A New Proof of Inequality for Continuous Linear Functionals

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Abstract

Gavrea and Ivan (2010) obtained an inequality for a continuous linear functional which annihilates all polynomials of degree at most Open image in new window for some positive integer Open image in new window . In this paper, a new functional proof by Riesz representation theorem is provided. Related results and further applications of the inequality are also brought together.

Keywords

Hilbert Space Quadrature Rule Gaussian Measure Continuous Linear Bernoulli Number 

1. Introduction

Let Open image in new window   be an integer and Open image in new window . Denote by Open image in new window the set of all polynomials of degree not exceeding Open image in new window . Let Open image in new window be a continuous linear functional which annihilates all polynomials of degree at most Open image in new window ; that is,
It is well known that a continuous linear functional is bounded, and finding the bound or norm of a continuous linear functional is a fundamental task in functional analysis. Recently, in light of the Taylor formula and the Cauchy-Schwarz inequality, Gavrea and Ivan in [1] obtained an inequality for the continuous linear functional Open image in new window satisfying (1.1). In order to state their result, we need some more symbols. Recall that the Open image in new window norm of a square integrable function Open image in new window on Open image in new window is defined by

and denote by Open image in new window the truncated power function. The notation Open image in new window means that the functional Open image in new window is applied to Open image in new window considered as a function of Open image in new window . The main result of [1] can now be stated as follows.

Theorem 1.1.

The functional Open image in new window satisfies the following inequality:
are the best possible constants. The equality is attained if and only if Open image in new window is of the form

where Open image in new window is an arbitrary constant and the symbol Open image in new window denotes a Open image in new window th antiderivative of Open image in new window .

Remark 1.2.

Usually, the functional Open image in new window is allowed an interchange with the integral (this is silently assumed throughout this paper). This is true in most interesting cases when, for example, Open image in new window is an integral or a derivative or a linear combination of them. If the interchange is permitted, then it is easily verified

It should be pointed out that the inequality (1.3) can be found in Wang and Han [2, Lemma 1] (see also [3]). In this note, we will give a short account of historical background on inequality (1.3). A new functional proof based on the Riesz representation theorem [4, 5] is also given. Furthermore, some related results are brought together, and further applications are also included.

2. Historical Background

It is well known that a Hilbert space can be given a Gaussian measure. Let Open image in new window be a Hilbert space equipped with Gaussian measure and Open image in new window a continuous linear functional acting on Open image in new window . Smale in [6] (a pioneering work on continuous complexity theory) defined an average (with respect to the Gaussian measure) error for quadrature rules. A result of Smale [6] says that the average error is proportional to Open image in new window . More precisely,

Using (2.1), Smale was able to compute the average error for right rectangle rule, the trapezoidal rule, and Simpson's rule (see [6, Theorem D]).

Later on, Wang and Han in [2] extended and unified results in [6, Theorem D], and they also simplified the corresponding analysis given in [6]. The main observations in [2] are

(i)any quadrature rule has its algebraic precision, or equivalently, the corresponding quadrature error functional annihilates some polynomials,

(ii)and hence the Peano kernel theorem applies.

In fact, more can be stated. The quadrature rule in the above observations can be replaced by any continuous linear one. The main result and its elegant proof deserve to be better known. For reader's convenience, they are recorded here. To do this, we need more notations. For brevity, let Open image in new window . Denote by Open image in new window the square integrable functions on Open image in new window , and
The inner product on Open image in new window is defined by

Then, Open image in new window is a Hilbert space of functions. The result in [2] can be now stated as follows.

Theorem 2.1.

Let Open image in new window be a continuous linear functional satisfying Open image in new window , for all Open image in new window . Then,

It is easily seen that Theorem 1.1 is a rediscovery of Theorem 2.1. For completeness, we record the original short but beautiful proof of (2.4) in [2].

Proof.

We have by the Peano kernel theorem
And hence,
Obviously,
Solving the above equality, we have

Applying the functional Open image in new window to both sides of (2.10) and noting (2.7) and (2.8), we obtain (2.4) as required.

3. A Functional Proof

It seems that the original proof of Wang and Han recorded in the previous section does not fully utilize the space Open image in new window . We now provide an alternative functional proof.

First, we define an equivalence relation ~ on Open image in new window with respect to its subspace Open image in new window since Open image in new window vanishes on Open image in new window . We say that Open image in new window if Open image in new window . It is easy to check that the quotient space Open image in new window is still a Hilbert space. For any Open image in new window , there must exist a function Open image in new window such that Open image in new window , Open image in new window . For example,
may serve this purpose. So, we may assume that Open image in new window , Open image in new window , for any Open image in new window and, the inner product on Open image in new window is
The functional Open image in new window can be viewed as acting on Open image in new window , since it vanishes on Open image in new window . The Peano kernel theorem can be rewritten as
where Open image in new window is defined by (2.6). By the Riesz representation theorem (see, e.g., [4] or [5]), there exists a unique Open image in new window such that
From (3.2) and (3.4)
From (3.3) and (3.6), we have
which gives
Applying the linear functional Open image in new window to both sides of the above equality gives
which together with (3.5) yields

as desired.

Remark 3.1.

From the above proof, we see that Open image in new window given by (3.8) is the representer of the Hilbert space Open image in new window .

4. Related Results and Further Applications

Numerical integration and quadrature rules are classical topics in numerical analysis while quadrature error functionals are typical continuous linear functionals on function spaces. It was quadrature error functionals that stimulated study of Smale [6] and Wang and Han [2]. So it is natural to consider the applications of (2.4) to quadrature error estimates.

Example 4.1.

Let Open image in new window be a positive integer, Open image in new window and Open image in new window . Let the Euler-Maclaurin remainder functional Open image in new window be defined by
where Open image in new window is the Open image in new window th Bernoulli polynomial. It is not hard to verify that Open image in new window vanishes on Open image in new window . So the norm of Open image in new window can be calculated according to (2.4). It can be found in [2] (cf. [7]) which gave a bound in terms of Bernoulli number Open image in new window ; that is,

Example 4.2.

Let Open image in new window , Open image in new window be positive integers and Open image in new window . Suppose that the following quadrature rule
is exact for any polynomial of degree Open image in new window for some positive integer Open image in new window . Then,
defines a composite quadrature error functional which annihilates any Open image in new window . So Theorem 3 applies. The expression for the norm of Open image in new window can be found in [2]. A different but easy-to-use expression can also be found in [7]

where Open image in new window is the Bernoulli function, defined by Open image in new window . Here Open image in new window stands for the fractional part of Open image in new window .

Example 4.3.

The error functionals Open image in new window , Open image in new window , and Open image in new window for the midpoint rule, trapezoidal quadrature and Simpson's rule are, respectively,
They vanishes on Open image in new window , Open image in new window , and Open image in new window , respectively. So, (4.5) applies (see [7] for details). It is a routine computation to find their norms and they can be found in [2] (some of them can also be found in [6, 7, 8]). In the following, Open image in new window stands for the dual space of Open image in new window
From these and (1.4), or equivalently (2.4), we immediately obtain

Note that there is a mistake in Example 9 in [1]. The constant Open image in new window in the last inequality is mistaken to be Open image in new window there.

Recently, there is a flurry of interest in the so-called Ostrowski-Grüss-type inequalities. Some authors, for example, see Ujević [9], consider to bound a quadrature error functional in terms of the Chebyshev functional, that is, Open image in new window , for some appropriate integer Open image in new window (see, e.g., [9]). It is worth mentioning that these Ostrowski-Grüss-type inequalities are related to inequality (1.3). Actually, we have the following general result.

Proposition 4.4.

Suppose that a continuous linear functional Open image in new window vanishes on Open image in new window . Then for any nonnegative Open image in new window , we have

Proof.

Moreover, by noting Open image in new window , we have
It is trivial to check that

From (4.12)–(4.14) and (2.4), follows (4.9). This completes the proof.

Note that Proposition 4.4 shows that we have a corresponding inequality (4.9) for every Open image in new window whenever we have inequality (2.4). It should be mentioned, however, (4.9) does not hold for Open image in new window in general especially when the kernel of Open image in new window is exactly Open image in new window .

Proposition 4.4 can be reformulated in a slightly different language as follows.

Corollary 4.5.

Suppose that Open image in new window is a continuous linear functional acting on Open image in new window and Open image in new window . Then for any nonnegative Open image in new window , both (2.4) and (4.9) hold while only (2.4) is also valid for Open image in new window .

Finally, we end this paper with an inequality of the above-mentioned Grüss-type. More examples are left to the interested readers.

Example 4.6 (see also [7]).

From Example 4.3 and Proposition 4.4, we have

In view of Proposition 4.4 or Corollary 4.5, the above inequality is still valid with Open image in new window replaced by Open image in new window and Open image in new window , respectively, and with obvious change in the coefficients. We omit the details.

Notes

Acknowledgment

The work is supported by Zhejiang Provincial Natural Science Foundation of China (Y6100126).

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Copyright information

© F. Cui and S. Yang. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.College of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina

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