Copyright © 2009 Yingfang Du and Huajie Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We determine the weakly asymptotically orders for the average errors
of the Grünwald interpolation sequences based on the Tchebycheff nodes
in the Wiener space. By these results we know that for the Lp-norm
(2≤q≤4) approximation, the p-average (1≤p≤4) error of some Grünwald interpolation sequences is weakly equivalent to the p-average
errors of the best polynomial approximation sequence.
1. Introduction and Main Results
Let 
be a real separable Banach space equipped with a probability measure 
on the Borel sets of 
. Let 
be another normed space such that 
is continuously embedded in 
. By 
we denote the norm in 
. Any 
such that 
is a measurable mapping is called an approximation operator (or just approximation). The 
-average error of 
is defined as
(1.1)
Since in practice the underlying function is usually given via its (exact or noisy) values at finitely many points, the approximation operator 
is often considered depending on some function values about 
only. Many papers such as [1–4] studied the complexity of computing an 
-approximation in average case setting. Noticed that the polynomial interpolation operators are important approximation tool in the continuous functions space, and they are depending on some function values about 
only, we want to know the average error for some polynomial interpolation operators in the Wiener measure. Now we turn to describe the contents in detail.
Let 
be the space of continuous function 
defined on 
such that 
. The space 
is equipped with the sup norm. The Wiener measure 
is uniquely defined by the following property:
(1.2)
for every 
, where 
denote the set class of all Borel measurable subsets of 
, and 
with 
. Its mean is zero, and its correlation operator is given by 
for 
, that is,
(1.3)
In this paper, we specify 
, and for every measurable subset 
, we define
(1.4)
For 
, denote by 
the linear normed space of 
-integrable function 
on 
with the following finite norm:
(1.5)
Let
(1.6)
be the zeros of 
the 
th degree Tchebycheff polynomial of the first kind. The well-known Grünwald interpolation polynomial of 
based on 
is given by (see [5])
(1.7)
where
(1.8)
Theorem 1.1.
Let
be defined as above. Then
(1.9)
where in the following
means that there exists
independent of
such that
, and the constant
may be different in the same expression.
By Hölder inequality, combining Theorem 1.1 with paper [2] we know that for 
(1.10)
Remark 1.2.
Denote by 
the set of algebraic polynomials of degree 
. For 
, let 
denote the best 
-approximation polynomial of 
from 
. Then the 
-average error of the best 
-approximation of continuous functions by polynomials from 
over the Wiener space is given by
(1.11)
By Theorem 1.1 and paper [6] we can obtain that for 
and 
, we have
(1.12)
Remark 1.3.
Let us recall some fundamental notions about the information-based complexity in the average case setting. Let 
be a set with a probability measure 
, and let 
be a normed linear space with norm 
. Let 
be a measurable mapping from 
into 
which is called a solution operator. Let 
be a measurable mapping from 
into 
, and let 
be a measurable mapping from 
into 
which are called an information operator and an algorithm, respectively. For 
, the 
-average error of the approximation 
with respect to the measure 
is defined by
(1.13)
and the 
-average radius of information 
with respect to 
is defined by
(1.14)
where 
ranges over the set of all algorithms. Futhermore, let 
denote a class of permissible information functional and denote 
the set of nonadaptive information operators 
from 
of cardinality 
, that is,
(1.15)
Let
(1.16)
denote the 
th minimal 
-average radius of nonadaptive information in the class 
.
For example, if 
and 
are defined as above, 
is the identity mapping 
, and 
is consist of function evaluations at fixed point; then by [2] we know that for 
-norm approximation, if 
, we have
(1.17)
It is easy to understand that 
can be viewed as a composition of a nonadaptive information operator from 
and a linear algorithm, and for 
(1.18)
In comparison with the result of Theorem 1.1, we consider the following Grünwald interpolation. Let
(1.19)
be the zeros of 
, the 
th Tchebycheff polynomial of the second kind. The Grünwald interpolation polynomial of 
based on 
is given by
(1.20)
where
(1.21)
Theorem 1.4.
Let
be defined as above. Then
(1.22)
2. The Proof of Theorem 1.1
We consider the upper estimate first. From [7, page 107, 
] we obtain
(2.1)
where 
is the 
th absolute moment of the standard normal distribution. It is easy to verify
(2.2)
From (2.2) and Hölder inequality we can obtain
(2.3)
By (1.3) we obtain
(2.4)
Let 
, then it is easy to verify
(2.5)
By (2.4), (2.5), and a simple computation we can obtain
(2.6)
By (1.3), it is easy to verify that for 
(2.7)
Let 
. From (2.7) and a simple computation we know that for 
(2.8)
From [8] we know 
, hence
(2.9)
From (1.8) it follows that
(2.10)
From (2.7) and (2.10) it follows that
(2.11)
From (2.10) it follows that
(2.12)
Let 
, we have
(2.13)
By 
we know that for 
, thus
(2.14)
It is easy to know
(2.15)
By 
, (2.16), (2.17), and (2.18) we can obtain
(2.16)
From (2.12) and (2.16) we can obtain
(2.17)
Similarly
(2.18)
From (2.3), (2.8), (2.11), (2.17), and (2.18) we can obtain
(2.19)
By (2.1), (2.3), (2.6), and (2.19) we can obtain the upper estimate.
Now we consider the lower estimate. For 
, we can obtain the lower estimate from [2]. For 
, from (2.4) we know that
(2.20)
Let 
, then from (2.5) we know that
(2.21)
Hence we can verify that for 
(2.22)
From (2.20) and (2.22) and a simple computation we can obtain
(2.23)
From (2.2), (2.3), and (2.19) it follows that
(2.24)
From (2.1), (2.2), (2.23), and (2.24) we can obtain the lower estimate for 
.
3. The Proof of Theorem 1.4
Let
(3.1)
be the quasi-Hermite-Fejer interpolation polynomial of degree 
based on the extended Tchebycheff nodes of the second kind (see [8]); then by a simple computation we obtain
(3.2)
Denote
(3.3)
By (3.2) and the unique of the Hermite interpolation polynomial 
which satisfies interpolation conditions,
(3.4)
we obtain
(3.5)
By (3.5) and 
we know that
(3.6)
From [8] we know that for every 
(3.7)
where 
is the modulus of continuity of 
on 
defined for every 
, and 
is independent of 
and 
. By (3.7) and [6] we can obtain
(3.8)
By using 
and 
we obtain
(3.9)
By (1.3) we obtain
(3.10)
From (3.9) and (3.10) we obtain
(3.11)
Similar to (3.11), we have
(3.12)
By (3.3) and 
we obtain
(3.13)
By [8] we know that
(3.14)
By (1.3), (3.13), (3.14), and 
, we obtain
(3.15)
By (3.6), (3.8), (3.11), (3.12), and (3.15) we can obtain the upper estimate of Theorem 1.4. On the other hand, by (3.5) we can verify that
(3.16)
From (3.16), (3.8), (3.11), (3.12), and (3.15) we can obtain the lower estimate of Theorem 1.4.
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