A New Proof of Inequality for Continuous Linear Functionals
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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.
KeywordsHilbert Space Quadrature Rule Gaussian Measure Continuous Linear Bernoulli Number
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  can now be stated as follows.
It should be pointed out that the inequality (1.3) can be found in Wang and Han [2, Lemma 1] (see also ). 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
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]).
(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.
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 .
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.
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  and Wang and Han . So it is natural to consider the applications of (2.4) to quadrature error estimates.
Recently, there is a flurry of interest in the so-called Ostrowski-Grüss-type inequalities. Some authors, for example, see Ujević , 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., ). It is worth mentioning that these Ostrowski-Grüss-type inequalities are related to inequality (1.3). Actually, we have the following general result.
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.
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 ).
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.
The work is supported by Zhejiang Provincial Natural Science Foundation of China (Y6100126).
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