aarjaneiro/planetmath huggingface dataset:
approximation

approximation

An

approximation

of some mathematical object (such as the number

2

, the

function

sin

⁡

(

x

)

, a circle, or something else) is another mathematical object (such as value 1.414, a function

x

-

x

3

/

6

, a

regular polygon

with
64 sides) that has almost the same value or

function values

or shape as the original object.

There are many different reasons for making an approximation. It can be that we have no way to know the exact value,
for example if we solve an equation by plotting a graph on a piece of paper and read the value from the x-axis, then
we only get an approximation of the exact value. It can be that we know the exact value, say

e

+

2

, but we
can’t use that exact value, perhaps we need to draw a line of length

e

+

2

cm, then we need a numerical approximation of the original value.

10

7

⋅

π

is an approximation of the number of seconds in a year.
Perhaps we don’t know how to draw the graph of the

sin

function, then we can make an approximation with another function that has approximately the same function values within some range of the

argument

.

In every approximation we introduce an error by definition, since we are not using the correct value. This error can sometimes be controlled; for instance we make a smaller error if we use 3.1416 instead of 4 as the value of

π

.
The magnitude of the error that can be accepted depends on what we wish to do with the approximation.

When we wish to approximate a function by another function, we can have different requirements on the approximation. Sometimes, the approximating function may be required to

pass through

the actual points of the function, in which case we are dealing with a class of approximation techniques known as

interpolation

. When merely having common points is not enough, one may require that the approximating function to have the same

derivatives

as the actual function (see

Taylor series

). Other times, we may wish for the values of the approximation to differ from the exact values by less than a
certain amount within a certain

interval

(see

uniform convergence

), or the approximations to be as close to the actual amounts as possible (see least square approximations).

Title

approximation

Canonical name

Approximation

Date of creation

2013-03-22 14:22:19

Last modified on

2013-03-22 14:22:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 41A99

Generated on Fri Feb 9 21:28:10 2018 by

LaTeXML
countable basis

countable basis

A

countable basis

β

of a vector space

V

over a field

F

is a

countable

subset

β

⊂

V

with the

property

that every

element

v

∈

V

can be written as an

infinite series

v

=

∑

x

∈

β

a

x

⁢

x

in exactly one way (where

a

x

∈

F

). We are implicitly assuming, without further comment, that the vector space

V

has been given a topological structure or normed

structure

in which the above infinite sum is

absolutely convergent

(so that it

converges

to

v

regardless of the order in which the terms are summed).

The archetypical example of a countable basis is the Fourier series of a

function

: every

continuous

real-valued periodic function

f

on the unit circle

S

1

=

ℝ

/

2

⁢

π

can be written as a Fourier series

f

⁢

(

x

)

=

∑

n

=

0

∞

a

n

⁢

cos

⁡

(

n

⁢

x

)

+

∑

n

=

1

∞

b

n

⁢

sin

⁡

(

n

⁢

x

)

in exactly one way.

Note: A countable basis is a

countable set

, but it is not usually a basis.

Title

countable basis

Canonical

name

CountableBasis

Date of creation

2013-03-22 12:10:37

Last modified on

2013-03-22 12:10:37

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 42-00

Classification

msc 15A03

Synonym

Schauder basis

Generated on Fri Feb 9 21:28:25 2018 by

LaTeXML
every normed space with Schauder basis is separable

every normed space with Schauder basis is separable

\PMlinkescapephrase

dense in

Here we show that every

normed space

that has a Schauder basis is

separable

(

http://planetmath.org/Separable

).
Note that we are (implicitly) assuming that the normed spaces in question
are spaces over the field

K

where

K

is either

ℝ

or

ℂ

.
So let

(

X

,

∥

⋅

∥

)

be a normed space with Schauder basis,
say

S

=

{

e

1

,

e

2

,

…

}

.
Notice that our notation implies that

S

is

infinite

.
In finite dimensional case,
the same proof with a slight modification will yield the result.

Now, set

Q

to be the set of all finite sums

q

1

⁢

e

1

+

⋯

+

q

n

⁢

e

n

such that each

q

j

=

a

j

+

b

j

⁢

i

where

a

j

,

b

j

∈

ℚ

.
Clearly

Q

is countable.
It remains to show that

Q

is dense (

http://planetmath.org/Dense

) in

X

.

Let

ϵ

>

0

. Let

x

∈

X

.
By definition of Schauder basis,
there is a

sequence

of scalars

(

α

n

)

and there exists

N

such that for all

n

≥

N

we have,

∥

∑

j

=

1

n

α

j

⁢

e

j

-

x

∥

<

ϵ

/

2

But then in particular,

∥

∑

j

=

1

N

α

j

⁢

e

j

-

x

∥

<

ϵ

/

2

Furthermore, by

density

of

ℚ

in

ℝ

,
we know that there exist constants

a

1

,

…

,

a

N

,

b

1

,

…

,

b

N

in

ℚ

such that,

∥

∑

j

=

1

N

(

a

j

+

b

j

⁢

i

)

⁢

e

j

-

∑

j

=

1

N

α

j

⁢

e

j

∥

<

ϵ

/

2

By

triangle inequality

we obtain:

∥

∑

j

=

1

N

(

a

j

+

b

j

⁢

i

)

⁢

e

j

-

x

∥

≤

∥

∑

j

=

1

N

(

a

j

+

b

j

⁢

i

)

⁢

e

j

-

∑

j

=

1

N

α

j

⁢

e

j

∥

+

∥

∑

j

=

1

N

α

j

⁢

e

j

-

x

∥

<

ϵ

Noting that

∑

j

=

1

N

(

a

j

+

b

j

⁢

i

)

⁢

e

j

is an element of

Q

(by construction of

Q

)
and that

x

and

ϵ

were arbitrary,
we conclude that every

neighborhood

of

x

contains an element of

Q

,
for all

x

in

X

.
This proves that

Q

is dense in

X

and

completes

the proof.

Title

every normed space with Schauder basis is separable

Canonical name

EveryNormedSpaceWithSchauderBasisIsSeparable

Date of creation

2013-03-22 17:36:07

Last modified on

2013-03-22 17:36:07

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 42-00

Classification

msc 15A03

Classification

msc 46B15

Generated on Fri Feb 9 21:28:33 2018 by

LaTeXML
existence of Laplace transform

existence of Laplace transform

Theorem 1

.

For every

measurable function

f

:

[

0

,

∞

)

→

C

, if there exists a

real number

t

0

such that

∫

0

∞

e

-

s

⁢

t

0

⁢

|

f

⁢

(

s

)

|

⁢

𝑑

s

converges, then the

Laplace transform

L

⁢

(

f

)

is
a

well-defined

function

from

{

t

∈

C

⁢

∣

ℜ

⁡

t

>

⁢

t

0

}

to

C

. Furthermore, the Laplace transform function is

analytic

.

Title

existence of Laplace transform

Canonical

name

ExistenceOfLaplaceTransform

Date of creation

2013-03-22 16:31:12

Last modified on

2013-03-22 16:31:12

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 42-01

Generated on Fri Feb 9 21:28:35 2018 by

LaTeXML
Laplace transform of a Gaussian function

Laplace transform of a Gaussian function

We evaluate the

Laplace transform

1

1

cf.

Gaussian function

, wikipedia.org

ℒ

⁢

{

e

-

t

2

}

=

∫

0

∞

e

-

s

⁢

t

⁢

e

-

t

2

⁢

𝑑

t

=

F

⁢

(

s

)

.

(1)

In fact,

ℒ

⁢

{

e

-

t

2

}

=

∫

0

∞

e

-

(

t

2

+

2

⁢

s

2

⁢

t

+

s

2

4

-

s

2

4

)

⁢

𝑑

t

=

e

s

2

4

⁢

∫

0

∞

e

-

(

t

+

s

2

)

2

⁢

𝑑

t

.

By making the change of variable

t

+

s

2

=

u

, we have (by the second equality in (1), the variable on operator’s argument is immaterial)

ℒ

⁢

{

e

-

t

2

}

=

e

s

2

4

⁢

∫

s

2

∞

e

-

u

2

⁢

𝑑

u

.

That is,

ℒ

⁢

{

e

-

t

2

}

=

F

⁢

(

s

)

=

π

2

⁢

e

s

2

4

⁢

erfc

⁢

(

s

2

)

,

where

erfc

⁢

(

⋅

)

is the

complementary error function

. Its path of integration is subject to the

restriction

arg

⁡

(

u

)

→

θ

, with

|

θ

|

≤

π

/

4

as

u

→

∞

along the path, with equality only if

ℜ

⁡

(

u

2

)

remains bounded to the left.

Title

Laplace transform of a Gaussian function

Canonical name

LaplaceTransformOfAGaussianFunction

Date of creation

2013-03-22 16:03:21

Last modified on

2013-03-22 16:03:21

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

5

Author

perucho (2192)

Entry type

Application

Classification

msc 42-01

Generated on Fri Feb 9 21:28:37 2018 by

LaTeXML
common Fourier series

common Fourier series

This entry gives some examples of commonly encountered

periodic functions

and their

Fourier series

, with graphs to show the speed of

convergence

.

We take

[

0

,

1

]

as the model

interval

,
with the

n

th Fourier coefficient of a

function

f

defined as

f

^

⁢

(

n

)

=

∫

0

1

f

⁢

(

t

)

⁢

e

-

2

⁢

π

⁢

i

⁢

n

⁢

t

⁢

𝑑

t

,

n

∈

ℤ

.

The

parameters

of the functions in the examples
have been chosen to attempt to minimize
the complexity of

f

^

⁢

(

0

)

. But the Fourier coefficients for
the most common of the functions given below are easily derived
by taking the appropriate linear transformations on the

coefficients

given.

We do not dwell on the convergence of the Fourier series for each function,
although we note that by a theorem of Dirichlet, the Fourier series for each function (each of

bounded variation

)

converges uniformly

on any compact interval where the function is

continuous

.

1

Square wave

f

⁢

(

t

)

=

{

-

1

2

,

t

<

1

2

+

1

2

,

t

>

1

2

.

{makeimage}

f

^

⁢

(

n

)

=

{

0

,

n

=

0

1

-

e

i

⁢

π

⁢

n

-

2

⁢

π

⁢

i

⁢

n

,

n

≠

0

=

{

0

,

n

is even

-

1

π

⁢

i

⁢

n

,

n

is odd.

f

⁢

(

x

)

∼

-

2

π

⁢

∑

n

∈

ℕ

⁢

odd

1

n

⁢

sin

⁡

2

⁢

π

⁢

n

⁢

t

.

Figure 1:

Square wave function

2

Sawtooth wave

f

⁢

(

t

)

=

t

-

1

2

.

{makeimage}

f

^

⁢

(

n

)

=

{

0

,

n

=

0

-

1

2

⁢

π

⁢

i

⁢

n

,

n

≠

0

.

f

⁢

(

t

)

∼

-

1

π

⁢

∑

n

∈

ℕ

1

n

⁢

sin

⁡

2

⁢

π

⁢

n

⁢

t

.

Figure 2:

Sawtooth wave function

3

Triangular wave

f

⁢

(

t

)

=

{

t

-

1

4

,

t

<

1

2

-

t

+

3

4

,

t

>

1

2

.

{makeimage}

f

^

⁢

(

n

)

=

{

0

,

n

=

0

-

1

-

e

i

⁢

π

⁢

n

2

⁢

π

2

⁢

n

2

,

n

≠

0

.

=

{

0

,

n

is even

-

1

π

2

⁢

n

2

,

n

is odd

.

f

⁢

(

t

)

∼

-

2

π

2

⁢

∑

n

∈

ℕ

⁢

odd

1

n

2

⁢

cos

⁡

2

⁢

π

⁢

n

⁢

t

.

Figure 3:

Triangular wave function

Title

common Fourier series

Canonical name

CommonFourierSeries

Date of creation

2013-03-22 15:41:51

Last modified on

2013-03-22 15:41:51

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Example

Classification

msc 42A16

Related topic

SawBladeFunction

Related topic

MinimalityPropertyOfFourierCoefficients

Generated on Fri Feb 9 21:28:40 2018 by

LaTeXML
example of Fourier series

example of Fourier series

Here we present an example of

Fourier series

:

Example:

Let

f

:

(

-

π

,

π

)

→

ℝ

be the “identity”

function

,
defined by

f

⁢

(

x

)

=

x

,

for all

⁢

x

∈

(

-

π

,

π

)

.

We will compute the Fourier coefficients for this function. Notice
that

cos

⁡

(

n

⁢

x

)

is an

even function

, while

f

and

sin

⁡

(

n

⁢

x

)

are

odd functions

.

a

0

f

=

1

2

⁢

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

𝑑

x

=

1

2

⁢

π

⁢

∫

-

π

π

x

⁢

𝑑

x

=

0

a

n

f

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

(

n

⁢

x

)

⁢

𝑑

x

=

1

π

⁢

∫

-

π

π

x

⁢

cos

⁡

(

n

⁢

x

)

⁢

𝑑

x

=

0

b

n

f

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

(

n

⁢

x

)

⁢

𝑑

x

=

1

π

⁢

∫

-

π

π

x

⁢

sin

⁡

(

n

⁢

x

)

⁢

𝑑

x

=

=

2

π

∫

0

π

x

sin

(

n

x

)

d

x

=

2

π

(

[

-

x

⁢

cos

⁡

(

n

⁢

x

)

n

]

0

π

+

[

sin

⁡

(

n

⁢

x

)

n

2

]

0

π

=

)

=

(

-

1

)

n

+

1

2

n

Notice that

a

0

f

,

a

n

f

are

0

because

x

and

x

⁢

cos

⁡

(

n

⁢

x

)

are
odd functions. Hence the Fourier series for

f

⁢

(

x

)

=

x

is:

f

⁢

(

x

)

=

x

=

a

0

f

+

∑

n

=

1

∞

(

a

n

f

⁢

cos

⁡

(

n

⁢

x

)

+

b

n

f

⁢

sin

⁡

(

n

⁢

x

)

)

=

=

∑

n

=

1

∞

(

-

1

)

n

+

1

⁢

2

n

⁢

sin

⁡

(

n

⁢

x

)

,

∀

x

∈

(

-

π

,

π

)

For an application of this Fourier series, see value of the

Riemann zeta function

at

s

=

2

.

Title

example of Fourier series

Canonical name

ExampleOfFourierSeries

Date of creation

2013-03-22 13:57:13

Last modified on

2013-03-22 13:57:13

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Example

Classification

msc 42A16

Synonym

example of Fourier coefficients

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Related topic

FourierSineAndCosineSeries

Generated on Fri Feb 9 21:28:43 2018 by

LaTeXML
Fourier series of function of bounded variation

Fourier series of function of bounded variation

If the

real function

f

is of

bounded variation

on the

interval

[

-

π

,

+

π

]

,  then its

Fourier series

expansion

a

0

2

+

∑

n

=

1

∞

(

a

n

⁢

cos

⁡

n

⁢

x

+

b

n

⁢

sin

⁡

n

⁢

x

)

(1)

with the

coefficients

(

http://planetmath.org/FourierCoefficients

)

{

a

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

b

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

x

⁢

d

⁢

x

(2)

converges at every point of the interval. The sum of the series is at the

interior points

x

equal to the

arithmetic mean

of the left-sided (

http://planetmath.org/OneSidedLimit

) and the

right-sided limit

of

f

at

x

and at the end-points of the interval equal to

1

2

⁢

(

lim

x

→

-

π

+

⁡

f

⁢

(

x

)

+

lim

x

→

+

π

-

⁡

f

⁢

(

x

)

)

.

Remark 1.

Because of the

periodicity

of the terms of the terms, the expansion (1) converges for all real values of

x

and it represents a

periodic function

with the period

2

⁢

π

.

Remark 2.

If the

function

f

is of bounded variation, instead of

[

-

π

,

+

π

]

,  on the interval

[

-

p

,

+

p

]

the equations (1) and (2) may be converted via the change of variable

x

:=

p

⁢

t

π

to

a

0

2

+

∑

n

=

1

∞

(

a

n

⁢

cos

⁡

n

⁢

π

⁢

t

p

+

b

n

⁢

sin

⁡

n

⁢

π

⁢

t

p

)

(3)

and

{

a

n

=

1

p

⁢

∫

-

p

p

f

⁢

(

t

)

⁢

cos

⁡

n

⁢

π

⁢

t

p

⁢

d

⁢

t

b

n

=

1

p

⁢

∫

-

p

p

f

⁢

(

t

)

⁢

sin

⁡

n

⁢

π

⁢

t

p

⁢

d

⁢

t

.

(4)

Correspondingly, the sum of (3) at the points of

[

-

p

,

+

p

]

is expressed via the left-sided and righr-sided limits of

f

⁢

(

t

)

.

Title

Fourier series of function of bounded variation

Canonical name

FourierSeriesOfFunctionOfBoundedVariation

Date of creation

2013-03-22 17:58:00

Last modified on

2013-03-22 17:58:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 42A16

Classification

msc 42A20

Classification

msc 26A45

Related topic

DirichletConditions

Related topic

FourierCoefficients

Generated on Fri Feb 9 21:28:46 2018 by

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generalized Riemann-Lebesgue lemma

generalized Riemann-Lebesgue lemma

Generalized Riemann-Lebesgue lemmaFernando Sanz Gamiz

Lemma 1

.

Let

h

:

R

→

C

be a bounded

measurable function

.
If

h

satisfies

the

averaging condition

lim

c

→

+

∞

⁡

1

c

⁢

∫

0

c

h

⁢

(

t

)

⁢

𝑑

t

=

0

then

lim

ω

→

∞

⁡

∫

a

b

f

⁢

(

t

)

⁢

h

⁢

(

ω

⁢

t

)

⁢

𝑑

t

=

0

with

-

∞

<

a

<

b

<

+

∞

for any

f

∈

L

1

⁢

[

a

,

b

]

Proof.

Obviously we only need to prove the lemma when both

h

and

f

are
real and

0

=

a

<

b

<

∞

.

Let

𝟏

[

a

,

b

]

be the

indicator function

of the

interval

[

a

,

b

]

. Then

lim

ω

→

∞

⁡

∫

0

b

𝟏

[

a

,

b

]

⁢

h

⁢

(

ω

⁢

t

)

⁢

𝑑

t

=

lim

ω

→

∞

⁡

1

ω

⁢

∫

0

ω

⁢

b

h

⁢

(

t

)

⁢

𝑑

t

=

0

by the

hypothesis

. Hence, the lemma is valid for indicators,
therefore for

step functions

.

Now let

C

be a bound for

h

and choose

ϵ

>

0

.
As step functions are dense in

L

1

, we can find, for any

f

∈

L

1

⁢

[

a

,

b

]

, a step function

g

such that

∥

f

-

g

∥

1

<

ϵ

,
therefore

lim

ω

→

∞

⁡

|

∫

a

b

f

⁢

(

t

)

⁢

h

⁢

(

ω

⁢

t

)

⁢

𝑑

t

|

⩽

lim

ω

→

∞

⁡

∫

a

b

|

f

⁢

(

t

)

-

g

⁢

(

t

)

|

⁢

|

h

⁢

(

ω

⁢

t

)

|

⁢

𝑑

t

+

lim

ω

→

∞

⁡

|

∫

a

b

g

⁢

(

t

)

⁢

h

⁢

(

ω

⁢

t

)

⁢

𝑑

t

|

⩽

lim

ω

→

∞

⁡

C

⁢

∥

f

-

g

∥

1

<

C

⁢

ϵ

because

lim

ω

→

∞

⁡

|

∫

a

b

g

⁢

(

t

)

⁢

h

⁢

(

ω

⁢

t

)

⁢

𝑑

t

|

=

0

by what we have proved for step
functions. Since

ϵ

is arbitrary, we are done.

∎

Title

generalized Riemann-Lebesgue lemma

Canonical name

GeneralizedRiemannLebesgueLemma

Date of creation

2013-03-22 17:06:03

Last modified on

2013-03-22 17:06:03

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

13

Author

fernsanz (8869)

Entry type

Theorem

Classification

msc 42A16

Related topic

RiemannLebesgueLemma

Related topic

FourierCoefficients

Related topic

Integral2

Generated on Fri Feb 9 21:28:49 2018 by

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Poisson summation formula

Poisson summation formula

Let

f

:

ℝ

→

ℝ

be an integrable function and let

f

^

⁢

(

ξ

)

=

∫

ℝ

e

-

2

⁢

π

⁢

i

⁢

ξ

⁢

x

⁢

f

⁢

(

x

)

⁢

𝑑

x

,

ξ

∈

ℝ

.

be its

Fourier transform

. The

Poisson summation formula

is the assertion that

∑

n

∈

ℤ

f

⁢

(

n

)

=

∑

n

∈

ℤ

f

^

⁢

(

n

)

.

(1)

whenever

f

is such that both of the above infinite sums are

absolutely convergent

.

Equation (

1

) is useful because it establishes a
correspondence between

Fourier series

and

Fourier integrals

. To see
the connection, let

g

⁢

(

x

)

=

∑

n

∈

ℤ

f

⁢

(

x

+

n

)

,

x

∈

ℝ

,

be the

periodic
function

obtained by pseudo-averaging

1

1

This terminology is at best a metaphor. The operation in question is not a genuine mean, in the technical sense of that word.

f

relative to

ℤ

acting
as the discrete group of translations on

ℝ

. Since

f

was
assumed to be integrable,

g

is defined almost everywhere, and is
integrable over

[

0

,

1

]

with

∥

g

∥

L

1

⁢

[

0

,

1

]

≤

∥

f

∥

L

1

⁢

(

ℝ

)

.

Since

f

is integrable, we may interchange integration and

summation

to obtain

f

^

⁢

(

k

)

=

∑

n

∈

ℤ

∫

0

1

f

⁢

(

x

+

n

)

⁢

e

-

2

⁢

π

⁢

i

⁢

k

⁢

x

⁢

𝑑

x

=

∫

0

1

e

-

2

⁢

π

⁢

i

⁢

k

⁢

x

⁢

g

⁢

(

x

)

⁢

𝑑

x

for every

k

∈

ℤ

. In other words, the restriction of the Fourier
transform of

f

to the

integers

gives the Fourier coefficients of the
averaged, periodic function

g

. Since we have assumed that the

f

^

⁢

(

k

)

form an absolutely convergent series, we have that

g

⁢

(

x

)

=

∑

k

∈

ℤ

f

^

⁢

(

k

)

⁢

e

2

⁢

π

⁢

i

⁢

k

⁢

x

in the sense of

uniform convergence

. Evaluating the above equation at

x

=

0

, we
obtain the Poisson summation formula (

1

).

Title

Poisson summation formula

Canonical name

PoissonSummationFormula

Date of creation

2013-03-22 13:27:25

Last modified on

2013-03-22 13:27:25

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

16

Author

rmilson (146)

Entry type

Theorem

Classification

msc 42A16

Classification

msc 42A38

Synonym

Poisson summation

Generated on Fri Feb 9 21:28:55 2018 by

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Riemann-Lebesgue lemma

Riemann-Lebesgue lemma

.

Let

f

:

[

a

,

b

]

→

ℂ

be a

measurable function

. If

f

is

L

1

integrable

, that is to say if the

Lebesgue integral

of

|

f

|

is
finite, then

∫

a

b

f

⁢

(

x

)

⁢

e

i

⁢

n

⁢

x

⁢

𝑑

x

→

0

,

a

⁢

s

n

→

±

∞

.

The above result, commonly known as the

Riemann-Lebesgue lemma

, is of
basic importance in

harmonic analysis

. It is

equivalent

to the
assertion that the

Fourier coefficients

f

^

n

of a

periodic

, integrable
function

f

⁢

(

x

)

, tend to

0

as

n

→

±

∞

.

The proof can be organized into 3 steps.

Step 1.

An

elementary

calculation shows that

∫

I

e

i

⁢

n

⁢

x

⁢

𝑑

x

→

0

,

a

⁢

s

n

→

±

∞

for every

interval

I

⊂

[

a

,

b

]

. The

proposition

is therefore true
for all

step functions

with

support

(

http://planetmath.org/SupportOfFunction

) in

[

a

,

b

]

.

Step 2.

By the

monotone convergence theorem

, the proposition is true for all

positive

functions

, integrable on

[

a

,

b

]

.

Step 3.

Let

f

be an arbitrary measurable function,
integrable on

[

a

,

b

]

. The proposition is true for such a general

f

, because one can always write

f

=

g

-

h

,

where

g

and

h

are positive functions, integrable on

[

a

,

b

]

.

Title

Riemann-Lebesgue lemma

Canonical name

RiemannLebesgueLemma

Date of creation

2013-03-22 13:08:04

Last modified on

2013-03-22 13:08:04

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Theorem

Classification

msc 42A16

Generated on Fri Feb 9 21:28:57 2018 by

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Dirichlet conditions

Dirichlet conditions

Let

f

be a

piecewise

http://planetmath.org/node/1146

regular

real-valued function defined on some

interval

[

a

,

b

]

, such that

f

has only a finite number of

discontinuities

and

extrema

in

[

a

,

b

]

. Then the

Fourier series

of this function converges to

f

when

f

is

continuous

and to the

arithmetic mean

of the left-handed and right-handed

http://planetmath.org/node/2950

limit of

f

at a point where it is

discontinuous

.

Title

Dirichlet conditions

Canonical name

DirichletConditions

Date of creation

2013-03-22 13:21:53

Last modified on

2013-03-22 13:21:53

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 42A20

Related topic

FourierSeriesOfFunctionOfBoundedVariation

Related topic

FourierCoefficients

Generated on Fri Feb 9 21:28:59 2018 by

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uniqueness of Fourier expansion

uniqueness of Fourier expansion

If a

real function

f

,

Riemann integrable

on the

interval

[

-

π

,

π

]

,  may be expressed as sum of a trigonometric series

f

⁢

(

x

)

=

a

0

2

+

∑

m

=

1

∞

(

a

m

⁢

cos

⁡

m

⁢

x

+

b

m

⁢

sin

⁡

m

⁢

x

)

(1)

where the series

a

1

+

b

1

+

a

2

+

b

2

+

a

3

+

b

3

+

…

of the

coefficients

converges absolutely

, then the series (1)

converges uniformly

on the interval and can be integrated termwise (

http://planetmath.org/SumFunctionOfSeries

).  The same concerns apparently the series which are gotten by multiplying the equation (1) by

cos

⁡

n

⁢

x

and by

sin

⁡

n

⁢

x

;  the results of the integrations determine for the coefficients

a

n

and

b

n

the unique values

a

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

,

b

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

x

⁢

d

⁢

x

for any

n

.  So the

Fourier series

of

f

is unique.

As a consequence, we can infer that the well-known

goniometric formula

sin

2

⁡

x

=

1

-

cos

⁡

2

⁢

x

2

presents the Fourier series of the

even function

sin

2

⁡

x

.

Title

uniqueness of Fourier expansion

Canonical name

UniquenessOfFourierExpansion

Date of creation

2013-03-22 18:22:16

Last modified on

2013-03-22 18:22:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Result

Classification

msc 42A20

Classification

msc 42A16

Classification

msc 26A06

Synonym

uniqueness of Fourier series

Related topic

FourierSineAndCosineSeries

Related topic

MinimalityPropertyOfFourierCoefficients

Related topic

DeterminationOfFourierCoefficients

Related topic

ComplexSineAndCosine

Related topic

UniquenessOfDigitalRepresentation

Related topic

UniquenessOfLaurentExpansion

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