periodicity as the Bravais lattice, i.e.

f
(

r

)
=
f
(

R

+

r

)

{displaystyle f(mathbf {r} )=f(mathbf {R} +mathbf {r} )}

for any lattice vector

R

{displaystyle mathbf {R} }

. This situation frequently occurs in solid-state physics where

f
(

r

)

{displaystyle f(mathbf {r} )}

might, for example, represent the effective potential that an electron “feels” inside a periodic crystal. In presence of such a periodic potential, the quantum-mechanical description of the electron results in a periodically modulated plane-wave commonly known as Bloch state.

In order to develop

f
(

r

)

{displaystyle f(mathbf {r} )}

in a Fourier series, it is convenient to introduce an auxiliary function

g
(

x

1

,

x

2

,

x

3

)
≜
f
(

r

)
=
f

(

x

1

a

1

a

1

+

x

2

a

2

a

2

+

x

3

a

3

a

3

)

.

{displaystyle g(x_{1},x_{2},x_{3})triangleq f(mathbf {r} )=fleft(x_{1}{frac {mathbf {a} _{1}}{a_{1}}}+x_{2}{frac {mathbf {a} _{2}}{a_{2}}}+x_{3}{frac {mathbf {a} _{3}}{a_{3}}}right).}

Both

f
(

r

)

{displaystyle f(mathbf {r} )}

and

g
(

x

1

,

x

2

,

x

3

)

{displaystyle g(x_{1},x_{2},x_{3})}

contain essentially the same information. However, instead of the position vector

r

{displaystyle mathbf {r} }

, the arguments of

g

{displaystyle g}

are coordinates

x

1
,
2
,
3

{displaystyle x_{1,2,3}}

along the unit vectors

a

i

/

a

i

{displaystyle mathbf {a} _{i}/{a_{i}}}

of the Bravais lattice, such that

g

{displaystyle g}

is an ordinary periodic function in these variables,

g
(

x

1

,

x

2

,

x

3

)
=
g
(

x

1

+

a

1

,

x

2

,

x

3

)
=
g
(

x

1

,

x

2

+

a

2

,

x

3

)
=
g
(

x

1

,

x

2

,

x

3

+

a

3

)

∀

x

1

,

x

2

,

x

3

.

{displaystyle g(x_{1},x_{2},x_{3})=g(x_{1}+a_{1},x_{2},x_{3})=g(x_{1},x_{2}+a_{2},x_{3})=g(x_{1},x_{2},x_{3}+a_{3})quad forall ;x_{1},x_{2},x_{3}.}

This trick allows us to develop

g

{displaystyle g}

as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are

c
(

m

1

,

m

2

,

m

3

)
=

1

a

3

∫

0

a

3

d

x

3

1

a

2

∫

0

a

2

d

x

2

1

a

1

∫

0

a

1

d

x

1

g
(

x

1

,

x

2

,

x

3

)

e

−
i
2
π

(

m

1

a

1

x

1

+

m

2

a

2

x

2

+

m

3

a

3

x

3

)

,

{displaystyle {begin{aligned}c(m_{1},m_{2},m_{3})={frac {1}{a_{3}}}int _{0}^{a_{3}}dx_{3}{frac {1}{a_{2}}}int _{0}^{a_{2}}dx_{2}{frac {1}{a_{1}}}int _{0}^{a_{1}}dx_{1},g(x_{1},x_{2},x_{3}),e^{-i2pi left({tfrac {m_{1}}{a_{1}}}x_{1}+{tfrac {m_{2}}{a_{2}}}x_{2}+{tfrac {m_{3}}{a_{3}}}x_{3}right)}end{aligned}},}

where

m

1

,

m

2

,

m

3

{displaystyle m_{1},m_{2},m_{3}}

are all integers.

c
(

m

1

,

m

2

,

m

3

)

{displaystyle c(m_{1},m_{2},m_{3})}

plays the same role as the coefficients

c

j
,
k

{displaystyle c_{j,k}}

in the previous section but in order to avoid double subscripts we note them as a function.

Once we have these coefficients, the function

g

{displaystyle g}

can be recovered via the Fourier series

g
(

x

1

,

x

2

,

x

3

)
=

∑

m

1

,

m

2

,

m

3

∈

Z

c
(

m

1

,

m

2

,

m

3

)

e

i
2
π

(

m

1

a

1

x

1

+

m

2

a

2

x

2

+

m

3

a

3

x

3

)

.

{displaystyle g(x_{1},x_{2},x_{3})=sum _{m_{1},m_{2},m_{3}in mathbb {Z} },c(m_{1},m_{2},m_{3}),e^{i2pi left({tfrac {m_{1}}{a_{1}}}x_{1}+{tfrac {m_{2}}{a_{2}}}x_{2}+{tfrac {m_{3}}{a_{3}}}x_{3}right)}.}

We would now like to abandon the auxiliary coordinates

x

1
,
2
,
3

{displaystyle x_{1,2,3}}

and to return to the original position vector

r

{displaystyle mathbf {r} }

. This can be achieved by means of the reciprocal lattice whose vectors

b

1
,
2
,
3

{displaystyle mathbf {b} _{1,2,3}}

are defined such that they are orthonormal (up to a factor

2
π

{displaystyle 2pi }

) to the original Bravais vectors

a

1
,
2
,
3

{displaystyle mathbf {a} _{1,2,3}}

,

a

i

⋅

b

j

=
2
π

δ

i
j

,

{displaystyle mathbf {a} _{i}cdot mathbf {b_{j}} =2pi delta _{ij},}

with

δ

i
j

{displaystyle delta _{ij}}

the Kronecker delta. With this, the scalar product between a reciprocal lattice vector

Q

{displaystyle mathbf {Q} }

and an arbitrary position vector

r

{displaystyle mathbf {r} }

written in the Bravais lattice basis becomes

Q

⋅

r

=

(

m

1

b

1

+

m

2

b

2

+

m

3

b

3

)

⋅

(

x

1

a

1

a

1

+

x

2

a

2

a

2

+

x

3

a

3

a

3

)

=
2
π

(

x

1

m

1

a

1

+

x

2

m

2

a

2

+

x

3

m

3

a

3

)

,

{displaystyle mathbf {Q} cdot mathbf {r} =left(m_{1}mathbf {b} _{1}+m_{2}mathbf {b} _{2}+m_{3}mathbf {b} _{3}right)cdot left(x_{1}{frac {mathbf {a} _{1}}{a_{1}}}+x_{2}{frac {mathbf {a} _{2}}{a_{2}}}+x_{3}{frac {mathbf {a} _{3}}{a_{3}}}right)=2pi left(x_{1}{frac {m_{1}}{a_{1}}}+x_{2}{frac {m_{2}}{a_{2}}}+x_{3}{frac {m_{3}}{a_{3}}}right),}

which is exactly the expression occurring in the Fourier exponents. The Fourier series for

f
(

r

)
=
g
(

x

1

,

x

2

,

x

3

)

{displaystyle f(mathbf {r} )=g(x_{1},x_{2},x_{3})}

can therefore be rewritten as a sum over the all reciprocal lattice vectors

Q

=

m

1

b

1

+

m

2

b

2

+

m

3

b

3

{displaystyle mathbf {Q} =m_{1}mathbf {b} _{1}+m_{2}mathbf {b} _{2}+m_{3}mathbf {b} _{3}}

,

f
(

r

)
=

∑

Q

c
(

Q

)

e

i

Q

⋅

r

,

{displaystyle f(mathbf {r} )=sum _{mathbf {Q} }c(mathbf {Q} ),e^{imathbf {Q} cdot mathbf {r} },}

and the coefficients are

c
(

Q

)
=

1

a

3

∫

0

a

3

d

x

3

1

a

2

∫

0

a

2

d

x

2

1

a

1

∫

0

a

1

d

x

1

f

(

x

1

a

1

a

1

+

x

2

a

2

a

2

+

x

3

a

3

a

3

)

e

−
i

Q

⋅

r

.

{displaystyle c(mathbf {Q} )={frac {1}{a_{3}}}int _{0}^{a_{3}}dx_{3},{frac {1}{a_{2}}}int _{0}^{a_{2}}dx_{2},{frac {1}{a_{1}}}int _{0}^{a_{1}}dx_{1},fleft(x_{1}{frac {mathbf {a} _{1}}{a_{1}}}+x_{2}{frac {mathbf {a} _{2}}{a_{2}}}+x_{3}{frac {mathbf {a} _{3}}{a_{3}}}right)e^{-imathbf {Q} cdot mathbf {r} }.}

The remaining task will be to convert this integral over lattice coordinates back into a volume integral. The relation between the lattice coordinates

x

1
,
2
,
3

{displaystyle x_{1,2,3}}

and the original cartesian coordinates

r

=
(
x
,
y
,
z
)

{displaystyle mathbf {r} =(x,y,z)}

is a linear system of equations,

r

=

x

1

a

1

a

1

+

x

2

a

2

a

2

+

x

3

a

3

a

3

,

{displaystyle mathbf {r} =x_{1}{frac {mathbf {a} _{1}}{a_{1}}}+x_{2}{frac {mathbf {a} _{2}}{a_{2}}}+x_{3}{frac {mathbf {a} _{3}}{a_{3}}},}

which, when written in matrix form,

[

x

y

z

]

=

J

[

x

1

x

2

x

3

]

=

[

a

1

a

1

,

a

2

a

2

,

a

3

a

3

]

[

x

1

x

2

x

3

]

,

{displaystyle {begin{bmatrix}x\y\zend{bmatrix}}=mathbf {J} {begin{bmatrix}x_{1}\x_{2}\x_{3}end{bmatrix}}={begin{bmatrix}{frac {mathbf {a} _{1}}{a_{1}}},{frac {mathbf {a} _{2}}{a_{2}}},{frac {mathbf {a} _{3}}{a_{3}}}end{bmatrix}}{begin{bmatrix}x_{1}\x_{2}\x_{3}end{bmatrix}},,}

involves a constant matrix

J

{displaystyle mathbf {J} }

whose columns are the unit vectors

a

j

/

a

j

{displaystyle mathbf {a} _{j}/a_{j}}

of the Bravais lattice. When changing variables from

r

{displaystyle mathbf {r} }

to

(

x

1

,

x

2

,

x

3

)

{displaystyle (x_{1},x_{2},x_{3})}

in an integral, the same matrix

J

{displaystyle mathbf {J} }

appears as a Jacobian matrix

J

=

[

∂
x

∂

x

1

∂
x

∂

x

2

∂
x

∂

x

3

∂
y

∂

x

1

∂
y

∂

x

2

∂
y

∂

x

3

∂
z

∂

x

1

∂
z

∂

x

2

∂
z

∂

x

3

]

.

{displaystyle mathbf {J} ={begin{bmatrix}{dfrac {partial x}{partial x_{1}}}&{dfrac {partial x}{partial x_{2}}}&{dfrac {partial x}{partial x_{3}}}\[12pt]{dfrac {partial y}{partial x_{1}}}&{dfrac {partial y}{partial x_{2}}}&{dfrac {partial y}{partial x_{3}}}\[12pt]{dfrac {partial z}{partial x_{1}}}&{dfrac {partial z}{partial x_{2}}}&{dfrac {partial z}{partial x_{3}}}end{bmatrix}},.}

Its determinant

J

{displaystyle J}

is therefore also constant and can be inferred from any integral over any domain; here we choose to calculate the volume of the primitive unit cell

Γ

{displaystyle Gamma }

in both coordinate systems:

V

Γ

=

∫

Γ

d

3

r
=
J

∫

0

a

1

d

x

1

∫

0

a

2

d

x

2

∫

0

a

3

d

x

3

=
J

a

1

a

2

a

3

{displaystyle V_{Gamma }=int _{Gamma }d^{3}r=Jint _{0}^{a_{1}}dx_{1}int _{0}^{a_{2}}dx_{2}int _{0}^{a_{3}}dx_{3}=J,a_{1}a_{2}a_{3}}

The unit cell being a parallelepiped, we have

V

Γ

=

a

1

⋅
(

a

2

×

a

3

)

{displaystyle V_{Gamma }=mathbf {a} _{1}cdot (mathbf {a} _{2}times mathbf {a} _{3})}

and thus

d

3

r
=
J
d

x

1

d

x

2

d

x

3

=

a

1

⋅
(

a

2

×

a

3

)

a

1

a

2

a

3

d

x

1

d

x

2

d

x

3

.

{displaystyle d^{3}r=Jdx_{1}dx_{2}dx_{3}={frac {mathbf {a} _{1}cdot (mathbf {a} _{2}times mathbf {a} _{3})}{a_{1}a_{2}a_{3}}}dx_{1}dx_{2}dx_{3}.}

This allows us to write

c
(

Q

)

{displaystyle c(mathbf {Q} )}

as the desired volume integral over the primitive unit cell

Γ

{displaystyle Gamma }

in ordinary cartesian coordinates:

c
(

Q

)
=

1

a

1

⋅
(

a

2

×

a

3

)

∫

Γ

d

3

r

f
(

r

)
⋅

e

−
i

Q

⋅

r

.

{displaystyle c(mathbf {Q} )={frac {1}{mathbf {a} _{1}cdot (mathbf {a} _{2}times mathbf {a} _{3})}}int _{Gamma }d^{3}r,f(mathbf {r} )cdot e^{-imathbf {Q} cdot mathbf {r} },.}

Hilbert space

As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of Hilbert spaces. For example, the space of square-integrable functions on

[
−
π
,
π
]

{displaystyle [-pi ,pi ]}

forms the Hilbert space

L

2

(
[
−
π
,
π
]
)

{displaystyle L^{2}([-pi ,pi ])}

. Its inner product, defined for any two elements

f

{displaystyle f}

and

g

{displaystyle g}

, is given by:

⟨
f
,
g
⟩
=

1

2
π

∫

−
π

π

f
(
x
)

g
(
x
)

¯

d
x
.

{displaystyle langle f,grangle ={frac {1}{2pi }}int _{-pi }^{pi }f(x){overline {g(x)}},dx.}

This space is equipped with the orthonormal basis

{

e

n

=

e

i
n
x

:
n
∈

Z

}

{displaystyle left{e_{n}=e^{inx}:nin mathbb {Z} right}}

.
Then the (generalized) Fourier series expansion of

f
∈

L

2

(
[
−
π
,
π
]
)

{displaystyle fin L^{2}([-pi ,pi ])}

, given by

f
(
x
)
=

∑

n
=
−
∞

∞

c

n

e

i
n
x

,

{displaystyle f(x)=sum _{n=-infty }^{infty }c_{n}e^{inx},}

can be written as^[42]

f
=

∑

n
=
−
∞

∞

⟨
f
,

e

n

⟩

e

n

.

{displaystyle f=sum _{n=-infty }^{infty }langle f,e_{n}rangle ,e_{n}.}

The sine-cosine form follows in a similar fashion. Indeed, the sines and cosines form an orthogonal set:

∫

−
π

π

cos
⁡
(
m
x
)

cos
⁡
(
n
x
)

d
x
=

1
2

∫

−
π

π

cos
⁡
(
(
n
−
m
)
x
)
+
cos
⁡
(
(
n
+
m
)
x
)

d
x
=
π

δ

m
n

,

m
,
n
≥
1
,

{displaystyle int _{-pi }^{pi }cos(mx),cos(nx),dx={frac {1}{2}}int _{-pi }^{pi }cos((n-m)x)+cos((n+m)x),dx=pi delta _{mn},quad m,ngeq 1,}

∫

−
π

π

sin
⁡
(
m
x
)

sin
⁡
(
n
x
)

d
x
=

1
2

∫

−
π

π

cos
⁡
(
(
n
−
m
)
x
)
−
cos
⁡
(
(
n
+
m
)
x
)

d
x
=
π

δ

m
n

,

m
,
n
≥
1

{displaystyle int _{-pi }^{pi }sin(mx),sin(nx),dx={frac {1}{2}}int _{-pi }^{pi }cos((n-m)x)-cos((n+m)x),dx=pi delta _{mn},quad m,ngeq 1}

(where δ_mn is the Kronecker delta), and

∫

−
π

π

cos
⁡
(
m
x
)

sin
⁡
(
n
x
)

d
x
=

1
2

∫

−
π

π

sin
⁡
(
(
n
+
m
)
x
)
+
sin
⁡
(
(
n
−
m
)
x
)

d
x
=
0
;

{displaystyle int _{-pi }^{pi }cos(mx),sin(nx),dx={frac {1}{2}}int _{-pi }^{pi }sin((n+m)x)+sin((n-m)x),dx=0;}

Hence, the set

{

1

2

,

cos
⁡
x

2

,

sin
⁡
x

2

,
…
,

cos
⁡
(
n
x
)

2

,

sin
⁡
(
n
x
)

2

,
…

}

,

{displaystyle left{{frac {1}{sqrt {2}}},{frac {cos x}{sqrt {2}}},{frac {sin x}{sqrt {2}}},dots ,{frac {cos(nx)}{sqrt {2}}},{frac {sin(nx)}{sqrt {2}}},dots right},}

also forms an orthonormal basis for

L

2

(
[
−
π
,
π
]
)

{displaystyle L^{2}([-pi ,pi ])}

. The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.

Least squares property

The earlier Eq.2:

s

N

(
x
)
=

∑

n
=
−
N

N

S
[
n
]
 

e

i
2
π

n
P

x

,

{displaystyle s_{N}(x)=sum _{n=-N}^{N}S[n] e^{i2pi {tfrac {n}{P}}x},}

is a trigonometric polynomial of degree

N

{displaystyle N}

that can be generally expressed as:

p

N

(
x
)
=

∑

n
=
−
N

N

p
[
n
]
 

e

i
2
π

n
P

x

.

{displaystyle p_{N}(x)=sum _{n=-N}^{N}p[n] e^{i2pi {tfrac {n}{P}}x}.}

Parseval’s theorem implies that:

.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}

Convergence theorems

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.

If

s

{displaystyle s}

is continuously differentiable, then

(
i
n
)
S
[
n
]

{displaystyle (in)S[n]}

is the

n

{displaystyle n}

th Fourier coefficient of the first derivative

s
′

{displaystyle s’}

. Since

s
′

{displaystyle s’}

is continuous, and therefore bounded, it is square-integrable and its Fourier coefficients are square-summable. Then, by the Cauchy–Schwarz inequality,

(

∑

n
≠
0

|

S
[
n
]

|

)

2

≤

∑

n
≠
0

1

n

2

⋅

∑

n
≠
0

|

n
S
[
n
]

|

2

.

{displaystyle {biggl (}sum _{nneq 0}{bigl |}S[n]{bigr |}{biggr )}^{2}leq sum _{nneq 0}{frac {1}{n^{2}}}cdot sum _{nneq 0}{bigl |}nS[n]{bigr |}^{2}.}

This means that

s

{displaystyle s}

is absolutely summable. The sum of this series is a continuous function, equal to

s

{displaystyle s}

, since the Fourier series converges in

L

1

{displaystyle L^{1}}

to

s

{displaystyle s}

:

This result can be proven easily if

s

{displaystyle s}

is further assumed to be

C

2

{displaystyle textstyle C^{2}}

, since in that case

n

2

S
[
n
]

{displaystyle textstyle n^{2}S[n]}

tends to zero as

n
→
∞

{displaystyle nrightarrow infty }

. More generally, the Fourier series is absolutely summable, thus converges uniformly to

s

{displaystyle s}

, provided that

s

{displaystyle s}

satisfies a Hölder condition of order {tfrac {1}{2}}}”>

α
>

1
2

{displaystyle alpha >{tfrac {1}{2}}}

{tfrac {1}{2}}}”>. In the absolutely summable case, the inequality:

N}{bigl |}S[n]{bigr |}}”>

sup

x

|

s
(
x
)
−

s

N

(
x
)

|

≤

∑

|

n

|

>
N

|

S
[
n
]

|

{displaystyle sup _{x}{bigl |}s(x)-s_{N}(x){bigr |}leq sum _{|n|>N}{bigl |}S[n]{bigr |}}

N}{bigl |}S[n]{bigr |}}”>

proves uniform convergence.

Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at

x

{displaystyle x}

if

s

{displaystyle s}

is differentiable at

x

{displaystyle x}

, to more sophisticated results such as Carleson’s theorem which states that the Fourier series of an

L

2

{displaystyle textstyle L^{2}}

function converges almost everywhere.

Divergence

Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact.

In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere.^[48]

It is 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 [0,π] by^[49]

f
(
x
)
=

∑

n
=
1

∞

1

n

2

sin
⁡

[

(

2

n

3

+
1

)

x
2

]

.

{displaystyle f(x)=sum _{n=1}^{infty }{frac {1}{n^{2}}}sin left[left(2^{n^{3}}+1right){frac {x}{2}}right].}

Because the function is even the Fourier series contains only cosines:

∑

m
=
0

∞

C

m

cos
⁡
(
m
x
)
.

{displaystyle sum _{m=0}^{infty }C_{m}cos(mx).}

The coefficients are:

C

m

=

1
π

∑

n
=
1

∞

1

n

2

{

2

2

n

3

+
1
−
2
m

+

2

2

n

3

+
1
+
2
m

}

{displaystyle C_{m}={frac {1}{pi }}sum _{n=1}^{infty }{frac {1}{n^{2}}}left{{frac {2}{2^{n^{3}}+1-2m}}+{frac {2}{2^{n^{3}}+1+2m}}right}}

As m increases, the coefficients will be positive and increasing until they reach a value of about

C

m

≈
2

/

(

n

2

π
)

{displaystyle C_{m}approx 2/(n^{2}pi )}

at

m
=

2

n

3

/

2

{displaystyle m=2^{n^{3}}/2}

for some n and then become negative (starting with a value around

−
2

/

(

n

2

π
)

{displaystyle -2/(n^{2}pi )}

) and getting smaller, before starting a new such wave. At

x
=
0

{displaystyle x=0}

the Fourier series is simply the running sum of

C

m

,

{displaystyle C_{m},}

and this builds up to around

1

n

2

π

∑

k
=
0

2

n

3

/

2

2

2
k
+
1

∼

1

n

2

π

ln
⁡

2

n

3

=

n
π

ln
⁡
2

{displaystyle {frac {1}{n^{2}pi }}sum _{k=0}^{2^{n^{3}}/2}{frac {2}{2k+1}}sim {frac {1}{n^{2}pi }}ln 2^{n^{3}}={frac {n}{pi }}ln 2}

in the nth wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks. Note that though the function is continuous, it is not differentiable.