On best one sided approximation by multivariate lagrange

Mathematical Theory and Modeling www.iiste.org

ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)

Vol.4, No.9, 2014

10

On best one-sided approximation by multivariate Lagrange

Interpolation in weighted spaces

Saheb AL-Saidy Jawad K. Judy

E-Mail:[email protected] E-Mail:[email protected]

Dept. of Mathematics / College of Science / Al-Mustansiriya University- Iraq-July-2014.

Abstract:

Degree of best one-sided approximation of multivariate function that lies

in weighted space ( -space) by construct an operator which is dependent on

multivariate Lagrange Interpolation that satisfies:

‖ ‖ ∑

[ ]

Where the domain [ ] has been studied in this work. The result which

we end in it for the function (i.e. lies in Sobolov Space)

that the degree of best one-sided approximation of derivative of a function

which lies in the Sobolev space is minimized when the domain of is

restricted into rectangle domain in where:

∏ [ ] and ,direct theorem is proved in this work

by construct a new form of modulus of continuity and with benefit from

properties of simplex spline . Finally we shall prove invers

theorem of best one sided approximation of the function by using an operator

, the same result of best approximation of the function and the relation

between best and best one sided approximation.

1. Introduction:-

Throughout this paper, we use the weight function ∏ which

is non-negative measurable function on For the weighted space

is define by:

{ } Such that for the function ,

‖ ‖ [

]

, also

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Vol.4, No.9, 2014

11

‖ ‖ { { }} , [ ] and

degree of best one sided approximation of the function is defined as:

[ ] ⏟

‖

‖ (Where is the space of all

polynomials of degree and variables and

) and

degree of best approximation of is defined as:

[ ] ⏟

‖ ‖ . For we have

.Now for there are

(

) monomials

which have total degree ,a natural basis for is formed by {

}.Let { } be a sequence of pairwise distinct points in also that

{ } if

and if there is a unique polynomial

such that:

for each then we say that the Lagrange

interpolation problem is poised with respect to in and we denoted for the

polynomial by ,we call the (possibly infinite) sequence poised in

block if for any the Lagrange interpolation problem is poised with

respect to in where

.For

we have a whole block of

monomials of degree in several variables called the monomials , if

we arrange the multi-indices in lexicographical order, we can number

the monomials of total degree as [ ]

[ ] and is spanned by :

{ [ ]

[ ]

[ ]

| [ ]

[ ]| |

[ ]

[ ]| }

From this block wise viewpoint, it is only natural to group the interpolation

points according to the structure and we rewrite them as:

{ [ ]

[ ]

[ ]

| [ ]

[ ]| |

[ ]

[ ]| }

We refer that whenever is poised in block, the pointe can be

arranged in such a way. If is poised in block, the nth Newton fundamental

polynomial, denoted by [ ]

,

and is uniquely defined by the

conditions:




Vol.4, No.9, 2014

12

[ ]

(

) [ ]

( [ ]

) . Also we introduce

the vectors:

[ ]

This means that for each level there are Newton fundamental polynomials of

degree exactly n which vanish on all points of lower level and all points of the

th level except the one which has the same index Next, we recall the definition

and some properties of simplex spline as in [1], given knots

the simplex spline is defined by the condition:

…………… (1-1)

Where { } .To exclude

cases of degeneration which can be handled similarly, let us assume here that the

convex hull of the knots, [ ] has dimension . Then the simplex spline

is a nonnegative piecewise polynomial of degree ,

supported on [ ]. The order of differentiability depends on the position

of the knots; if, e.g., the knots are in general position (i.e. any subset of

knots spans a proper simplex, then the simplex spline has maximal order of

differentiability, namely . The most important property for our present

purposes is the formula for directional derivatives namely,

∑ ( | ) where ∑

∑

in

particular for .

( | ) ( | ).Now

let be poised in blocked, we introduce the vectors

[

] And [ ] so we define the finite difference

in which denoted by [

] as:

[ ]

[ ] [

]

∑ [

]

[ ]

.




Vol.4, No.9, 2014

13

If and if we have ordered points then there exists

one fundamental polynomial, given by:

[ ]

, so the classical divided difference of a function of

one variable is defined:

[ ] [ ] .Also in [1] we

establish a representation of our finite difference in terms of simplex splines let:

{ ( ) } be an index set each

define a path among the components of which we denoted by

, {

} we note that by definition thus the path

described by starts from

,passes through

and ends at

,the collection { } contains all paths from the sole point

of

level to all the pointe of level .For any path we define the th

directional derivative along that path as:

And we will need the

values ∏

[ ](

)

.

In this search we introduce a new form of modulus of smoothness in

multivariate case as the following:

Where

is the usual th forward

difference of step length with respect to .

Also, modulus of smoothness is defined as:

⏟

‖ ‖ .

In this search we define a new form of difference as:

Where

‖

‖

[

]




Vol.4, No.9, 2014

14

It is clear that

‖ ‖ ‖ ‖ ‖ ‖ .

In [3] suppose that (where is positive integer numbers) be abounded

and satisfy if and Then , we define { } and we assume that is rectangle in with side length in the -th direction , , .

Also In [3] we define: ∑ ⏟

‖ ‖ .

And If Satisfies: (if for all ) and ⋃

where each is rectangle and for each , there is a rectangle

with one of its vertices such that: ⋂ Implies …….............................……………… (1-2).

2. Auxiliary Results: Theorem 2.1:[1]

Let the interpolation problem is based on the points be poised.

Then the Lagrange interpolation polynomial

is given by:

∑ ∑ [

] [ ]

.

Theorem 2.2:[1]

For and we have:

∑

[ ]

Theorem 2.3:

For , [ ] we get:

.

Proof:

By using properties of norm, properties of partial derivative and by




Vol.4, No.9, 2014

15

Assume ∑ ( )

we get:

( )

⏟

{‖ (

)‖

}

⏟

{‖ ‖ }

⏟

{

∑ (

)

}

⏟

{

∑ (

)

}

⏟

{

∑ (

)

⏟

{∑ (

) ( )

}

⏟

{ ∑ ( )

}

⏟

{ }

⏟

{ ⏟

}

⏟

{‖ ‖ }

.

Theorem 2.4:

For and we have:




Vol.4, No.9, 2014

16

∑

( )

.

Proof:

Suppose that

and by using (1-1) and theorem 2.3 we

have:

∑

∑

∑

∑

∑ ∑

(

)

∑

.

∑ ( )

∑ ( )

.

Theorem 2.5:

For [ ] and [ ] and

we have:

[ ]

.

Proof:

[ ]

∏

∏

∏

∏ .

Now, we construct an operator as we mention before as:




Vol.4, No.9, 2014

17

For the polynomial [ ]

it is clear that

[ ] is

algebraic polynomial of degree less than .And for [ ]

[ ]

(

) And [ ]

(

)

Is the Th Newton fundamental polynomial [ ]

and:

[ ]

(

) ∏ [ ]

Assume that ∏

‖∑

[ ] ‖

[ ]

(

)

Now, we define -opearator as:

∑ ‖ ‖ .

Where:

{ } [ ]

[ ]

Theorem 2.6:

[ ].

Proof:

As in theorem (2.5) we have:

∏ ∏

‖∑

[ ]

‖

[ ]

(

)

∏ ∏

⏟

∑

[ ]

[ ](

)




Vol.4, No.9, 2014

18

∏ ∏

(

) ( )

( ) ( )

∏∏

( ) ( )

∏

∏

.

Theorem 2.6:

For [ ] then

, [ ] .

Proof:

∑ ‖ ‖

‖ ‖

With the same way we can prove: .

Then

, [ ] .

Theorem 2.7:

For the polynomial [ ]

we have:

‖ ∏

[ ]

(

)

‖

.

Proof:

By using properties of limit we get:

‖ ∏

[ ]

(

)

‖

‖

∏

[ ]

(

)

‖




Vol.4, No.9, 2014

19

‖

∏

‖

‖

∏

‖

‖

∏ ( )

‖

‖

∏ ( )

‖

‖

‖

.

3. Main Results:

Theorem 3.1:

For the function [ ] we have:

[ ] ∑ .

Proof:

By using theorems 2.6, 2.2, properties of norm and theorem 2.7 we get:

[ ] ‖

‖

‖∑ ‖ ‖ ‖

‖ ∑ ∏ ∏

‖∑

[ ]

‖

[ ](

)

‖ ‖

‖

‖∏ ∑ ∏

‖∑

[ ]

‖

[ ]

(

)

‖ ∑

[ ]

‖

‖




Vol.4, No.9, 2014

20

‖∏ ∑ ∏

‖∑

[ ]

‖

[ ]

(

)

⏟

∑

[ ]

‖

‖∏ ∑ ∏

‖∑

[ ]

‖

[ ]

(

)

⏟

∑

[ ]

‖

‖∏ ∑ ∏ ( )

‖∑ [ ]

‖

( [ ]

(

( )))

∑

[ ]

⏟

‖

‖∏ ∑ ∏

‖∑

[ ]

‖

[ ]

(

)

∑ ⏟

|

[ ]

| ⏟

|

|

‖

‖∏ ∑ ∏

‖∑

[ ]

‖

[ ]

(

)

∑ ‖

[ ]

‖

‖

‖

‖

‖∏ ∑ ∏

‖∑

[ ] ‖

[ ](

)

∑ ‖

[ ]

‖

∑ ‖

‖

‖

‖∏ ∑ ∏

[ ]

(

)

∑‖

‖

‖

‖∏ ∑ ∏

[ ]

(

)

‖

∑ ‖

‖

∑ ‖ ‖




Vol.4, No.9, 2014

21

∑

∑ .

Now, we discuss the best on sided approximate of derivative of the function

[ ] and minimize degree of this approximation when domain of

the function is restrict to rectangle domain in where ∏ [ ]

and .

Theorem 3.2:[3]

If satisfies (1-2) and is rectangle with side length vector then for

:

‖ ‖ ∑ ‖ ‖

Theorem 3.3:

For [ ] we have:

[ ]

.

Proof:

Since

, [ ]

Then either

Or,

where

So, by using Bernstein equality in multivariate case and theorem 3.1 we get:

[ ] ‖

‖

‖

‖

‖

‖

∑ .

The other cases:




Vol.4, No.9, 2014

22

And since lies in weighted space then lies in weighted space.

So, by using properties of norm and theorems 3.2.

There exist [ ] such that:

‖ ‖

‖ ‖

‖ ‖ ‖ ‖

‖ ‖

‖ ‖ ∑ ‖ ‖

.

In this section we try to estimate invers theorem of one sided in multivariate

case by benefit from invers theorem in multivariate case and relation between

best approximation and best one approximation.

Theorem 3.4:[4]

For [ ] we have:

.

It's easy to prove above theorem in multivariate case

Theorem 3.5:[2 ](Invers Theorem)

Let be Freud weights for be an integer then for

then:

{‖ ‖ ∑

}.

Theorem 3.6:

For [ ] we have:

{‖ ‖ ∑

}.




Vol.4, No.9, 2014

23

Proof:

By using theorems 3.4 and 3.5 we get:

Then:

{‖ ‖ ∑

}

{‖ ‖ ∑

}

So, we have:

{‖ ‖ ∑

}.

References:

[1] Thomas Sauer and Yuan Xu "On Multivariate Lagrange Interpolation

"copyright 1995 American Mathematical Society, Mathematics of Computation,

July 1995, Pages 1147-1170.

[2] H.N.Mhaskar "On the Degree of Approximation in Multivariate Weighted

Approximation" Department Of Mathematics, California State University, Los

Angeles California, 90032, U.S.A.

[3] W.Dahmen, R.DeVore and K.Scherer"Multi-Dimensional Spline

Approxmation"0036-1429/80/1703-0007501.00/0 siamj,number.anal. vol.17

No.3, June 1980.

[4] Al-Saad Hamsa Ali "ON One Sided Approximation of Functions "a thesis,

College Of Science, AL-Mustansiriya University/2014.


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On best one sided approximation by multivariate lagrange