aarjaneiro/planetmath huggingface dataset:
Fourier transform

Fourier transform

The

Fourier transform

F

⁢

(

s

)

of a

function

f

⁢

(

t

)

is defined as follows:

F

⁢

(

s

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

e

-

i

⁢

s

⁢

t

⁢

f

⁢

(

t

)

⁢

𝑑

t

.

The Fourier transform exists if

f

is

Lebesgue integrable

on the whole real axis.

If

f

is Lebesgue integrable and can be divided into a finite number of

continuous

,

monotone functions

and at every point both

one-sided limits

exist, the Fourier transform can be inverted:

f

⁢

(

t

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

e

i

⁢

s

⁢

t

⁢

F

⁢

(

s

)

⁢

𝑑

s

.

Sometimes the Fourier transform is also defined without the factor

1

2

⁢

π

in one direction, but therefore giving the

transform

into the other direction a factor

1

2

⁢

π

. So when looking a transform up in a table you should find out how it is defined in that table.

The Fourier transform has some important

properties

, that can be used when solving

differential equations

. We denote the Fourier transform of

f

with respect to

t

in terms of

s

by

ℱ

t

⁢

(

f

)

.

•

ℱ

t

⁢

(

a

⁢

f

+

b

⁢

g

)

=

a

⁢

ℱ

t

⁢

(

f

)

+

b

⁢

ℱ

t

⁢

(

g

)

,

where

a

and

b

are constants.

•

ℱ

t

⁢

(

∂

∂

⁡

t

⁢

f

)

=

i

⁢

s

⁢

ℱ

t

⁢

(

f

)

.

•

ℱ

t

⁢

(

∂

∂

⁡

x

⁢

f

)

=

∂

∂

⁡

x

⁢

ℱ

t

⁢

(

f

)

.

•

We define the bilateral

convolution

of two functions

f

1

and

f

2

as:

(

f

1

∗

f

2

)

⁢

(

t

)

:=

1

2

⁢

π

⁢

∫

-

∞

∞

f

1

⁢

(

τ

)

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

τ

.

Then the following equation holds:

ℱ

t

⁢

(

(

f

1

∗

f

2

)

⁢

(

t

)

)

=

ℱ

t

⁢

(

f

1

)

⋅

ℱ

t

⁢

(

f

2

)

.

If

f

⁢

(

t

)

is some signal (maybe a wave) then the frequency domain of

f

is given as

ℱ

t

⁢

(

f

)

. Rayleigh’s

theorem

states that then the energy

E

carried by the signal

f

given by:

E

=

∫

-

∞

∞

|

f

⁢

(

t

)

|

2

⁢

𝑑

t

can also be expressed as:

E

=

∫

-

∞

∞

|

ℱ

t

⁢

(

f

)

⁢

(

s

)

|

2

⁢

𝑑

s

.

In general we have:

∫

-

∞

∞

|

f

⁢

(

t

)

|

2

⁢

𝑑

t

=

∫

-

∞

∞

|

ℱ

t

⁢

(

f

)

⁢

(

s

)

|

2

⁢

𝑑

s

,

also known as the

first Parseval theorem

.

Title

Fourier transform

Canonical

name

FourierTransform

Date of creation

2013-03-22 12:34:28

Last modified on

2013-03-22 12:34:28

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

17

Author

mathwizard (128)

Entry type

Definition

Classification

msc 42A38

Related topic

Wavelet

Related topic

ProgressiveFunction

Related topic

DiscreteFourierTransform

Related topic

FourierSeriesInComplexFormAndFourierIntegral

Related topic

TwoDimensionalFourierTransforms

Related topic

TableOfGeneralizedFourierAndMeasuredGroupoidTransforms

Defines

first Parseval theorem

Generated on Fri Feb 9 21:29:10 2018 by

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microlocal analysis

microlocal analysis

In

mathematical analysis

,

microlocal analysis

is a term use to describe techniques developed from the 1950s onwards based on

Fourier transforms

related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes

generalized functions

, pseudo-differential operators, wave front sets, Fourier integral operators, and paradifferential operators.

The term microlocal implies localisation not just at a point, but in terms of cotangent space directions at a given point. This gains in importance on manifolds of

dimension

greater

than one.

This entry was adapted from the Wikipedia article

http://en.wikipedia.org/wiki/Microlocal_analysis

Microlocal analysis as of January 17, 2007.

Title

microlocal analysis

Canonical

name

MicrolocalAnalysis

Date of creation

2013-03-22 16:35:25

Last modified on

2013-03-22 16:35:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 42A38

Generated on Fri Feb 9 21:29:12 2018 by

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table of Fourier transforms

table of Fourier transforms

Below are

tables of Fourier transforms

(

http://planetmath.org/FourierTransform

); one lists some of the common properties, and the other lists some common examples.

Properties

Original

Transformed

comment

derivation

a

⁢

f

⁢

(

t

)

+

b

⁢

g

⁢

(

t

)

a

⁢

ℱ

⁢

{

f

⁢

(

t

)

}

+

b

⁢

ℱ

⁢

{

g

⁢

(

t

)

}

linearity

f

⁢

(

t

)

*

g

⁢

(

t

)

ℱ

⁢

{

f

⁢

(

t

)

}

⁢

ℱ

⁢

{

g

⁢

(

t

)

}

convolution

property

f

⁢

(

t

+

α

)

F

⁢

(

s

)

⁢

exp

⁡

(

-

i

⁢

α

⁢

s

)

time shift, where

F

⁢

(

s

)

=

ℱ

⁢

{

f

⁢

(

t

)

}

f

′

⁢

(

t

)

i

⁢

s

⁢

ℱ

⁢

{

f

⁢

(

t

)

}

differentiation

f

⁢

(

t

)

¯

F

⁢

(

-

s

)

¯

conjugation

, where

F

⁢

(

s

)

=

ℱ

⁢

{

f

⁢

(

t

)

}

f

⁢

(

α

⁢

t

)

1

|

α

|

⁢

F

⁢

(

s

α

)

scaling, where

F

⁢

(

s

)

=

ℱ

⁢

{

f

⁢

(

t

)

}

with

α

≠

0

Examples

⁢
f

(
t
)

⁢
F

{

⁢
f

(
t
)

}

conditions
explanation
derivation

⁢
δ

(
t
)

1

Dirac delta function

1

⁢
2
π
δ

(
s
)

e

⁢
i
a
t

⁢
2
π
δ

(

-
s
α

)

∈
a
R

cos

(

⁢
a
t

)

⁢
π

(

+

⁢
δ

(

+
s
a

)

⁢
δ

(

-
s
a

)

)

∈
a
R

sin

(

⁢
a
t

)

⁢
i
π

(

-

⁢
δ

(

+
s
a

)

⁢
δ

(

-
s
a

)

)

∈
a
R

Title
table of Fourier transforms

Canonical name
TableOfFourierTransforms

Date of creation
2013-03-22 18:08:12

Last modified on
2013-03-22 18:08:12

Owner
CWoo (3771)

Last modified by
CWoo (3771)

Numerical id
7

Author
CWoo (3771)

Entry type
Feature

Classification
msc 42A38

Synonym
Groupoid Transforms

Defines
Fourier-Stieltjes generalization of FT

Generated on Fri Feb 9 21:29:27 2018 by

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almost periodic function (classical definition)

almost periodic function (classical definition)

A

continuous function

f

:

ℝ

→

ℝ

is said to be

almost

periodic

if, for every

ϵ

>

0

, there exists an a number

L

ϵ

>

0

such that for every

interval

I

of length

L

ϵ

there exists a number

ω

I

∈

I

such that

|

f

⁢

(

x

+

ω

I

)

-

f

⁢

(

x

)

|

<

ϵ

whenever

x

∈

ℝ

.

Intuition: we want the

function

to have an ”approximate period”.
However, it is easy to write too weak condition. First, we want
uniform estimate in

x

. If we allow

ω

to be small than
the condition degenerates to

uniform continuity

. If we require
a single

ω

, than the condition still is too weak (it
allows pretty wide changes). For periodic function every

multiple

of a period is still a period. So, if the length of an interval
is longer than the period, then the interval contains a period.
The definition of almost periodic functions mimics the above
property of periodic functions: every sufficiently long interval
should contain an approximate period.

It is possible to generalize this notion. The range of the function can be taken to be a

normed vector space

— in the first definition, we merely need to replace the

absolute value

with the norm:

∥

f

⁢

(

x

+

ω

)

-

f

⁢

(

x

)

∥

<

ϵ

In the second definition, interpret uniform convergence as uniform convergence with respect to the norm. A common case of this is the case where the range is the

complex numbers

. It is worth noting that if the

vector space

is finite dimensional, a function is almost periodic if and only if each of its

components

with respect to a basis is almost periodic.

Also the domain may be taken to be a group

G

. A function is called almost periodic iff set of its translates is pre-compact (

compact

after

completion

).
Equivalently, a continuous function

f

on a topological group

G

is
almost periodic iff there is a compact group

K

, a continuous
function

g

on

K

and a (

continuous

)

homomorphism

h

form

G

to

K

such that

f

is the composition of

g

and

h

.
The classical case described above arises when the group is the

additive group

of the

real number

field. Almost periodic functions with respect to groups play a role in the

representation theory

of non-compact Lie algebras. (In the compact case, they are

trivial

— all continuous functions are almost periodic.)

The notion of an almost periodic function should not be confused with the notion of quasiperiodic function.

Title

almost periodic function (classical definition)

Canonical name

AlmostPeriodicFunctionclassicalDefinition

Date of creation

2013-03-22 14:53:14

Last modified on

2013-03-22 14:53:14

Owner

drini (3)

Last modified by

drini (3)

Numerical id

13

Author

drini (3)

Entry type

Definition

Classification

msc 42A75

Synonym

almost periodic function

Related topic

ExampleOfNonSeperableHilbertSpace

Defines

almost periodic

Generated on Fri Feb 9 21:29:35 2018 by

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almost periodic function (equivalent definition)

almost periodic function (equivalent definition)

There is an

equivalent

definition

of almost periodic function due to Bochner:

A

function

f

:

ℝ

→

ℝ

is

almost periodic

if
every

sequence

of translates of

f

has a uniformly convergent

subsequence

.

Not only is this definition simpler to state than that of Bohr,
but it also generalizes to functions on groups.

Title

almost periodic function (equivalent definition)

Canonical

name

AlmostPeriodicFunctionequivalentDefinition

Date of creation

2013-03-22 15:23:03

Last modified on

2013-03-22 15:23:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 42A75

Generated on Fri Feb 9 21:29:37 2018 by

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progressive function

progressive function

Definition

A

function

f

∈

L

2

⁢

(

ℝ

)

is called

progressive

iff its

Fourier transform

is supported by

positive

frequencies only:

supp

f

^

⊆

ℝ

+

.

It is called

regressive

iff the time reversed function

f

⁢

(

-

t

)

is progressive, or equivalently, if

supp

f

^

⊆

ℝ

-

.

Title

progressive function

Canonical

name

ProgressiveFunction

Date of creation

2013-03-22 14:28:20

Last modified on

2013-03-22 14:28:20

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

4

Author

swiftset (1337)

Entry type

Definition

Classification

msc 42A99

Related topic

FourierTransform

Defines

regressive function

Generated on Fri Feb 9 21:29:49 2018 by

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Ingham Inequality

Ingham Inequality

Let

(

t

j

)

j

∈

ℤ

a

increasing sequence

of positive real numbers such that

t

j

+

1

-

t

j

≥

γ

>

1

,

j

∈

ℤ

.

Then for all

n

∈

ℕ

and for all complex

sequences

(

c

j

)

j

=

-

n

n

, we have

m

⁢

∑

j

=

-

n

n

|

c

j

|

2

≤

∫

-

π

π

|

∑

j

=

-

n

n

1

2

⁢

π

⁢

c

j

⁢

e

i

⁢

t

j

⁢

x

|

2

⁢

𝑑

x

,

where

m

=

2

π

⁢

(

1

-

1

γ

2

)

.

Title

Ingham Inequality

Canonical name

InghamInequality

Date of creation

2013-03-22 15:54:58

Last modified on

2013-03-22 15:54:58

Owner

ncrom (8997)

Last modified by

ncrom (8997)

Numerical id

8

Author

ncrom (8997)

Entry type

Theorem

Classification

msc 42B05

Generated on Fri Feb 9 21:29:51 2018 by

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Parseval equality

Parseval equality

0.1

Parseval’s Equality

Theorem. –

If

{

e

j

:

j

∈

J

}

is an

orthonormal basis

of an

Hilbert space

H

, then for every

x

∈

H

the following equality holds:

∥

x

∥

2

=

∑

j

∈

J

|

⟨

x

,

e

j

⟩

|

2

.

The above theorem is a more sophisticated form of Bessel’s

inequality

(where the inequality is in fact an equality). The difference is that for Bessel’s inequality it is only required that the set

{

e

j

:

j

∈

J

}

is an

orthonormal set

, not necessarily an orthonormal basis.

0.2

Parseval’s Theorem

Applying Parseval’s equality on the Hilbert space

L

2

⁢

(

[

-

π

,

π

]

)

(

http://planetmath.org/LpSpace

), of

square integrable

functions

on the

interval

[

-

π

,

π

]

, with the orthonormal basis consisting of

trigonometric functions

, we obtain

Theorem (Parseval’s theorem). –

Let

f

be a Riemann square integrable function from

[

-

π

,

π

]

to

ℝ

.  The following equality holds

1

π

⁢

∫

-

π

π

f

2

⁢

(

x

)

⁢

𝑑

x

=

(

a

0

f

)

2

2

+

∑

k

=

1

∞

[

(

a

k

f

)

2

+

(

b

k

f

)

2

]

,

where

a

0

f

,

a

k

f

,

b

k

f

are the Fourier coefficients of the function

f

.

The function

f

can be a Lebesgue-integrable function, if we use the

Lebesgue integral

in place of the

Riemann integral

.

Title

Parseval equality

Canonical name

ParsevalEquality

Date of creation

2013-03-22 13:57:10

Last modified on

2013-03-22 13:57:10

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 42B05

Synonym

Parseval equation

Synonym

Parseval identity

Synonym

Lyapunov equation

Related topic

BesselInequality

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Defines

Parseval theorem

Generated on Fri Feb 9 21:29:54 2018 by

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proof of Ingham Inequality

proof of Ingham Inequality

Let

f

⁢

(

x

)

=

1

2

⁢

π

⁢

∑

j

=

-

n

n

c

j

⁢

e

i

⁢

t

j

⁢

x

.

If

k

⁢

(

x

)

is a integrable function in

(

-

∞

,

∞

)

, let

K

⁢

(

u

)

=

∫

-

∞

∞

k

⁢

(

x

)

⁢

e

i

⁢

x

⁢

u

⁢

𝑑

x

.

It is easy to prove the equalities.

∫

-

∞

∞

k

⁢

(

x

)

⁢

|

f

⁢

(

x

)

|

2

⁢

𝑑

x

=

∑

j

,

h

=

-

n

n

c

j

⁢

c

h

¯

⁢

K

⁢

(

t

j

-

t

h

)

.

∫

-

∞

∞

k

⁢

(

x

)

⁢

f

⁢

(

x

)

⁢

e

-

i

⁢

x

⁢

t

h

⁢

𝑑

x

=

∑

j

=

-

n

n

c

j

⁢

K

⁢

(

t

j

-

t

h

)

.

For the rest of the proof we make the following choices:

k

⁢

(

x

)

=

{

cos

⁡

(

x

/

2

)

,

|

x

|

≤

π

0

,

|

x

|

>

π

Then, after computation, we have

K

⁢

(

u

)

=

4

⁢

cos

⁡

(

π

⁢

u

)

1

-

4

⁢

u

2

.

Let

T

=

π

and

γ

=

1

+

δ

(

δ

=

ε

/

π

>

0

). Since

|

t

j

+

1

-

t

j

|

≥

γ

, we have that

|

t

j

-

t

h

|

≥

|

j

-

h

|

⁢

γ

. If we suppose that

h

is fixed, we have

∑

j

≠

h

|

K

⁢

(

t

j

-

t

h

)

|

=

∑

j

≠

h

|

4

⁢

cos

⁡

π

⁢

(

t

j

-

t

h

)

1

-

4

⁢

(

t

j

-

t

h

)

2

|

≤

∑

j

≠

h

4

4

⁢

(

j

-

h

)

2

⁢

γ

2

-

1

≤

≤

4

γ

2

⁢

∑

j

≠

h

1

4

⁢

(

j

-

h

)

2

-

1

<

8

γ

2

⁢

∑

r

=

1

∞

1

4

⁢

r

2

-

1

=

=

8

2

⁢

γ

2

⁢

∑

r

=

1

∞

(

1

2

⁢

r

-

1

-

1

2

⁢

r

+

1

)

=

4

γ

2

=

K

⁢

(

0

)

γ

2

.

But, since

2

⁢

|

c

j

⁢

c

h

¯

|

≤

|

c

j

|

2

+

|

c

h

|

2

and

K

⁢

(

t

j

-

t

h

)

=

K

⁢

(

t

h

-

t

j

)

, we find

∫

-

∞

∞

k

⁢

(

x

)

⁢

|

f

⁢

(

x

)

|

2

⁢

𝑑

x

=

∑

j

,

h

=

-

n

n

c

j

⁢

c

h

¯

⁢

K

⁢

(

t

j

-

t

h

)

=

=

∑

j

|

c

j

|

2

⁢

K

⁢

(

0

)

+

∑

j

,

h

;

j

≠

h

𝒪

⁢

(

j

,

h

)

⁢

|

c

j

|

2

+

|

c

h

|

2

2

⁢

|

K

⁢

(

t

j

-

t

h

)

|

=

∑

j

|

c

j

|

2

⁢

(

K

⁢

(

0

)

+

∑

h

≠

j

𝒪

⁢

(

j

,

h

)

⁢

|

K

⁢

(

t

j

-

t

h

)

|

)

,

where

|

𝒪

⁢

(

j

,

h

)

|

≤

1

. Using the

definition

of

k

and the previous inequality, we have

∫

-

π

π

|

f

⁢

(

x

)

|

2

⁢

𝑑

x

≥

∑

j

|

c

j

|

2

⁢

K

⁢

(

0

)

⁢

(

1

-

1

γ

2

)

,

and we have obtained the conclusion.

Title

proof of Ingham Inequality

Canonical

name

ProofOfInghamInequality

Date of creation

2013-03-22 15:55:18

Last modified on

2013-03-22 15:55:18

Owner

ncrom (8997)

Last modified by

ncrom (8997)

Numerical id

8

Author

ncrom (8997)

Entry type

Proof

Classification

msc 42B05

Generated on Fri Feb 9 21:29:58 2018 by

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Wirtinger’s inequality

Wirtinger’s inequality

Theorem

:

Let

f

:

ℝ

→

ℝ

be a periodic function of period

2

⁢

π

, which is

continuous

and has a continuous

derivative

throughout

ℝ

, and such
that

∫

0

2

⁢

π

f

⁢

(

x

)

=

0

.

(1)

Then

∫

0

2

⁢

π

f

′

⁣

2

⁢

(

x

)

⁢

𝑑

x

≥

∫

0

2

⁢

π

f

2

⁢

(

x

)

⁢

𝑑

x

(2)

with equality if and only if

f

⁢

(

x

)

=

a

⁢

cos

⁡

x

+

b

⁢

sin

⁡

x

for some

a

and

b

(or equivalently

f

⁢

(

x

)

=

c

⁢

sin

⁡

(

x

+

d

)

for some

c

and

d

).

Proof:

Since Dirichlet’s conditions are met, we
can write

f

⁢

(

x

)

=

1

2

⁢

a

0

+

∑

n

≥

1

(

a

n

⁢

sin

⁡

n

⁢

x

+

b

n

⁢

cos

⁡

n

⁢

x

)

and moreover

a

0

=

0

by (

1

). By Parseval’s

identity

,

∫

0

2

⁢

π

f

2

⁢

(

x

)

⁢

𝑑

x

=

∑

n

=

1

∞

(

a

n

2

+

b

n

2

)

and

∫

0

2

⁢

π

f

′

⁣

2

⁢

(

x

)

⁢

𝑑

x

=

∑

n

=

1

∞

n

2

⁢

(

a

n

2

+

b

n

2

)

and since the summands are all

≥

0

, we get (

2

),
with equality if and only if

a

n

=

b

n

=

0

for all

n

≥

2

.

Hurwitz used Wirtinger’s

inequality

in his tidy 1904
proof of the

isoperimetric inequality

.

Title

Wirtinger’s inequality

Canonical name

WirtingersInequality

Date of creation

2013-03-22 14:02:38

Last modified on

2013-03-22 14:02:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 42B05

Synonym

Wirtinger inequality

Generated on Fri Feb 9 21:30:01 2018 by

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Chebyshev polynomial

Chebyshev polynomial

The

Chebyshev polynomials of first kind

are defined by the simple
formula

T

n

⁢

(

x

)

=

cos

⁡

(

n

⁢

t

)

,

where

x

=

cos

⁡

t

.

It is an example of a

trigonometric polynomial

.

This can be seen to be a

polynomial

by expressing

cos

⁡

(

k

⁢

t

)

as a polynomial of

cos

⁡

(

t

)

, by using the formula for cosine of angle-sum:

cos

⁡

(

1

⁢

t

)

=

cos

⁡

(

t

)

cos

⁡

(

2

⁢

t

)

=

cos

⁡

(

t

)

⁢

cos

⁡

(

t

)

-

sin

⁡

(

t

)

⁢

sin

⁡

(

t

)

=

2

⁢

(

cos

⁡

(

t

)

)

2

-

1

cos

⁡

(

3

⁢

t

)

=

4

⁢

(

cos

⁡

(

t

)

)

3

-

3

⁢

cos

⁡

(

t

)

⋮

So we have

T

0

⁢

(

x

)

=

1

T

1

⁢

(

x

)

=

x

T

2

⁢

(

x

)

=

2

⁢

x

2

-

1

T

3

⁢

(

x

)

=

4

⁢

x

3

-

3

⁢

x

⋮

These polynomials obey the recurrence relation:

T

n

+

1

⁢

(

x

)

=

2

⁢

x

⁢

T

n

⁢

(

x

)

-

T

n

-

1

⁢

(

x

)

for

n

=

1

,

2

,

…

Related are the

Chebyshev polynomials

of the

second kind

that are
defined as

U

n

-

1

⁢

(

cos

⁡

t

)

=

sin

⁡

(

n

⁢

t

)

sin

⁡

(

t

)

,

which
can similarly be seen to be polynomials through either a similar process as the
above or by the

relation

U

n

-

1

⁢

(

t

)

=

n

⁢

T

n

′

⁢

(

t

)

.

The first few are:

U

0

⁢

(

x

)

=

1

U

1

⁢

(

x

)

=

2

⁢

x

U

2

⁢

(

x

)

=

4

⁢

x

2

-

1

U

3

⁢

(

x

)

=

8

⁢

x

3

-

4

⁢

x

⋮

The same recurrence relation also holds for

U

:

U

n

+

1

⁢

(

x

)

=

2

⁢

x

⁢

U

n

⁢

(

x

)

-

U

n

-

1

⁢

(

x

)

for

n

=

1

,

2

,

…

.

Title

Chebyshev polynomial

Canonical name

ChebyshevPolynomial

Date of creation

2013-03-22 12:22:56

Last modified on

2013-03-22 12:22:56

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Definition

Classification

msc 42C05

Classification

msc 42A05

Classification

msc 33C45

Related topic

Polynomial

Defines

Chebyshev polynomial of first kind

Defines

Chebyshev polynomial of second kind

Generated on Fri Feb 9 21:30:11 2018 by

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Christoffel-Darboux formula

Christoffel-Darboux formula

Let

{

ϕ

i

}

i

=

0

n

be orthonormal

polynomials

(the degree of

ϕ

k

is

k

) and let

k

n

be the

coefficient

of

x

n

in

ϕ

n

. Then

∑

k

=

0

n

ϕ

k

⁢

(

x

)

⁢

ϕ

k

⁢

(

y

)

=

k

n

k

n

+

1

⁢

(

ϕ

n

⁢

(

y

)

⁢

ϕ

n

+

1

⁢

(

x

)

-

ϕ

n

⁢

(

x

)

⁢

ϕ

n

+

1

⁢

(

y

)

x

-

y

)

The reason this formula is interesting is that the left-hand side is
the integral kernel for the projection operator to the

subspace

spanned
by the polynomials

{

ϕ

i

}

i

=

0

n

.

Title

Christoffel-Darboux formula

Canonical name

ChristoffelDarbouxFormula

Date of creation

2013-03-22 16:20:32

Last modified on

2013-03-22 16:20:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 42C05

Classification

msc 33D45

Generated on Fri Feb 9 21:30:14 2018 by

LaTeXML
Balian-Low

Balian-Low

Theorem 1

(Balian-Low)

Suppose

g

∈

L

2

⁢

(

R

)

and

g

m

,

n

⁢

(

x

)

=

e

2

⁢

π

⁢

i

⁢

m

⁢

x

⁢

g

⁢

(

x

-

n

)

, where

m

,

n

∈

Z

.
If

{

g

m

,

n

:

m

,

n

∈

Z

}

is an

orthonormal basis

for

L

2

⁢

(

R

)

, then either

∫

-

∞

∞

x

2

⁢

|

g

⁢

(

x

)

|

2

⁢

𝑑

x

=

∞

⁢

or

⁢

∫

-

∞

∞

ξ

2

⁢

|

g

^

⁢

(

ξ

)

|

2

⁢

𝑑

ξ

=

∞

.

Title

Balian-Low

Canonical name

BalianLow

Date of creation

2013-03-22 15:35:12

Last modified on

2013-03-22 15:35:12

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

4

Author

swiftset (1337)

Entry type

Theorem

Classification

msc 42C10

Generated on Fri Feb 9 21:30:23 2018 by

LaTeXML
equivalent condition for the translates of an L_2 function to form a Riesz sequence, an

equivalent condition for the translates of an

L

2

function to form a Riesz sequence, an

Theorem 1

Let

ϕ

∈

L

2

⁢

(

R

)

,

ϕ

k

⁢

(

x

)

=

ϕ

⁢

(

x

-

k

)

and

ϕ

^

be the

Fourier transform

of

ϕ

. Let

A

and

B

be

positive

constants

. Then the following are

equivalent

:

(i)

∀

c

⁢

(

k

)

∈

l

2

,

A

⁢

∥

c

∥

l

2

2

≤

∥

∑

k

∈

ℤ

c

⁢

(

k

)

⁢

ϕ

k

∥

2

≤

B

⁢

∥

c

∥

l

2

2

(ii)

A

≤

∑

k

∈

ℤ

|

ϕ

^

⁢

(

ω

+

2

⁢

π

⁢

k

)

|

2

≤

B

The first of the above conditions is the

definition

for

{

ϕ

k

}

k

∈

ℤ

to form a

Riesz sequence

.

Title

equivalent condition for the translates of an

L

2

function

to form a Riesz sequence, an

Canonical

name

EquivalentConditionForTheTranslatesOfAnL2FunctionToFormARieszSequenceAn

Date of creation

2013-03-22 15:20:07

Last modified on

2013-03-22 15:20:07

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

15

Author

Gorkem (3644)

Entry type

Theorem

Classification

msc 42C99

Related topic

RieszSequence

Generated on Fri Feb 9 21:30:29 2018 by

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