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 be a positive integer, and . Let the Euler-Maclaurin remainder functional be defined by

where is the th Bernoulli polynomial. It is not hard to verify that vanishes on . So the norm of can be calculated according to (2.4). It can be found in [2] (cf. [7]) which gave a bound in terms of Bernoulli number ; that is,

Example 4.2.

Let , be positive integers and . Suppose that the following quadrature rule

is exact for any polynomial of degree for some positive integer . Then,

defines a composite quadrature error functional which annihilates any . So Theorem 3 applies. The expression for the norm of can be found in [2]. A different but easy-to-use expression can also be found in [7]

where is the Bernoulli function, defined by . Here stands for the fractional part of .

Example 4.3.

The error functionals , , and for the midpoint rule, trapezoidal quadrature and Simpson's rule are, respectively,

They vanishes on , , and , 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 [6–8]). In the following, stands for the dual space of

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 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 [9], consider to bound a quadrature error functional in terms of the Chebyshev functional, that is, , for some appropriate integer (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 vanishes on . Then for any nonnegative , we have

Proof.

Let be a polynomial in such that . Let

Then, and

since and vanishes on . Obviously,

Moreover, by noting , 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 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.

Corollary 4.5.

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 [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 replaced by and , respectively, and with obvious change in the coefficients. We omit the details.