- Research Article
- Open Access
A New Proof of Inequality for Continuous Linear Functionals
© F. Cui and S. Yang. 2011
- Received: 23 January 2011
- Accepted: 2 March 2011
- Published: 14 March 2011
Gavrea and Ivan (2010) obtained an inequality for a continuous linear functional which annihilates all polynomials of degree at most for some positive integer . 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.
- Hilbert Space
- Quadrature Rule
- Gaussian Measure
- Continuous Linear
- Bernoulli Number
and denote by the truncated power function. The notation means that the functional is applied to considered as a function of . The main result of  can now be stated as follows.
where is an arbitrary constant and the symbol denotes a th antiderivative of .
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.
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.
Then, is a Hilbert space of functions. The result in  can be now stated as follows.
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 to both sides of (2.10) and noting (2.7) and (2.8), we obtain (2.4) as required.
It seems that the original proof of Wang and Han recorded in the previous section does not fully utilize the space . We now provide an alternative functional proof.
From the above proof, we see that given by (3.8) is the representer of the Hilbert space .
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.
where is the Bernoulli function, defined by . Here stands for the fractional part of .
Note that there is a mistake in Example 9 in . The constant in the last inequality is mistaken to be there.
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, , for some appropriate integer (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 whenever we have inequality (2.4). It should be mentioned, however, (4.9) does not hold for in general especially when the kernel of is exactly .
Proposition 4.4 can be reformulated in a slightly different language as follows.
Suppose that is a continuous linear functional acting on and . Then for any nonnegative , both (2.4) and (4.9) hold while only (2.4) is also valid for .
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 replaced by and , 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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