aarjaneiro/planetmath huggingface dataset:
Cesàro summability

Cesàro summability

Cesàro summability is a generalized

convergence

criterion for

infinite
series

. We say that a series

∑

n

=

0

∞

a

n

is
Cesàro summable if the Cesàro means of the

partial sums

converge

to
some limit

L

. To be more precise, letting

s

N

=

∑

n

=

0

N

a

n

denote the

N

th

partial sum, we say that

∑

n

=

0

∞

a

n

Cesàro converges to a limit

L

, if

1

N

+

1

⁢

(

s

0

+

…

+

s

N

)

→

L

as

N

→

∞

.

Cesàro summability is a

generalization

of the usual definition of the
limit of an infinite series.

Proposition 1

Suppose that

∑

n

=

0

∞

a

n

=

L

,

in the usual sense that

s

N

→

L

as

N

→

∞

. Then, the series in
question Cesàro converges to the same
limit.

The

converse

, however is false. The standard example of a

divergent
series

, that is nonetheless Cesàro summable is

∑

n

=

0

∞

(

-

1

)

n

.

The sequence of partial sums

1

,

0

,

1

,

0

,

…

does not converge. The Cesàro means, namely

1

1

,

1

2

,

2

3

,

2

4

,

3

5

,

3

6

,

…

do converge, with

1

/

2

as the limit. Hence the series in
question is Cesàro summable.

There is also a

relation

between Cesàro summability and

Abel
summability

1

1

This and

similar

results are often called Abelian
theorems.

.

Theorem 2

(Frobenius)

A series that is Cesàro summable is also Abel summable. To be more
precise, suppose that

1

N

+

1

⁢

(

s

0

+

…

+

s

N

)

→

L

𝑎𝑠

N

→

∞

.

Then,

f

⁢

(

r

)

=

∑

n

=

0

∞

a

n

⁢

r

n

→

L

𝑎𝑠

r

→

1

-

as well.

Title

Cesàro summability

Canonical name

CesaroSummability

Date of creation

2013-03-22 13:07:01

Last modified on

2013-03-22 13:07:01

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 40G05

Related topic

CesaroMean

Generated on Fri Feb 9 21:24:45 2018 by

LaTeXML
proof of Abel’s convergence theorem

proof of Abel’s convergence theorem

Suppose that

∑

n

=

0

∞

a

n

=

L

is a

convergent series

, and set

f

⁢

(

r

)

=

∑

n

=

0

∞

a

n

⁢

r

n

.

Convergence

of the first series implies that

a

n

→

0

, and
hence

f

⁢

(

r

)

converges

for

|

r

|

<

1

. We will show that

f

⁢

(

r

)

→

L

as

r

→

1

-

.

Let

s

N

=

a

0

+

⋯

+

a

N

,

N

∈

ℕ

,

denote the corresponding

partial sums

. Our proof relies on the
following

identity

f

⁢

(

r

)

=

∑

n

a

n

⁢

r

n

=

(

1

-

r

)

⁢

∑

n

s

n

⁢

r

n

.

(1)

The above identity obviously works at the level of

formal power
series

. Indeed,

a

0

+

(

a

1

+

a

0

)

⁢

r

+

(

a

2

+

a

1

+

a

0

)

⁢

r

2

+

⋯

-

(

a

0

⁢

r

+

(

a

1

+

a

0

)

⁢

r

2

+

⋯

)

=

a

0

+

a

1

⁢

r

+

a

2

⁢

r

2

+

⋯

Since the partial sums

s

n

converge to

L

, they are

bounded

, and
hence

∑

n

s

n

⁢

r

n

converges for

|

r

|

<

1

. Hence for

|

r

|

<

1

, identity
(

1

) is also a genuine functional equality.

Let

ϵ

>

0

be given. Choose an

N

sufficiently large so that
all partial sums,

s

n

with

n

>

N

, satisfy

|

s

n

-

L

|

≤

ϵ

. Then, for all

r

such that

0

<

r

<

1

, one obtains

|

∑

n

=

N

+

1

∞

(

s

n

-

L

)

⁢

r

n

|

≤

ϵ

⁢

r

N

+

1

1

-

r

.

Note that

f

⁢

(

r

)

-

L

=

(

1

-

r

)

⁢

∑

n

=

0

N

(

s

n

-

L

)

⁢

r

n

+

(

1

-

r

)

⁢

∑

n

=

N

+

1

∞

(

s

n

-

L

)

⁢

r

n

.

As

r

→

1

-

, the first term tends to

0

. The

absolute value

of the
second term is estimated by

ϵ

⁢

r

N

+

1

≤

ϵ

. Hence,

lim sup

r

→

1

-

⁡

|

f

⁢

(

r

)

-

L

|

≤

ϵ

.

Since

ϵ

>

0

was
arbitrary, it follows that

f

⁢

(

r

)

→

L

as

r

→

1

-

.
QED

Title

proof of Abel’s convergence theorem

Canonical name

ProofOfAbelsConvergenceTheorem

Date of creation

2013-03-22 13:07:39

Last modified on

2013-03-22 13:07:39

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Proof

Classification

msc 40G10

Related topic

ProofOfAbelsLimitTheorem

Generated on Fri Feb 9 21:24:50 2018 by

LaTeXML
proof of Tauber’s convergence theorem

proof of Tauber’s convergence theorem

Let

f

⁢

(

z

)

=

∑

n

=

0

∞

a

n

⁢

z

n

,

be a

complex power

series,

convergent

in the

open disk

|

z

|

<

1

. We suppose that

1.

n

⁢

a

n

→

0

as

n

→

∞

, and that

2.

f

⁢

(

r

)

converges

to some finite

L

as

r

→

1

-

;

and wish to show that

∑

n

a

n

converges to the same

L

as well.

Let

s

n

=

a

0

+

⋯

+

a

n

, where

n

=

0

,

1

,

…

, denote the

partial
sums

of the series in question. The enabling idea in Tauber’s

convergence

result (as well as other

Tauberian theorems

) is the
existence of a correspondence in the evolution of the

s

n

as

n

→

∞

, and the evolution of

f

⁢

(

r

)

as

r

→

1

-

. Indeed we shall show that

|

s

n

-

f

⁢

(

n

-

1

n

)

|

→

0

as

n

→

∞

.

(1)

The desired result then follows in an obvious fashion.

For every real

0

<

r

<

1

we have

s

n

=

f

⁢

(

r

)

+

∑

k

=

0

n

a

k

⁢

(

1

-

r

k

)

-

∑

k

=

n

+

1

∞

a

k

⁢

r

k

.

Setting

ϵ

n

=

sup

k

>

n

⁡

|

k

⁢

a

k

|

,

and noting that

1

-

r

k

=

(

1

-

r

)

⁢

(

1

+

r

+

⋯

+

r

k

-

1

)

<

k

⁢

(

1

-

r

)

,

we have that

|

s

n

-

f

⁢

(

r

)

|

≤

(

1

-

r

)

⁢

∑

k

=

0

n

k

⁢

a

k

+

ϵ

n

n

⁢

∑

k

=

n

+

1

∞

r

k

.

Setting

r

=

1

-

1

/

n

in the above

inequality

we get

|

s

n

-

f

⁢

(

1

-

1

/

n

)

|

≤

μ

n

+

ϵ

n

⁢

(

1

-

1

/

n

)

n

+

1

,

where

μ

n

=

1

n

⁢

∑

k

=

0

n

|

k

⁢

a

k

|

are the Cesàro
means of the

sequence

|

k

⁢

a

k

|

,

k

=

0

,

1

,

…

Since the
latter sequence converges to zero, so do the means

μ

n

, and the
suprema

ϵ

n

. Finally,
Euler’s formula for

e

gives

lim

n

→

∞

⁡

(

1

-

1

/

n

)

n

=

e

-

1

.

The validity of (

1

) follows immediately. QED

Title

proof of Tauber’s convergence theorem

Canonical name

ProofOfTaubersConvergenceTheorem

Date of creation

2013-03-22 13:08:20

Last modified on

2013-03-22 13:08:20

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Proof

Classification

msc 40G10

Generated on Fri Feb 9 21:24:53 2018 by

LaTeXML
left hand rule

left hand rule

The

left hand rule

for computing the

Riemann integral

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

is

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

lim

n

→

∞

⁡

∑

j

=

1

n

f

⁢

(

a

+

(

j

-

1

)

⁢

(

b

-

a

n

)

)

⁢

(

b

-

a

n

)

.

If the Riemann integral is considered as a

measure

of area under a curve, then the expressions

f

⁢

(

a

+

(

j

-

1

)

⁢

(

b

-

a

n

)

)

the of the

rectangles

, and

b

-

a

n

is the common of the rectangles.

The Riemann integral can be approximated by using a definite value for

n

rather than taking a limit. In this case, the

partition

is

{

[

a

,

a

+

b

-

a

n

)

,

…

,

[

a

+

(

b

-

a

)

⁢

(

n

-

1

)

n

,

b

]

}

, and the

function

is evaluated at the left

endpoints

of each of these

intervals

. Note that this is a special case of a

left Riemann sum

in which the

x

j

’s are evenly spaced.

Title

left hand rule

Canonical name

LeftHandRule

Date of creation

2013-03-22 15:57:38

Last modified on

2013-03-22 15:57:38

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 41-01

Classification

msc 28-00

Classification

msc 26A42

Related topic

RightHandRule

Related topic

MidpointRule

Related topic

RiemannSum

Related topic

ExampleOfEstimatingARiemannIntegral

Generated on Fri Feb 9 21:26:28 2018 by

LaTeXML
midpoint rule

midpoint rule

The

midpoint rule

for computing the

Riemann integral

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

is

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

lim

n

→

∞

⁡

∑

j

=

1

n

f

⁢

(

a

+

(

j

-

1

2

)

⁢

(

b

-

a

n

)

)

⁢

(

b

-

a

n

)

.

If the Riemann integral is considered as a

measure

of area under a curve, then the expressions

f

⁢

(

a

+

(

j

-

1

2

)

⁢

(

b

-

a

n

)

)

the of the

rectangles

, and

b

-

a

n

is the common of the rectangles.

The Riemann integral can be approximated by using a definite value for

n

rather than taking a limit. In this case, the

partition

is

{

[

a

,

a

+

b

-

a

n

)

,

…

,

[

a

+

(

b

-

a

)

⁢

(

n

-

1

)

n

,

b

]

}

, and the function is evaluated at the

midpoints

of each of these

intervals

. Note that this is a special case of a

Riemann sum

in which the

x

j

’s are evenly spaced and the

c

j

’s chosen are the midpoints.

If

f

is Riemann integrable on

[

a

,

b

]

such that

|

f

′′

⁢

(

x

)

|

≤

M

for every

x

∈

[

a

,

b

]

, then

|

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

-

∑

j

=

1

n

f

⁢

(

a

+

(

j

-

1

2

)

⁢

(

b

-

a

n

)

)

⁢

(

b

-

a

n

)

|

≤

M

⁢

(

b

-

a

)

3

24

⁢

n

2

.

Title

midpoint rule

Canonical name

MidpointRule

Date of creation

2013-03-22 15:57:44

Last modified on

2013-03-22 15:57:44

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 41-01

Classification

msc 28-00

Classification

msc 26A42

Related topic

LeftHandRule

Related topic

RightHandRule

Related topic

RiemannSum

Related topic

ExampleOfEstimatingARiemannIntegral

Generated on Fri Feb 9 21:26:30 2018 by

LaTeXML
right hand rule

right hand rule

The

right hand rule

for computing the

Riemann integral

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

is

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

lim

n

→

∞

⁡

∑

j

=

1

n

f

⁢

(

a

+

j

⁢

(

b

-

a

n

)

)

⁢

(

b

-

a

n

)

.

If the Riemann integral is considered as a

measure

of area under a curve, then the expressions

f

⁢

(

a

+

j

⁢

(

b

-

a

n

)

)

the of the

rectangles

, and

b

-

a

n

is the common of the rectangles.

The Riemann integral can be approximated by using a definite value for

n

rather than taking a limit. In this case, the

partition

is

{

[

a

,

a

+

b

-

a

n

)

,

…

,

[

a

+

(

b

-

a

)

⁢

(

n

-

1

)

n

,

b

]

}

, and the

function

is evaluated at the

endpoints

of each of these

intervals

. Note that this is a special case of a

right Riemann sum

(

http://planetmath.org/RightRiemannSum

) in which the

x

j

’s are evenly spaced.

Title

right hand rule

Canonical name

RightHandRule

Date of creation

2013-03-22 15:57:41

Last modified on

2013-03-22 15:57:41

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 41-01

Classification

msc 28-00

Classification

msc 26A42

Related topic

LeftHandRule

Related topic

MidpointRule

Related topic

RiemannSum

Related topic

ExampleOfEstimatingARiemannIntegral

Generated on Fri Feb 9 21:26:32 2018 by

LaTeXML
composite trapezoidal rule

composite trapezoidal rule

Definition

The

composite trapezoidal rule

is a method for approximating a

definite integral

by evaluating the

integrand

at

n

points. Let

[

a

,

b

]

be the interval of integration with a

partition

a

=

x

0

<

x

1

<

…

<

x

n

=

b

. Then the formal rule is given by

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

≈

1

2

⁢

∑

j

=

1

n

(

x

j

-

x

j

-

1

)

⁢

[

f

⁢

(

x

j

-

1

)

+

f

⁢

(

x

j

)

]

.

The

composite trapezoidal rule

can also be applied to a partition which is uniformly spaced (i.e. (

http://planetmath.org/Ie

)

x

j

-

x

j

-

1

=

h

for all

j

∈

{

1

,

…

,

n

}

). In this case, the formal rule is given by

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

≈

h

2

⁢

[

f

⁢

(

a

)

+

2

⁢

∑

j

=

1

n

-

1

f

⁢

(

a

+

j

⁢

h

)

+

f

⁢

(

b

)

]

.

Both

expressions

of the composite trapezoidal rule come from determining the areas of the figures in the corresponding graph. These figures are usually

right trapezoids

, but may also be

right triangles

or

line segments

on the

x

. See the entry on the

trapezoidal rule

for more details. See the

section

of the entry on examples of estimating a

Riemann integral

which deals with the composite trapezoidal rule for an illustration.

Remark:

The composite trapezoidal rule uses the trapezoidal rule on each

subinterval

, which is readily observed from

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

⁢

∑

j

=

1

n

∫

x

j

-

1

x

j

f

⁢

(

x

)

⁢

𝑑

x

≈

⁢

1

2

⁢

∑

j

=

1

n

(

x

j

-

x

j

-

1

)

⁢

[

f

⁢

(

x

j

-

1

)

+

f

⁢

(

x

j

)

]

.

Proposition

:

If

f

is Riemann integrable on

[

a

,

b

]

,

|

f

′′

⁢

(

x

)

|

≤

M

for all

x

∈

[

a

,

b

]

, and

n

is the number of

intervals

of the partition used to approximate

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

, then

|

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

-

1

2

⁢

∑

j

=

1

n

(

x

j

-

x

j

-

1

)

⁢

[

f

⁢

(

x

j

-

1

)

+

f

⁢

(

x

j

)

]

|

≤

M

⁢

(

b

-

a

)

3

12

⁢

n

2

.

Title

composite trapezoidal rule

Canonical

name

CompositeTrapezoidalRule

Date of creation

2013-03-22 16:05:16

Last modified on

2013-03-22 16:05:16

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 41A05

Classification

msc 41A55

Synonym

composite trapezoid rule

Related topic

TrapezoidalRule

Generated on Fri Feb 9 21:26:35 2018 by

LaTeXML
interpolation

interpolation

Interpolation

is a set of techniques in

approximation

where, given a set of paired data points

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

,

…

,

(

x

n

,

y

n

)

,

…

one is often interested in

•

finding a relation (usually in the form of a function

f

) that passes through (or is satisfied by) every one of these points, if the relation is unknown at the beginning,

•

finding a simplified relation to replace the original known relation that is very complicated and difficult to use,

•

finding other paired data points

(

x

α

,

y

α

)

in

addition

to the existing ones.

The data points

(

x

i

,

y

i

)

are called the

breakpoints

, and the function

f

is the

interpolating function

such that

f

⁢

(

x

i

)

=

y

i

for each

i

.

The choice of the interpolating function depends on what we wish to do with it. In some cases a

polynomial

is required, sometimes a

piecewise linear function

is prefered (

linear interpolation

), other times a

http://planetmath.org/node/4339

spline is of

interest

, when the interpolating function is required to not only to be

continuous

, but

differentiable

, or even smooth.

Even different strategies for finding the same interpolating function are of interest. The

Lagrange interpolation formula

is a direct way to calculate the interpolating polynomial. The Vandermonde interpolation formula is mainly of interest as a theoretical tool. Numerical implementation of Vandermonde interpolation involves solution of large ill conditioned

linear systems

, so numerical stability is questionable.

Title

interpolation

Canonical

name

Interpolation

Date of creation

2013-03-22 14:20:05

Last modified on

2013-03-22 14:20:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 41A05

Classification

msc 65D05

Defines

breakpoints

Defines

interpolating function

Generated on Fri Feb 9 21:26:37 2018 by

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Simpson’s 3/8 rule

Simpson’s 3/8 rule

Simpson’s

3

8

rule is a method for approximating a

definite integral

by evaluating the

integrand

at finitely many
points. The formal rule is given by

∫

x

0

x

3

f

⁢

(

x

)

⁢

𝑑

x

≈

3

⁢

h

8

⁢

[

f

⁢

(

x

0

)

+

3

⁢

f

⁢

(

x

1

)

+

3

⁢

f

⁢

(

x

2

)

+

f

⁢

(

x

3

)

]

where

x

1

=

x

0

+

h

,

x

2

=

x

0

+

2

⁢

h

,

x

3

=

x

0

+

3

⁢

h

.

Simpson’s

3

8

rule is the third Newton-Cotes quadrature
formula. It has degree of precision 3. This means it is exact for

polynomials

of degree less than or equal to three. Simpson’s

3

8

rule is an improvement to the traditional Simpson’s
rule. The extra function evaluation gives a slightly more accurate
approximation . We can see this with an example.

Using the fundamental theorem of the

calculus

, one shows

∫

0

π

sin

⁡

(

x

)

⁢

𝑑

x

=

2

.

In this case Simpson’s rule gives,

∫

0

π

sin

⁡

(

x

)

⁢

𝑑

x

≈

π

6

⁢

[

sin

⁡

(

0

)

+

4

⁢

sin

⁡

(

π

2

)

+

sin

⁡

(

π

)

]

=

2.094

However, Simpson’s

3

8

rule does slightly better.

∫

0

π

sin

⁡

(

x

)

⁢

𝑑

x

≈

(

3

8

)

⁢

π

3

⁢

[

sin

⁡

(

0

)

+

3

⁢

sin

⁡

(

π

3

)

+

3

⁢

sin

⁡

(

2

⁢

π

3

)

+

sin

⁡

(

π

)

]

=

2.040

Title

Simpson’s 3/8 rule

Canonical

name

Simpsons38Rule

Date of creation

2013-03-22 13:40:56

Last modified on

2013-03-22 13:40:56

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

11

Author

Daume (40)

Entry type

Definition

Classification

msc 41A05

Classification

msc 41A55

Generated on Fri Feb 9 21:26:44 2018 by

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Weierstrass approximation theorem

Weierstrass approximation theorem

If

f

is a

continuous

real-valued function on a

interval

[

a

,

b

]

then for all

ε

>

0

there exists a

polynomial

P

which

satisfies

|

f

⁢

(

x

)

-

P

⁢

(

x

)

|

<

ε

∀

x

∈

[

a

,

b

]

This

theorem

also holds for

compact subsets

of

ℝ

n

.

The

Stone-Weierstrass theorem

is a

generalization

to even more general situations.

Title

Weierstrass approximation theorem

Canonical name

WeierstrassApproximationTheorem

Date of creation

2013-03-22 14:34:52

Last modified on

2013-03-22 14:34:52

Owner

Tobi (6409)

Last modified by

Tobi (6409)

Numerical id

7

Author

Tobi (6409)

Entry type

Theorem

Classification

msc 41A10

Generated on Fri Feb 9 21:26:56 2018 by

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approximating sums of rational functions

approximating sums of rational functions

Given a sum of the form

∑

m

=

n

∞

f

⁢

(

m

)

where

f

is a

rational
function

, it is possible to approximate it by approximating

f

by another
rational function which can be summed in

closed form

. Furthermore, the

approximation

so obtained becomes better as

n

increases.

We begin with a simple illustrative example. Suppose that we want to sum

∑

m

=

n

∞

1

/

m

2

. We approximate

m

2

by

m

2

-

1

/

4

, which
factors as

(

m

+

1

/

2

)

⁢

(

m

-

1

/

2

)

. Then, upon separating the approximate summand
into

partial fractions

, the sum collapses:

∑

m

=

n

∞

1

(

m

+

1

/

2

)

⁢

(

m

-

1

/

2

)

=

∑

m

=

n

∞

(

1

m

-

1

/

2

-

1

m

+

1

/

2

)

=

∑

m

=

n

∞

1

m

-

1

/

2

-

∑

m

=

n

+

1

∞

1

m

-

1

/

2

=

1

n

-

1

/

2

Using a similar approach, we may estimate the error of our approximation.

[general method to come]

Title

approximating sums of rational functions

Canonical name

ApproximatingSumsOfRationalFunctions

Date of creation

2013-03-22 18:42:23

Last modified on

2013-03-22 18:42:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 41A20

Generated on Fri Feb 9 21:26:58 2018 by

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linear convergence

linear convergence

A

sequence

{

x

i

}

is said to

converge

linearly to

x

*

if there is a

constant

1

>

c

>

0

such that

||

x

i

+

1

-

x

*

||

≤

c

⁢

||

x

i

-

x

*

||

for all

i

>

N

for some

natural number

N

>

0

.

An alternative

definition

is that

||

x

i

+

1

-

x

i

||

≤

c

⁢

||

x

i

-

x

i

-

1

||

for all

i

.

Notice that if

N

=

1

, then by iterating the first inequality we have

||

x

i

+

1

-

x

*

||

≤

c

i

⁢

||

x

1

-

x

*

||

.

That is, the error decreases exponentially with the index

i

.

If the inequality holds for all

c

>

0

then we say that the sequence

{

x

i

}

has

superlinear convergence

.

Title

linear convergence

Canonical

name

LinearConvergence

Date of creation

2013-03-22 14:20:55

Last modified on

2013-03-22 14:20:55

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

13

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 41A25

Related topic

QuadraticConvergence

Defines

superlinear convergence

Generated on Fri Feb 9 21:26:59 2018 by

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quadratic convergence

quadratic convergence

A

sequence

{

x

i

}

in a

metric space

(

X

,

d

)

is said to

converge

quadratically to

x

*

if there is a constant

1

>

c

>

0

such that

d

⁢

(

x

i

+

1

,

x

*

)

≤

c

⁢

d

⁢

(

x

i

,

x

*

)

2

for all

i

.

The

convergence

is said to be of order

p

if

d

⁢

(

x

i

+

1

,

x

*

)

≤

c

⁢

d

⁢

(

x

i

,

x

*

)

p

for all

i

.

Title

quadratic convergence

Canonical name

QuadraticConvergence

Date of creation

2013-03-22 14:20:59

Last modified on

2013-03-22 14:20:59

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 41A25

Related topic

LinearConvergence

Generated on Fri Feb 9 21:27:01 2018 by

LaTeXML
superconvergence

superconvergence

A sequence

x

0

,

x

1

,

…

superconverges to 0

if, when the

x

i

are written in base 2, then each number

x

i

starts with

2

i

-

1

≈

2

i

zeroes.
For example, the following sequence is superconverging to 0.

x

n

+

1

=

x

n

2

(

x

n

)

10

(

x

n

)

2

x

0

=

1

2

.1

x

1

=

1

4

.01

x

2

=

1

16

.0001

x

3

=

1

256

.00000001

x

4

=

1

65536

.0000000000000001

In this case it is easy to see that the number of binary 0’s doubles each

x

n

.

A sequence

{

x

i

}

superconverges to

x

if

{

x

i

-

x

}

superconverges to 0, and a sequence

{

y

i

}

is said to be

superconvergent

if there exists a

y

to which the sequence superconverges.

Title

superconvergence

Canonical

name

Superconvergence

Date of creation

2013-03-22 11:58:12

Last modified on

2013-03-22 11:58:12

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

15

Author

mathcam (2727)

Entry type

Definition

Classification

msc 41A25

Synonym

superconverge

Related topic

NewtonsMethod

Related topic

KantorovitchsTheorem

Related topic

SuperincreasingSequence

Generated on Fri Feb 9 21:27:03 2018 by

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best approximation

best approximation

One of the problems in

approximation theory

is to determine points that minimize

distances

(to a given point or subset). More precisely,

Problem -

Let

X

be a metric space and

S

⊆

X

a subset. Given

x

0

∈

X

we want to know if there exists a point in

S

that minimizes the distance to

x

0

, i.e. if there exists

y

0

∈

S

such that

d

⁢

(

x

0

,

y

0

)

=

inf

y

∈

S

⁡

d

⁢

(

x

0

,

y

)

Definition -

A point

y

0

that the above conditions is called a

best

approximation

of

x

0

in

S

.

In general, best approximations do not exist. Thus, the problem is usually to identify classes of spaces

X

and

S

where the existence of best approximations can be assured.

Example :

When

S

is compact, best approximations of a given point

x

0

∈

X

in

S

always exist.

After one assures the existence of a best approximation, one can question about its uniqueness and how to calculate it explicitly.

Remark -

There is no reason to restrict to metric spaces. The definition of best approximation can be given for pseudo-metric spaces, semimetric spaces or any other space where a suitable notion of ”distance” can be given.

Title

best approximation

Canonical name

BestApproximation

Date of creation

2013-03-22 17:31:23

Last modified on

2013-03-22 17:31:23

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Definition

Classification

msc 41A50

Synonym

optimal approximation

Generated on Fri Feb 9 21:27:05 2018 by

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