Multivariate Lagrange Interpolation Approximation

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.9, 2014
On best one-sided approximation by multivariate Lagrange
Interpolation in weighted spaces
Saheb AL-Saidy Jawad K. Judy
E-Mail:dr.sahebmath@yahoo.com E-Mail:ali12mm@yahoo.com
Dept. of Mathematics / College of Science / Al-Mustansiriya University- Iraq-July-2014.
10
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:
‖ ‖ Σ
[ ]
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:
Π [ ]
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 Π
which
is non-negative measurable function on For the weighted space
is define by:
{ } Such that for the function ,
‖ ‖ [
]
, also

Vol.4, No.9, 2014
‖ ‖ { { }} , [ ] and
degree of best one sided approximation of the function is defined as:
11
̃ [ ] ⏟
‖
‖ (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,
Σ ( | )
where Σ
Σ
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:
[ ]
[ ] [ ]
Σ [
]
[ ] .

Vol.4, No.9, 2014
If and if we have ordered points then there exists
one fundamental polynomial, given by:
13
[ ]
, 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 Π
[ ] (
)
.
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:
Σ ⏟
‖ ‖ .
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:
Σ Σ [
]
[ ]
.
Theorem 2.2:[1]
For and we have:
Σ
[ ]
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 Σ (
)
we get:
( )
⏟
{‖
( )‖
}
⏟
{‖
‖ }
⏟
{
Σ (
)
}
⏟
{
Σ (
)
}
⏟
{
Σ (
)
⏟
{Σ (
) ( )
}
⏟
{ Σ (
)
}
⏟
{
}
⏟
{ ⏟
}
⏟
{‖
‖ }
.
Theorem 2.4:
For and we have:

Vol.4, No.9, 2014
16
Σ
( )
.
Proof:
Suppose that
and by using (1-1) and theorem 2.3 we
have:
Σ
Σ
Σ
Σ
Σ
Σ
(
)
Σ
.
Σ ( )
Σ ( )
.
Theorem 2.5:
For [ ] and [ ] and
we have:
[ ]
.
Proof:
[ ] Π
Π
Π
Π
.
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:
[ ] (
) Π
[ ]
Assume that Π
‖Σ
[ ]
‖
[ ] (
)
Now, we define
-opearator as:
Σ ‖ ‖ .
Where:
{ } [ ]
[ ]
Theorem 2.6:
[ ].
Proof:
As in theorem (2.5) we have:
Π
Π
‖Σ
[ ] ‖
[ ]
(
)
Π
Π
⏟
Σ
[ ]
[ ]
(
)

Vol.4, No.9, 2014
18
Π
Π
(
) ( )
(
) ( )
Π
Π
(
) ( )
Π
Π
.
Theorem 2.6:
For [ ] then
, [ ] .
Proof:
Σ ‖ ‖
‖ ‖
With the same way we can prove:
.
Then
, [ ] .
Theorem 2.7:
For the polynomial
[ ] we have:
‖
Π
[ ]
(
)
‖
.
Proof:
By using properties of limit we get:
‖
Π
[ ] (
)
‖
‖
Π
[ ] (
)
‖

Vol.4, No.9, 2014
19
‖
Π
‖
‖
Π
‖
‖
Π ( )
‖
‖
Π ( )
‖
‖
‖
.
3. Main Results:
Theorem 3.1:
For the function [ ] we have:
̃ [ ] Σ .
Proof:
By using theorems 2.6, 2.2, properties of norm and theorem 2.7 we get:
̃ [ ] ‖
‖
‖Σ ‖ ‖ ‖
‖ Σ Π
Π
‖Σ
[ ] ‖
[ ] (
)
‖ ‖
‖
‖Π Σ
Π
‖Σ
[ ] ‖
[ ] (
)
‖ Σ
[ ]
‖

Vol.4, No.9, 2014
20
‖Π Σ
Π
‖Σ
[ ] ‖
[ ]
(
)
⏟
Σ
[ ]
‖
‖Π Σ
Π
‖Σ
[ ] ‖
[ ]
(
)
⏟
Σ
[ ]
‖
‖Π Σ
Π ( )
‖Σ
[ ]
‖
(
[ ]
(
( )
))
Σ
[ ] ⏟
‖
‖Π Σ
Π
‖Σ
[ ] ‖
[ ] (
)
Σ ⏟
|
[ ] | ⏟
|
‖Π Σ
Π
‖Σ
[ ] ‖
[ ]
(
)
Σ ‖
[ ] ‖
‖
Π Σ
‖Π
‖Σ
[ ]
‖
[ ]
(
)
Σ ‖
[ ] ‖
Σ ‖
‖Π Σ
Π
[ ] (
)
Σ‖
‖
‖
‖Π Σ
Π
[ ]
(
)
‖
Σ ‖
‖
Σ ‖ ‖

Vol.4, No.9, 2014
21
Σ
Σ .
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 Π [ ]
and .
Theorem 3.2:[3]
If satisfies (1-2) and is rectangle with side length vector then for
:
‖ ‖ Σ ‖ ‖
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:
̃ [ ] ‖
‖
‖
‖
‖
‖
Σ .
The other cases:

Vol.4, No.9, 2014
And since lies in weighted space then lies in weighted space.
So, by using properties of norm and theorems 3.2.
22
There exist
[ ] such that:
̃ ‖
‖
‖
‖
‖ ‖ ‖
‖
‖ ‖
‖ ‖ Σ
‖ ‖
.
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:
{‖ ‖
Σ
}.
Theorem 3.6:
For [ ] we have:
{‖ ‖
Σ
̃ }.

Vol.4, No.9, 2014
23
Proof:
By using theorems 3.4 and 3.5 we get:
̃ Then:
{‖ ‖
Σ
}
{‖ ‖
Σ
̃ }
So, we have:
{‖ ‖
Σ
̃ }.
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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Multivariate Lagrange Interpolation Approximation

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