proof of Weierstrass approximation theorem in R^n

proof of Weierstrass approximation theorem in R^n

To show that the Weierstrass Approximaton

Theorem

holds in

ℝ

n

, we will use

induction

on

n

.

For the sake of simplicity, consider first the case of the cubical
region

0

≤

x

i

≤

1

,

1

≤

i

≤

n

. Suppose that

f

is a

continuous

, real valued

function

on this region. Let

ϵ

be an
arbitrary

positive

constant

.

Since a

continuous functions

on

compact

regions are

uniformly
continuous

,

f

is uniformly continuous. Hence, there exists an

integer

N

>

0

such that

|

f

⁢

(

a

)

-

f

⁢

(

b

)

|

<

ϵ

/

2

whenever

|

a

-

b

|

≤

1

/

N

and both

a

and

b

lie in the cubical region.

Define

ϕ

:

ℝ

→

ℝ

as follows:

ϕ

(

x

)

=

{

0

x

<

-

1

1

+

x

-

1

≤

x

≤

0

1

-

x

0

≤

x

≤

1

0

x

>

1

Consider the function

f

~

defined as follows:

f

~

⁢

(

x

1

,

…

⁢

x

n

)

=

∑

m

=

0

N

ϕ

⁢

(

N

⁢

x

1

+

m

)

⁢

f

⁢

(

m

/

N

,

x

2

,

…

⁢

x

n

)

We shall now show that

|

f

⁢

(

x

1

,

…

,

x

n

)

-

f

~

⁢

(

x

1

,

…

,

x

n

)

|

≤

ϵ

/

2

whenever

(

x

1

,

…

,

x

n

)

lies in the
cubical region. By way that

ϕ

was defined, only two of the terms
in the sum defining

f

~

will differ from zero for any
particular value of

x

1

, and hence

f

~

⁢

(

x

1

,

…

,

x

n

)

=

(

N

⁢

x

-

⌊

N

⁢

x

⌋

)

⁢

f

⁢

(

⌊

N

⁢

x

1

⌋

N

,

x

2

,

…

,

x

n

)

+

(

⌊

N

⁢

x

⌋

+

1

-

x

)

⁢

f

⁢

(

⌊

N

⁢

x

1

⌋

+

1

N

,

x

2

,

…

,

x

n

)

,

so

|

f

~

⁢

(

x

1

,

…

,

x

n

)

-

f

⁢

(

x

1

,

…

,

x

n

)

|

=

|

f

~

⁢

(

x

1

,

…

,

x

n

)

-

{

(

N

⁢

x

-

⌊

N

⁢

x

⌋

)

+

(

⌊

N

⁢

x

⌋

+

1

-

x

)

}

⁢

f

⁢

(

x

1

,

…

,

x

n

)

|

≤

(

N

⁢

x

-

⌊

N

⁢

x

⌋

)

⁢

|

f

⁢

(

⌊

N

⁢

x

1

⌋

N

,

x

2

,

…

,

x

n

)

-

f

⁢

(

x

1

,

X

2

⁢

…

,

x

n

)

|

+

(

⌊

N

⁢

x

⌋

+

1

-

x

)

⁢

|

(

⌊

N

⁢

x

1

⌋

+

1

N

,

x

2

,

…

,

x

n

)

-

f

⁢

(

x

1

,

x

2

⁢

…

,

x

n

)

|

≤

(

N

⁢

x

-

⌊

N

⁢

x

⌋

)

⁢

ϵ

2

+

(

⌊

N

⁢

x

⌋

+

1

-

x

)

⁢

ϵ

2

=

ϵ

2

.

Next, we will use the

Weierstrass approximation theorem

in

n

-

1

dimensions

and in one dimesnsion to approximate

f

~

by a

polynomial

. Since

f

is continuous and the cubical region is
compact,

f

must be

bounded

on this region. Let

M

be an

upper
bound

for the

absolute value

of

f

on the cubical region. Using the
Weierstrass approximation theorem in one dimension, we conclude that
there exists a polynoial

ϕ

˘

such that

|

ϕ

˘

⁢

(

a

)

-

ϕ

⁢

(

a

)

|

<

ϵ

/

(

4

⁢

M

⁢

N

)

for all

a

in the region. Using the
Weierstrass approximation theorem in

n

-

1

dimensions, we conclude
that there exist polynomials

p

m

,

0

≤

m

≤

N

such that

|

p

m

⁢

(

x

2

,

…

,

x

n

)

-

f

⁢

(

m

/

N

,

x

2

,

…

⁢

x

n

)

|

≤

ϵ

4

⁢

N

. Then one has the following

inequality

:

|

ϕ

˘

⁢

(

N

⁢

x

1

+

m

)

⁢

p

m

⁢

(

x

2

,

…

⁢

x

n

)

-

ϕ

⁢

(

N

⁢

x

1

+

m

)

⁢

f

⁢

(

m

/

N

,

x

2

,

…

⁢

x

n

)

|

=

|

ϕ

˘

(

N

x

1

+

m

)

p

m

(

x

2

,

…

x

n

)

-

ϕ

˘

(

N

x

1

+

m

)

f

(

m

/

N

,

x

2

,

…

x

n

)

+

ϕ

˘

(

N

x

1

+

m

)

f

(

m

/

N

,

x

2

,

…

x

n

)

-

ϕ

(

N

x

1

+

m

)

f

(

m

/

N

,

x

2

,

…

x

n

)

|

≤

|

ϕ

˘

⁢

(

N

⁢

x

1

+

m

)

|

⁢

|

p

m

⁢

(

x

2

,

…

⁢

x

n

)

-

f

⁢

(

m

/

N

,

x

2

,

…

⁢

x

n

)

|

+

|

f

⁢

(

m

/

N

,

x

2

,

…

⁢

x

n

)

|

⁢

|

ϕ

˘

⁢

(

N

⁢

x

1

+

m

)

-

ϕ

⁢

(

N

⁢

x

1

+

m

)

|

≤

ϵ

4

⁢

N

+

M

⁢

ϵ

4

⁢

M

⁢

N

=

ϵ

2

⁢

N

Define

f

˘

⁢

(

x

1

,

…

⁢

x

n

)

=

∑

m

=

0

N

ϕ

˘

⁢

(

N

⁢

x

1

+

m

)

⁢

p

m

⁢

(

x

2

,

…

⁢

x

n

)

.

As a finite sum of

products

of polynomials, this is a polynomial.
From the above inequality, we conclude that

|

f

˘

⁢

(

a

)

-

f

~

⁢

(

a

)

|

≤

ϵ

/

2

, hence

|

f

⁢

(

a

)

-

f

˘

⁢

(

a

)

|

≤

ϵ

.

It is a simple matter of rescaling variables to conclude the
Weirestrass

approximation theorem

for arbitrary parallelopipeds. Any
compact subset of

ℝ

n

can be embedded in some paralleloped
and any continuous function on the compact subset can be extended to a
continuous function on the parallelopiped. By approximating this
extended function, we conclude the Weierstrass approximation theorem
for arbitrary compact subsets of

ℝ

n

.

Title

proof of Weierstrass approximation theorem

in R^n

Canonical name

ProofOfWeierstrassApproximationTheoremInRn

Date of creation

2013-03-22 15:40:03

Last modified on

2013-03-22 15:40:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 41A10

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