Abstract
The works of Smale and Zhou (2003, 2007), Cucker and Smale (2002), and Cucker and Zhou (2007) indicate that approximation operators serve as cores of many machine learning algorithms. In this paper we study the Hermite-Fejér interpolation operator which has this potential of applications. The interpolation is defined by zeros of the Jacobi polynomials with parameters , . Approximation rate is obtained for continuous functions. Asymptotic expression of the -functional associated with the interpolation operators is given.
1. Introduction
Zhou and Jetter [1] used Bernstein-Durrmeyer operators for studying support vector machine classification algorithms. This work initiates the direction of applying more linear operators from approximation theory to learning theory. We will follow this direction and study Hermite-Fejér interpolation operator. It would be interesting to derive explicit learning rates by means of these operators for some specific learning algorithms.
Let denote the Jacobi polynomial of order . Let be the zeros of . We assume that . For any continuous function on , the Hermite-Fejér interpolation is a polynomial of order that satisfies for any . Let be the norm on (, ). Without introducing ambiguity we also use to denote the norm on (which is the totality of the continuous functions on with period , and in this case ).
One has (see e.g., [2]) if and only if .
Define Denote by the conjugate function of , and write . When , , one has (see [3, 4]) So when , , is saturation with and the saturation class is
Note that for all , the associated classes are identical ([4, Theorem 6]). See [2, 5–7] for related works.
Denote that and recursively with . Write . Define For any , one has [8] Let and . We use the following definition of -functional from [9]:
We cite the following three Theorems from [9].
Theorem 1. Let , be fixed. Then there is a constant such that for all and all , one has
in which the symbol does not rely on and .
Theorem 2. Let , be fixed. Then the following relation holds:
Theorem 3. Let , be fixed. Then there is a constant such that, for all and all , one has
Here the symbol does not rely on and .
The -functional for all , is characterized by the following
Theorem 4. Let , and , be fixed. Then, for all and all , the following holds:
in which
Moreover, if , , then, due to (see (2.6) of [9]), we have
Theorem 4 will be proved in Section 3. In Section 2 we will discuss some properties of Jacobi polynomials and make some remarks concerning the conjugate function .
2. Estimates for Jacobi Polynomials and Conjugate Functions
Chapter 8 of [10] gave the following.
Lemma 5. Let be fixed. For all and , one has
in which and . If , then there exists a making and . Moreover, let . Then
and, for , one has with .
Define
and we have
For we have a conclusion similar to (17) (see [11]) as follows.
Lemma 6. For all fixed and , there is such that
Denote by the set of -order algebraic polynomials and by the set of -order trigonometric polynomials. Denote further that
The following identity can be found in [9, 12] (see Lemma 5 of [12] and its proof).
Lemma 7. Let , be fixed. Then for all
For our purpose, we need the following result, which improves the estimate of [4] (see Lemma 4 of [4]).
Lemma 8. Let , be fixed. If and , then for all , the following holds:
Proof. Write . So and for
On the other hand,
Through integration by parts,
Moreover, we have
but
So we have
Thus, if , then
If , then and
In this case, for , we have
therefore,
Let in (35) and, with (32), we obtain
For , it is clear that if
then (31) and (37) imply that
On the other hand, it is easy to see that the above estimate also holds for . Thus, the desired inequality is obtained.
Next, we are going to prove (37). This time we define
Simple calculation shows that
With (36), we obtain
Hence,
Similar to the case of , we have
which obviously implies (37).
In what follows, we will give an estimate of the conjugate function defined in the saturation class .
Lemma 9. Let , be fixed. Then, for all and even , one has
in which does not depend on and . Moreover, let be the best approximation of . Then, for all , one gets
Proof. Since
we have
Integrating by parts, we obtain
Therefore,
To deal with the second term of the above estimate, we note that, if , then , and
Thus,
If , rewrite the previous term as
Obviously, we get
On the other hand, we have
Since
we obtain
Moreover, the following estimates hold:
Consequently, for all , we get
which proves the first assertion of the lemma.
The second estimate can be obtained from Lemma 8, the first estimate, and integration by parts.
Lemma 10. There exists an absolute constant such that, for all even ,
Proof. We may assume that and , . Thus, by [13, 14], for Fejér mean of , we have
therefore,
For , we have
Consequently, for with and , we obtain
We may assume that
Otherwise, choose and to make even and . Then
We conclude that
which gives the desired inequality.
3. Proof of Theorem 4
We need to prove the following Lemma before Theorem 4.
Lemma 11. Given , , there is such that, for all and all , one has
Proof. Denote that . Then from Theorem 3, we have
Next, let us estimate .
We know that with and . But (1) tells that . Hence, for . Lemma 5 tells that, for each , there is satisfying . Assume that . Then and further
We may assume that . Thus, the Bernstein inequality for trigonometric polynomials yields
But
So, we have
Combining this inequality with (68), we get
Now we need only to prove
Obviously,
Lemma 5 tells the following. Let be fixed, then for ,
Thus, for those ,
Since , following (23), (18), and Lemma 6, we have, for those ,
If and , then there is a satisfying (see Lemma 5). Hence, from Bernstein inequality , we have
Moreover,
Consequently, (78) obtained from Lemma 9 holds for all . Finally, from (78) and (75) we obtain (74).
Proof of Theorem 4. Firstly, we prove that
Let , , and . From Theorems 1 and 2 we conclude that, for some ,
We know that , so (see [14, page 43])
Thus, (18) and Lemmas 6 and 7 imply that
Consequently, by Lemma 11, we get
Obviously,
If , then Lemma 9 implies that
Since
therefore, for , we have
In the same way, if , we obtain
Consequently,
Therefore,
Next, we prove that
Firstly, we assume that and . Thus, for and , we have
For , denote that and let be the best approximation of . Following from (94), Lemmas 8 and 9, we have
Thus,
Therefore,
Next, suppose that . Clearly, we have
and for . Hence, let , , and . From Lemma 9 we obtain
for .
On the other hand, since , we have
We know that, for ,
Lemma 10 shows that the last three estimates imply that
So by Lemma 7 (see (78)) and Lemma 11 we obtain again
If , then, due to (see (2.6) of [9]), we have
Thus, (103) holds for all .
In the same way one can verify that
To complete the proof we have to verify that
where , as given by this theorem. Write . Theorem 2 shows that
Clearly, we have
where . We need only to prove that
Finally, let be even and . Notice that, for ,
It follows from Lemmas 10 and 9 that
If , then (94) implies (111). If , then
Thus, (111) is valid for all . Consequently, now let be the best approximation of and . From (111) and Lemmas 8 and 9, we have
When we use Lemma 9 for , the above is true for instead of . Thus,
Therefore,
which gives (109).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.