Applied Mathematics
Vol.08 No.09(2017), Article ID:79144,40 pages
10.4236/am.2017.89095
A Study of Weighted Polynomial Approximations with Several Variables (I)
Ryozi Sakai
Department of Mathematics, Meijo University, Nagoya, Japan
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 3, 2017; Accepted: September 16, 2017; Published: September 19, 2017
ABSTRACT
In this paper, we investigate the weighted polynomial approximations with several variables. Our study relates to the approximation for
by weighted polynomials. Then we will estimate the degree of approximation.
Keywords:
Weighted Polynomial Approximations with Several Variables, the Degree of Approximations
1. Introduction
Let
(
times,
integer) be the direct product space, and let
, where
are even weight functions. We suppose that for every nonnegative integer n,
In this paper, we will study to approximate the real-valued weighted function
by weighted polynomials
, where
. Here,
means a class of
all polynomials with at most n-degree for each variable
. We need to define the norms. Let
, and let
be measurable. Then we define
We assume that for
the integral is independent of the order of integration with respect to each
. When
, we write
. If
, we require that
is continuous and
, where
.
Our purpose in this paper is to approximate the weighted function
by weighted polynomials
. The paper is arranged as the following. In Section 2, we give the definition of the weights which are treated in this paper. In Section 3, we consider the approximation for the functions in
. In Section 4, we consider a property of higher order derivatives. In Section 5, we estimate the degree of approximations. In Section 6, we consider the approximation for the functions with bounded variation. In Section 7, we consider the approximation of the Lipschitz-type functions. In Section 8, we treat the functions with higher order derivatives.
2. Class of Weight Functions and Preliminaries
Throughout the paper
denote positive constants independent of
or polynomials
. The same symbol does not necessarily denote the same constant in different occurrences. Let
mean that there exists a constant
such that
holds for all
, where
is a subset.
We say that
is quasi-increasing if there exists
such that
for
. Hereafter we consider following weights.
Definition 2.1. Let
be a continuous and even function, and satisfy the following properties:
(a)
is continuous in
, with
.
(b)
exists and is positive in
.
(c)
(d) The function
is quasi-increasing in
, with
(e) There exists
such that
Then we write
.
Moreover, if there also exists a compact subinterval
of
, and
such that
then we write
. If
is bounded, then the weight
is called a Freud-type weight, and if
is unbounded, then w is called an Erdös-type weight.
For
, if there exists
such that for
,
(2.1)
and there exist
such that for
,
(2.2)
then we write
. Furthermore, if
(2.3)
and the inequality (2.2) with
hold, then we write
.
We have some examples satisfying Definition 2.1.
Example 2.2 (cf. [1] [2] ). (1) If an exponential
satisfies
where
are constants, then we call
the Freud weight. The class
contains the Freud weights.
(2) For
we define
where
. Moreover, we define
where
if
, and otherwise
. We note that
gives a Freud-type weight, that is,
is bounded..
(3) We define
(4) Let
, and let us define
If
, then we say that the weight w is regular. All weights in examples (1), (2) and (3) are regular.
(5) More generally we can give the examples of weights
. If the weight w is regular and if
satisfies (2.1), then for the regular weights we have
(see [3] , Corollary 5.5 (5.8)).
The following fact is very important for our study.
Proposition 2.3 ( [3] , Theorem 4.1 and (4.11)). Let
and
. Then for
, we can construct a new weight
such that
and for some
,
where
and
are MRS-numbers for the weight
and
, respectively, and
and
are correspond for
or
, respectively.
Let
be orthonormal polynomials with respect to a weight w, that is,
is the polynomial of degree n such that
For
, we denote by
the usual
space on
(here for
, if
, then we require
to be continuous, and
to have limit 0 at
). For
, we set
(2.4)
for
(the partial sum of Fourier-type series). The de la Vallée Poussin mean of order n is defined by
(2.5)
Let
. We need the Mhaskar-Rakhmanov-Saff numbers (MRS-numbers)
;
We easily see
and
For
, the degree of weighted polynomial approximation is defined by
3. Approximations for Lp-Functions
In this section, we treat the function such as
, where
), and if
, then we suppose that
is continuous and
. For any multivariate point
, we consider the weights;
As shown under, we will also use
. Let
,
,
. From Proposition 2.3 we see
. Then we admit to write
. For the weight
we construct the modulus of continuity of
. It involves the function
where
is defined by
where
is the MRS-number for the weight
. If
, then we have
. In the sequel, if
is an integer, then
will denote the
norm of
taken with respect to the j-th variable. This is a function of the remaining
variables. For each fixed
, we write
(3.1)
Using
we define the modulus of continuity. For the Freud-type weight, we define
If
is Erdös-type, then we define
We remark that if
is bounded, then we see
, so we do not need the definition for the Freud-type weight.
Let vn be the de la Vallée Poussin mean opetator, and let
denote the operation to f with respect to j-th co-ordinate, and
will denote the operator
applied to f with respect to each of the first j co-ordinates. Clearly,
(3.2)
Let
be the MRS-number for the weight
.
First, we consider the following Proposition.
Proposition 3.1 ( [4] , Theorem 3.14). For
,
is dence in
, where
is a set of all continuous functions with a compact support on
.
From this proposition, for any
there exist a constant
and a continuous function
with a compact support
such that
(3.3)
Then we give the following assumption:
Assumption 3.2. In (3.3) we suppose that for every co-ordinate
(3.4)
holds.
We define a new class of functions
as follows:
(3.5)
where if
, then
and we suppose that f is continuous and
(we write this fact as
). We state the theorem in this section.
Theorem 3.3. (1) We suppose
, and let
(3.6)
Let
. Then we have
(3.7)
where
(3.8)
and
. Especially
, then we have
(3.9)
(2) We suppose
, and let (3.6) holds. Let
. Then we have
(3.10)
Especially
, then we have
(3.11)
First we will show (3.7). We need some preliminaries.
Proposition 3.4 ( [5] , Theorem 1). Let
.
(1) We assume that
satisfies
. Then there exists a constant
such that when
, then
and so,
(2) We assume that
satisfies
. Then there exists a constant
such that if
, then
Proposition 3.5 ( [5] , Corollary 6.2 (6.5)). Let
.
(1) Let
, and
be an integer. Then
where C do not depend on g and n.
(2) Let
, and
be an integer. Then
where C do not depend on g and n.
Proposition 3.6. ( [6] ) Let
, and let
. Then there exist
and positive constants C1, C2 such that for every
(and for
, we require
to be continuous, and
to vanish at
) and every
,
where
and
do not depend on
and
, and
will be defined in Section 6.
We set
We need the infinite-finite inequality.
Theorem 3.7 (Infinite-finite inequality). Let
, and let
. Then
(3.12)
If
, then there exists
such that
(3.13)
where
.
To prove Theorem 3.7 we use the following proposition with
.
Proposition 3.8 ( [2] , Theorem 1.9). Let
, and let
. Then
(3.14)
If
, then there exists
such that
(3.15)
Proof of Theorem 3.7. For the proof of (3.12) we use (3.14). We put A for the left side of the above equation. Let
. By repeatedly applying Proposition 3.8 (3.14), we have
Next, we show the case of
.
Similarly, using Proposition 3.8 (3.15), we easily have (3.13). #
Lemma 3.9. Let
.
(1) We assume that
satisfies (3.6). Then there exists a constant
such that when
,
and
(2) We assume that
. Let
, then
Proof. (1) From Theorem 3.4 (1), for
Inductively,
For the second formula, using Theorem 3.7 and the above inequality, we have
Similarly we have the following:
(2) From Theorem 3.4 (2) for
,
Inductively,
#
Proof of (3.7) in Theorem 3.3. By Proposition 3.5 (2) and Proposition 3.6, we get
where the constant C1 and c1 is independent of
. Similarly, for
,
(3.16)
Using
, we get from Lemma 3.9 (2) and (3.2),
by (3.16)
where
for
. Hence we obtain (3.7).
Proof of (3.10) in Theorem 3.3. By Proposition 3.5 (1) and Proposition 3.6, we get
where the constant C1 and c1 is independent of
. Similarly, for
,
(3.17)
Using
, we get from Lemma 3.9 (1) and (3.2),
by (3.17)
Hence we obtain (3.10).
To prove (3.9) and (3.11) we need a lemma:
Lemma 3.10. Let
,
and
. We have
(3.18)
Proof. We may show
(3.19)
Let
. If we put
(3.20)
we will see
(3.21)
Then we conclude (3.19). Now, from (3.20)
Since
, we see
that is, we have
Then, using ( [2] , Lemmas 3.6, 3.7), we see
for some
and
large enough. We have (3.21). So we conclude (3.19). #
Proof of (3.9). We will estimate
To do so we may estimate
For
we take
large enough, and then by
we can select a continuous function
such that
We note
(see [7] , Lemma 7). If
from our assumption and Lemma 3.10 we have
If
, then by Lemma 3.10 we see
When we take
small enough, we see
because of the continuity of
. Therefore we have
. Consequently, we have
(3.22)
Finally, we see
Here, if we set
, then we see
for some
, that is,
Therefore
(3.23)
For given
if we take
large enough and then
small enough, then by (3.22) and (3.23) we have
Consequently, we have (3.9).
Proof of (3.11). If in the proof of (3.9) we set as
(constant), then we obtain (3.11). #
Corollary 3.11. We suppose that
are the Freud-type weights. Let
. Then we have
Especially
, then we have
Corollary 3.12. We suppose
, and let (3.6) hold. Let
. If
), then we have
Moreover, we suppose
, and let (3.6) hold. If
, then we have
4. A Property of Higher Order Derivatives
In this section we show an important theorem which is useful in approximation theory. We use the following notations for
. Let r be a positive integer, and
.
Then we see
Especially, if
, then
Theorem 4.1. Let
, and let
. Let a constant
be fixed. We suppose that
is absolutely continuous on
and
. Then we have
(4.1)
where we set for each
or 1,
,
Furthermore, let r be a positive integer, and let for each
,
. We suppose that
is absolutely continuous and
. Then we have
(4.2)
for each
with
(4.3)
and
(4.4)
Proposition 4.2 ( [8] , Theorem 9, cf. [9] , Lemma 3.4.4). Let
and a constant
be fixed.
(a) We have
(b) Let
, and let r be a positive integer. If g is absolutely continuous,
and
, then
When
(
), and
is absolutely continuous,
with
, we see
Proposition 4.3 ( [3] , Theorem 4.2). Let
. Then for
, we can construct a new weight
such that
on
,
(c is an absolutely constant) on
and
hold on
. Furthermore, we see
Proof of Theorem 4.1. For the proof of (4.1) we may put
with
in the proof of (4.4) below. So we prove only (4.4). We use Proposition 4.2 and 4.3 repeatedly.
where
by
where
We continue this manner with respect to
. Then we can easily obtain as follows:
where for each
. We set (4.3).
Let
. Then we need to show
We rearrange
as
if
, and as
if
, where
. Then we set
. We see
Then we have
#
We can generalize Theorem 4.1 easily. We give a class of nonnegative integers
, and set
. For
we set
. Then we consider the order as follows:
means
Corollary 4.4. Let
be classes of nonnegative integers, where
. For each
, we
suppose
(
). If
is absolutely continuous, and
, then we see
where for each
we set
We remark that
means
for
.
5. Degree of Approximation
We define the degree of approximation for
as follows:
Using this
, we can estimate the degree of approximation of
from
.
Theorem 5.1. (1) Let
and let
. Furthermore, we suppose (3.3). Then we have
(2) If
, and let
, then we have
(3) Let
and let
. Then we have
(4) Furthermore, let
. If
for some
, then we have
Proof. (1) There exists
such that
. Therefore, by Lemma 3.9 (1)
(2) We see
,
. Then, there exists
such that
. Therefore, by Lemma 3.9 (2)
(3) Similarly, we have (3).
(4) It follows from Theorem 3.3. #
Theorem 5.2. Let
, and let
. Then if
, we have
and
Proposition 5.3 ( [10] , Lemma 2.5, [2] , Corollary 10.2). Let
and
. Then there exists a constant
such that, if
(
),
and
Proof of Theorem 5.2. We use Proposition 5.3 and Lemma 3.7 (1).
and further, using Theorem A1 (the Markov-Bernstein inequality) in Appendix,
#
In the rest of only this section, we suppose
so
Let
In ( [7] , Corollary 8) we give the Favard-type inequalities:
Proposition 5.4 [7] . Let
, and let
be an integer. Let
, and let
. Then we have
and equivalently,
The following theorem is a generalization of Proposition 5.4.
Theorem 5.5. We suppose
, and let (3.3) satisfy, that is,
Let
for some positive integer r. Then we have
Equivalently,
Proof. Using
, we get from Lemma 3.9 (2) and (3.2),
We estimate each term. From Proposition 5.4 with the weight
,
Now, we use Theorem 4.1 and the fact
then we have
Consequently, we have
Corollary 5.6. Under the conditions of Theorem 5.5, if w is a Freud-type weight, then
Equivalently,
Let
. For
we define the K-functional
by
where the infimum is over all functions
which are absolutely
continuous and
. We have the following.
Theorem 5.7. We suppose
, and let
Let
, and let
. Then we have
Proof. We take g as
and for this g we select
such that
Then, from Theorem 5.5 we see
#
Corollary 5.8. Let
, and let
be a Freud-type weight. If
, then we have
Let
. Damelin [11] gives a K-functional as follows:
where
and
are chosen in advance and
We recall the r-th order of the modulus of smoothness
, which is defined as follows (cf. [6] and [11] ). Let r be a positive integer, and let
. We set
For the Freud-type weight,
For the Erdös-type weight,
where
We remark that if
is bounded then we see
. So, we may consider for only the Erdös-type weight. Then the following proposition holds.
Proposition 5.9 ( [11] , Theorem 1.2, 1.3). Let
, and let
(contains
). Let
for which
(for
, we require
to be continuous, and
to vanish at
). Then we have
On
we define
We see
. Then we have the following:
Theorem 5.10. We suppose
, and let
Then
Proof. Using
, we get from Proposition 3.3 (2) and (3.2),
#
6. Approximation for Functions with Bounded Variations
We define the modulus of continuity. For the Freud-type weight
(all of weights
are Freud-type), we define
If
is Erdös-type (some weights
are Erdös-type), then we define
It is sufficient to consider only the modulus for the Erdös-type.
Assumption 6.1. Let
, and let
. Suppose that
is continuous on
, and
has a bounded variation on any compact interval in
with
(6.1)
and if
, then we further suppose
(6.2)
In (6.7) and (6.8) below, we put
, where
. Especially, in (6.8) we set
(6.3)
Theorem 6.2. We suppose
, and let (3.5) hold. Let
. Then we have
(6.4)
Now Assumption 6.1 holds. Then we have
(6.5)
Proof of (6.4). By Proposition 3.3 (1) and Proposition 3.4, we get
where the constant
and
is independent of
. Similarly, for
,
(6.6)
Using
, we get from Lemma 3.7 (1) and (6.6),
Hence, we have (6.4). #
To prove Theorem 6.2 (6.5) we need the following two theorems.
Theorem 6.3 (cf. [12] , Proposition 3.2). Let
,
. Let
hold Assumption 6.1, especially (6.1) and (6.2) hold. Then there exists a constant
such that for every
and
,
(6.7)
and
(6.8)
Proof. Let
. For t > 0, we write
, and let
and
. Since
for
, if we take t small enough, then
(6.9)
On the other hand, by Proposition 4.2 with
,
Similarly,
Therefore, we see
(6.10)
Consequently, from (6.9) and (6.10) we have
that is, we have (6.7). We show (6.8). Let
for large
. Then
Next, for
we see
hence for
small enough,
On the other hand,
Hence for
small enough,
Consequently, we have
that is, (6.8). #
Theorem 6.4. Under Assumption 6.1, we have
where
and
Proof. Let
.
Similarly,
Hence,
Consequently, we have
where
#
Proof of Theorem 6.2 (6.5). Using Theorems 6.3 and 6.4 with
, we easily obtain (6.5). #
We will give an analogy of Theorem 6.2. To do so we use the following weights . They are guaranteed by Proposition 2.3.
(6.11)
Assumption 6.5. Let
. Suppose that
is continuous on
, and
has a bounded variation on any compact interval in
with
(6.12)
and if
, then we further suppose
(6.13)
where the weights
and
are defined by (6.11).
Theorem 6.6. We suppose
, and let
(6.14)
Let
. Then we have
(6.15)
Now Assumption 6.5 holds. Then we have
(6.16)
Proof of (6.15). By Proposition 3.3 (2) and Proposition 3.4, we get
where the constant C1 and c1 are independent of
. Similarly, for
,
(6.17)
Using
, we get from Lemma 3.7 (2) and (6.17),
by (3.11)
where
for
, that is, we have (6.15).
(6.16) follows from the following theorems. #
In Theorems 6.3 and 6.4 we replace
with
of (6.11), where
, then we easily have the following Theorem 6.7.
Theorem 6.7 (cf. [Theorem 6.3 in this paper]). Let
. Let
hold Assumption 6.5, especially (6.18) and (6.19) hold. Then there exists a constant
such that for every
and
,
(6.18)
and
(6.19)
Theorem 6.8. Under Assumption 6.5, we have
where
and
7. Approximation for Functions of the Lipschitz Class
Through this section we consider the weight
.
Theorem 7.1. (1) We suppose
,
, and let (3.6) hold. Let
. Then we have
(7.1)
where
is defined in (3.8). Now, we suppose that
is continuous on
. Let
for some
, and let
for some
, that is,
(7.2)
Then we have
(7.3)
(2) We suppose
,
, and hold (3.6). Let
. Then we have
(7.4)
Now, we suppose that
has the conditions as (1). Then we have
(7.5)
Proof. We suppose
.
(1) (7.2) follows from (3.7). We will show (7.3). Now, we use
, where
, and
. We will estimate
From
and
(see [3] , Lemma 7) we have
By (7.2) we see
On the other hand, for
we estimate
by the boundedness of
by the boundedness of
(note
) and Lemma 3.10
Hence we conclude
(7.6)
Now, we see
Here, if we set
, then we see
for some
, that is,
Therefore, we have
(7.7)
Consequently, by (7.6) and (7.7) we have
So we have (7.3), that is,
(2) (7.4) follows from (3.7). The estimate (7.5) follows as (1). We omit the proof. #
8. Approximation for Differentiable Functions
In this section, we treat the differentiable functions.
Let
be fixed integers, and let
. We suppose that the multivariate function
is r-times partial differentiable, and then with norm:
where
The class of all functions
with
will be denoted by
. In the sequel, if
is an integer, then
will denote the
-norm of
taken with respect to the i-th variable.
Theorem 8.1. We suppose
and
is an integer. Let
, and let
. Then we have
(8.1)
where
.
Remark 8.2. Especially, for
we have
Theorem 8.3 ( [7] , Cororally 8). We suppose
. Let
and
is an integer. If
be absolutely continuous and
, then we have
Equivalently,
Proof of Theorem 8.1. We use Proposition 2.3, that is,
. In view of Theorem 3.3 (1) and a repeated application of Theorem 8.3, we get
where the constant
is independent of
, and
denotes differentiation with respect to the first variable. Similarly, for
,
(8.2)
where
denotes the derivative with respect to the j-th variable.
Using Lemma 3.7 and (8.2), we obtain for integer
,
(8.3)
From (8.2) and (8.3), we get
Therefore, we conclude
#
Cite this paper
Sakai, R. (2017) A Study of Weighted Polynomial Approximations with Several Variables (I). Applied Mathematics, 8, 1267-1306. https://doi.org/10.4236/am.2017.89095
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Appendix
In this Appendix we state two inequalities which play important roles in the study of approximation theory. In fact, we use Theorem
in the proof of Theorem 5.2. Let
be the MRS number of
.
Theorem
(Markov-Bernstein inequalities). Let
, and let
be integers. There exists
such that for
and
.
(1) if
, then we have
(2) if
, then we have
The following, so called, Nikolskee-type inequality is useful.
Theorem
(Nikolskii-type inequality). Let
, and let
. For
, we have
and for
, we have
To prove Theorem
we need the Proposition 5.3.
Proof of Theorem
. To prove (1), we use the second inequality in Proposition 5.3, repeatedly. Then we easily obtain the result.
(2) Using the first inequality in Proposition 5.3, repeatedly, we have the result. #
The proof of Theorem
is obtained by repeatedly using the following proposition.
Proposition A3 ( [8] , Theorem 18). Let
, and let
. For
, we have
and for
, we have