Modified Trapezoidal Product Cubature Rules: Definiteness, Monotonicity, and a Posteriori Error Estimates

Abstract

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain ${[a,b]}^2=[a,b]\times [a,b]$. Our modified cubature formulae use mixed type data: except evaluations of the integrand on the points forming a uniform grid on ${[a,b]}^2$, they involve two or four univariate integrals. A useful property of these cubature formulae is that they are definite of order $(2,2)$, that is, they provide one-sided approximation to the double integral for real-valued integrands from the class ${\mathcal{C}}^{2,2}[a,b]=\{f(x,y):{\displaystyle \frac{{\partial }^{4}f}{\partial x^2\partial y^2}}\mathrm{continuous}\mathrm{and}\mathrm{does}\mathrm{not}\mathrm{change}\mathrm{sign}\mathrm{in}{(a,b)}^2\}$. For integrands from ${\mathcal{C}}^{2,2}[a,b]$ we prove monotonicity of the remainders and derive a posteriori error estimates.

1. Introduction and Statement of Main Result

A standard tool for the approximation of the definite integral ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$ is the (n-th) composite trapezium rule

$$Q_n^{Tr}\left[f\right]=h_n\underset{i=0}{\overset{n}{{\sum }^{\prime }}}f\left(x_{i,n}\right),x_{i,n}=a+i{h}_n,h_n={\displaystyle \frac{b-a}{n}}$$

(here and in what follows, the notation ${\sum }^{\prime }$ means that the first and last summands are halved). Denote by $R[Q_n^{Tr};f]$ the remainder functional,

$$R[Q_n^{Tr};f]:=\underset{a}{\overset{b}{\int }}f\left(x\right)dx-Q_n^{Tr}\left[f\right].$$

Assuming the integrand f is convex or concave in $[a,b]$, we have the following well-known properties of the trapezium rule:

(i)

Definiteness: $Q_n^{Tr}\left[f\right]$ provides one-sided approximation to $I\left[f\right]$, namely,

$$R[Q_n^{Tr};f]\le 0\mathrm{for}\mathrm{convex}f,R[Q_n^{Tr};f]\ge 0\mathrm{for}\mathrm{concave}f;$$

(ii)

Monotonicity:

$$\left|R[Q_{2n}^{Tr};f]\right|\le {\displaystyle \frac{1}{2}}\left|R[Q_n^{Tr};f]\right|;$$

(iii)

A posteriori error estimate:

$$\left|R[Q_{2n}^{Tr};f]\right|\le |Q_n^{Tr}\left[f\right]-Q_{2n}^{Tr}\left[f\right]|.$$

Definiteness and monotonicity are properties of quadrature rules which have been intensively studied (see e.g., [1,2,3,4,5] and the monographs [6,7]). These properties have found application in the software packages for numerical integration.

The product trapezium cubature rule

$$C_n^{Tr}\left[f\right]=h_n^2\underset{i=0}{\overset{n}{{\sum }^{\prime }}}\underset{j=0}{\overset{n}{{\sum }^{\prime }}}f(x_{i,n},y_{j,n}),x_{i,n}=y_{i,n}=a+i{h}_n,h_n={\displaystyle \frac{b-a}{n}},$$

is a natural candidate for approximating the double integral

$$I\left[f\right]:=\underset{{[a,b]}^2}{\int \int }f(x,y)dxdy,$$

where ${[a,b]}^2$ is the square $[a,b]\times [a,b]\subset {\mathbb{R}}^2$. While in the univariate case convexity/concavity of f is secured by the assumption $f\in C^2[a,b]$ with $f^{″}\ge 0$ (resp. $f^{″}\le 0$) in $(a,b)$, convexity/concavity of a bivariate function f is guaranteed by the semi-definiteness of its Hessian matrix ${\nabla }^{2}f$. However, our approach makes use a kind of iteration with respect to the two variables, therefore a more appropriate class of bivariate functions for us is

$${\mathcal{C}}^{2,2}[a,b]=\{f\in C^{2,2}\left({[a,b]}^2\right):D^{2,2}f\mathrm{does}\mathrm{not}\mathrm{change}\mathrm{sign}\mathrm{in}{(a,b)}^2\}.$$

Here and henceforth, for $k,l\in \mathbb{N}$,

$$C^{k,l}\left({[a,b]}^2\right):=\{f:{[a,b]}^2\mapsto \mathbb{R},D^{i,j}f\mathrm{is}\mathrm{cont}.\mathrm{in}{[a,b]}^2,0\le i\le k,0\le j\le l\},$$

where

$$D^{i,j}f:={\displaystyle \frac{{\partial }^{i+j}f}{\partial x^i\partial y^j}}.$$

In general, the remainder functional

$$R\left[C_n^{Tr};f\right]=I\left[f\right]-C_n^{Tr}\left[f\right]$$

does not possess properties like (i), (ii) and (iii) for integrands $f\in {\mathcal{C}}^{2,2}[a,b]$. By suitably modifying $C_n^{Tr}$, we obtain two families of cubature formulae which admit properties analogous to (i), (ii) and (iii). We pay a price for this: our modified cubature formulae involve evaluations of two or four univariate integrals of some traces of the integrand. With every bivariate function $f(x,y)$ defined on ${[a,b]}^2$ we associate six univariate functions:

$$\begin{array}{ccc}& & f_l\left(t\right)=f(a,t),f_r\left(t\right)=f(b,t),f_d\left(t\right)=f(t,a),f_u\left(t\right)=f(t,b),\hfill \\ & & f_v(t)=f({\displaystyle \frac{a+b}{2}},t),f_h(t)=f(t,{\displaystyle \frac{a+b}{2}}).\hfill \end{array}$$

Definition 1.

For every $n\in \mathbb{N}$, the modified product trapezoidal rules $S_n^{-}$ and $S_n^{+}$ are defined as follows:

$$S_n^{-}\left[f\right]=C_n^{Tr}\left[f\right]+(b-a)\left(R[Q_n^{Tr};f_v]+R[Q_n^{Tr};f_h]\right),$$

(1)

$$S_n^{+}\left[f\right]=C_n^{Tr}\left[f\right]+{\displaystyle \frac{b-a}{2}}\left(R[Q_n^{Tr};f_l]+R[Q_n^{Tr};f_r]+R[Q_n^{Tr};f_d]+R[Q_n^{Tr};f_u]\right).$$

(2)

The cubature formulae $S_n^{+}$ and $S_n^{-}$ have been introduced (with $[a,b]=[-1,1]$ and somewhat different notation) in [8], where some properties of these cubature formulae have been established. Set

$$R\left[S_n^{-};f\right]=I\left[f\right]-S_n^{-}\left[f\right],R\left[S_n^{+};f\right]=I\left[f\right]-S_n^{+}\left[f\right].$$

The following result has been proved in [8] (see Theorems 4.4 and 5.2 therein):

Theorem A.

(i)

For every $f\in C^{2,2}\left({[a,b]}^2\right)$, there exist $P\in {[a,b]}^2$ such that

$$R[S_n^{-};f]=c\left(S_n^{-}\right)D^{2,2}f\left(P\right),c\left(S_n^{-}\right)=-{\displaystyle \frac{{(b-a)}^6}{144n^2}}\left(1+{\displaystyle \frac{1}{n^2}}\right).$$

(3)

(ii)

For every $f\in C^{2,2}\left({[a,b]}^2\right)$, there exist $P\in {[a,b]}^2$ such that

$$R[S_n^{+};f]=c\left(S_n^{+}\right)D^{2,2}f\left(P\right),c\left(S_n^{+}\right)={\displaystyle \frac{{(b-a)}^6}{72n^2}}\left(1-{\displaystyle \frac{1}{2n^2}}\right).$$

(4)

Our main result here is monotonicity and a posteriori error estimates for the remainders of $S_n^{-}$ and $S_n^{+}$.

Theorem 1.

If $f\in {\mathcal{C}}^{2,2}[a,b]$, then

$$|R[S_{2n}^{-};f]|\le {\displaystyle \frac{1}{2}}|R[S_n^{-};f]|$$

(5)

and

$$|R[S_{2n}^{-};f]|\le |S_{2n}^{-}[f]-S_n^{-}[f]|.$$

(6)

Theorem 2.

If $f\in {\mathcal{C}}^{2,2}[a,b]$, then

$$|R[S_{2n}^{+};f]|\le \left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4(2n-1)}}\right)|R[S_n^{+};f]|$$

(7)

and

$$|R[S_{2n}^{+};f]|\le {\displaystyle \frac{4n-1}{4n-3}}|S_{2n}^{+}[f]-S_n^{+}[f]|.$$

(8)

Remark 1.

We note that Theorems 1 and 2 remain true in the wider class of bivariate functions

$${\tilde{\mathcal{C}}}^{2,2}[a,b]:=\left\{f:D^{2,2}f\mathit{has}\mathrm{a}\mathit{permanent}\mathit{sign}\mathrm{a}.\mathrm{e}.\mathit{in}{[a,b]}^2,I\left[|D^{2,2}f|\right]<\infty \right\}.$$

The requirement for continuity of $D^{2,2}f$ in ${[a,b]}^2$ is only essential for the error representations in Theorem A, and the assumption $f\in {\mathcal{C}}^{2,2}[a,b]$ in Theorems 1 and 2 was made for the sake of simplicity.

Let us briefly comment on the results stated in the above theorems. Set

$$\begin{array}{ccc}& & {\mathcal{C}}_{+}^{2,2}[a,b]=\{f\in {\mathcal{C}}^{2,2}[a,b]:D^{2,2}f\ge 0\mathrm{in}{[a,b]}^2\},\hfill \\ & & {\mathcal{C}}_{-}^{2,2}[a,b]=\{f\in {\mathcal{C}}^{2,2}:-f\in {\mathcal{C}}_{+}^{2,2}[a,b]\}.\hfill \end{array}$$

Theorem A implies that for every $n,m\in \mathbb{N}$ we have

$$S_n^{+}[f]\le I[f]\le S_m^{-}[f],f\in {\mathcal{C}}_{+}^{2,2}[a,b]$$

and the reversed inequalities hold when $f\in {\mathcal{C}}_{-}^{2,2}[a,b]$. Thus, for integrands from ${\mathcal{C}}^{2,2}[a,b]$ we obtain inclusions for the true value of $I[f]$. By analogy with the univariate case, we say that $S_n^{-}$ and $S_n^{+}$ are, respectively, negative definite and positive definite cubature formulae of order $(2,2)$.

Inequalities (5) and (7) in Theorems 1 and 2 imply that for integrands from ${\mathcal{C}}^{2,2}[a,b]$ doubling n results in reducing the error magnitude by a factor of at least two (or almost two). Inequalities (6) and (8) provide a posteriori error estimates, which may serve as stopping rules (rules for termination of calculations) in routines for automated numerical integration. Let us point out here that, although $S_n^{\pm }$ involve univariate integrals, the error bounds in (6) and (8) are in terms of only point evaluations of the integrand. In a certain sense, the inequalities in Theorems 1 and 2 are the best possible, as it will become clear from their proof.

The rest of the paper is structured as follows. In Section 2, we provide some necessary background, including Peano kernel representation of linear functionals, in particular, of the remainders of quadrature formulae, and the approach for construction of cubature formulae using mixed type data through interpolation by blending functions. At the end of this section we prove Theorem 4, which provides a method for derivation of a posteriori error estimates for definite cubature formulae. In Section 3, we apply Theorem 4 to prove Theorems 1 and 2. Section 4 contains numerical examples illustrating our theoretical results and some concluding remarks.

2. Preliminaries

2.1. Peano Kernel Representation of Linear Functionals

The notation ${\pi }_m$ will stand for the set of algebraic polynomials of degree not exceeding m. A classical result of Peano [9] asserts that if $\mathcal{L}$ is a linear functional defined in $C[a,b]$ which vanishes on ${\pi }_{r-1}$, then for functions g with $g^{(r-1)}$ absolutely continuous in $[a,b]$, $\mathcal{L}[g]$ admits the integral representation

$$\mathcal{L}[g]=\underset{a}{\overset{b}{\int }}{K}_r\left(t\right)g^{\left(r\right)}\left(t\right)dt,$$

where

$$K_r\left(t\right)=\mathcal{L}\left[{\displaystyle \frac{{(·-t)}_{+}^{r-1}}{(r-1)!}}\right],u_{+}=max\{u,0\}.$$

By $W_p^r[a,b]$, $(r\in \mathbb{N},p\ge 1)$, we denote the Sobolev class of functions

$$W_p^r[a,b]:=\{f\in C^{r-1}[a,b]:f^{(r-1)} \mathrm{abs}.\mathrm{continuous},{\int }_a^b|f^{\left(r\right)}\left(t\right){|}^{p}dt<\infty \}.$$

Note that $C^r[a,b]\subset W_p^r[a,b]$ for every $p\ge 1$.

A quadrature formula $Q_n$ approximating the integral ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$,

$$Q_n[f]=\sum _{i=1}^n{a}_{i,n}f\left({\tau }_{i,n}\right),a\le {\tau }_{1,n}<{\tau }_{2,n}<\cdots <{\tau }_{n,n}\le b,$$

is said to have algebraic degree of precision $r-1$ (in short, $ADP\left(Q_n\right)=r-1$), if its remainder

$$R[Q_n;f]=\underset{a}{\overset{b}{\int }}f\left(x\right)dx-Q_n[f]$$

vanishes identically on ${\pi }_{r-1}$ and $R[Q_n;x^r]\ne 0$. By Peano’s representation theorem, if $ADP\left(Q_n\right)=r-1$ and the integrand f belongs to $W_1^s[a,b]$, where $s\le r$, $s\in \mathbb{N}$, then the remainder of $Q_n$ admits the integral representation

$$R[Q_n;f]={\int }_a^b{K}_s(Q_n;t)f^{\left(s\right)}\left(t\right)dt$$

(9)

with

$$K_s(Q_n;t)=R\left[Q_n;{\displaystyle \frac{{(·-t)}_{+}^{s-1}}{(s-1)!}}\right].$$

$K_s[Q_n;·]$ will be referred to as the s-th Peano kernel of $Q_n$. Explicit representations of $K_s(Q_n;t)$ for $t\in [a,b]$ are

$$\begin{array}{cc}\hfill K_s(Q_n;t)& ={(-1)}^s\left[{\displaystyle \frac{{(t-a)}^s}{s!}}-{\displaystyle \frac{1}{(s-1)!}}\sum _{i=1}^n{a}_{i,n}{(t-{\tau }_{i,n})}_{+}^{s-1}\right]\hfill \\ \hfill & ={\displaystyle \frac{{(b-t)}^s}{s!}}-{\displaystyle \frac{1}{(s-1)!}}\sum _{i=1}^n{a}_{i,n}{({\tau }_{i,n}-t)}_{+}^{s-1}.\hfill \end{array}$$

(10)

Application of Hölder inequality to (9) yields sharp error estimates of the form

$$|R[Q_n;g]|\le c_{s,p}\left(Q_n\right){\parallel g^{\left(s\right)}\parallel }_p,s=1,\dots ,r,$$

in the corresponding Sobolev classes of functions, where ${\parallel ·\parallel }_p$ is the $L_p\left([a,b]\right)$-norm, $1\le p\le \infty$.

Of particular interest are the so-called definite quadrature formulae. Quadrature formula $Q_n$ is said to be definite of order r, if there exists a real non-zero constant $c_r\left(Q_n\right)$ such that for the remainder of $Q_n$ there holds

$$R[Q_n;f]=c_r\left(Q_n\right)f^{\left(r\right)}\left(\xi \right)$$

for every $f\in C^r[a,b]$, with some $\xi \in [a,b]$ depending on f. Furthermore, $Q_n$ is called positive definite (resp., negative definite) of order r, if $c_r\left(Q_n\right)>0$ ($c_r\left(Q_n\right)<0$). It is clear from (9) that $Q_n$ is a positive (negative) definite quadrature formula of order r if and only if $ADP\left(Q_n\right)=r-1$ and $K_r(Q_n;t)\ge 0$ (resp., $K_r(Q_n;t)\le 0$) on $(a,b)$. The interest in definite quadratures of order r lies in the one-sided approximation they provide to the integral for integrands having r-th derivative with a permanent sign in $[a,b]$.

We refer to [6,7] (Chapt. 4) for more details about application of Peano kernel theory for estimation of the error of quadrature formulae.

2.2. Cubature Formulae Using Mixed-Type Data

In [8] V. Gushev and one of the authors initiated study on adopting the Peano kernel theory to the error estimation of cubature formulae for approximate evaluation of double integrals on a rectangular region, a further development can be found in [10]. The approach applied there makes use of bivariate interpolation by blending functions (see, e.g., [11]). We present below the main idea and some of the results from [8,10].

For $m_1,m_2\in \mathbb{N}$, the set of blending functions $B^{m_1,m_2}\left({[a,b]}^2\right)$ is defined by

$$B^{m_1,m_2}\left({[a,b]}^2\right)=\left\{f\in C^{m_1,m_2}\left({[a,b]}^2\right):D^{m_1,m_2}f=0\right\}.$$

(11)

Given $X=\{x_1,x_2,\dots ,x_{m_1}\}$, $Y=\{y_1,y_2,\dots ,y_{m_2}\}$ with $a\le x_1<\cdots <x_{m_1}\le b$ and $a\le y_1<\cdots <y_{m_2}\le b$, we define a blending grid $G=G(X,Y)$ by

$$G(X,Y)=\left\{(x,y):\prod _{\mu =1}^{m_1}(x-x_{\mu })\prod _{\nu =1}^{m_2}(y-y_{\nu })=0\right\}.$$

For any function f defined on ${[a,b]}^2$ there exists a unique (Lagrange) blending interpolant $Bf=B_{G}f\in B^{m_1,m_2}\left({[a,b]}^2\right)$ satisfying $B{f}_{|G(X,Y)}=f_{|G(X,Y)}$, it is explicitly given by

$$Bf={\mathcal{L}}_{x}f+{\mathcal{L}}_{y}f-{\mathcal{L}}_x{\mathcal{L}}_{y}f,$$

where ${\mathcal{L}}_x$ and ${\mathcal{L}}_y$ are the Lagrange interpolation operators with respect to x and y with sets of interpolation nodes X and Y, respectively. Precisely, if ${\left\{{\ell }_{\mu }\right\}}_{\mu =1}^{m_1}$ and ${\left\{{\tilde{\ell }}_{\nu }\right\}}_{\nu =1}^{m_2}$ are the Lagrange basic polynomials for ${\pi }_{m_1-1}$ and ${\pi }_{m_2-1}$, defined by ${\ell }_{\mu }\left(x_j\right)={\delta }_{\mu ,j}(j=1,\dots ,m_1)$ and ${\tilde{\ell }}_{\nu }\left(y_j\right)={\delta }_{\nu ,j}(j=1,\dots ,m_2)$, respectively, with ${\delta }_{i,j}$ being the Kronecker symbol, then

$$Bf(x,y)=\sum _{\mu =1}^{m_1}{\ell }_{\mu }\left(x\right)f(x_{\mu },y)+\sum _{\nu =1}^{m_2}{\tilde{\ell }}_{\nu }\left(y\right)f(x,y_{\nu })-\sum _{\mu =1}^{m_1}\sum _{\nu =1}^{m_2}{\ell }_{\mu }\left(x\right){\tilde{\ell }}_{\nu }\left(y\right)f(x_{\mu },y_{\nu }).$$

Remark 2.

Blending grid $G=G(X,Y)$ can be defined also by sets X and Y containing multiple nodes, then ${\mathcal{L}}_x$ and ${\mathcal{L}}_y$ are replaced by the associated Hermite interpolation operators.

For $f\in C^{r,s}\left({[a,b]}^2\right)$ with $r,s\in \mathbb{N}$ satisfying $r\le m_1$ and $s\le m_2$, two iterated applications of the Peano theorem to $f-Bf=(Id-{\mathcal{L}}_x)(Id-{\mathcal{L}}_y)f$, where $Id$ is the identity operator, imply

$$f(x,y)-Bf(x,y)=\underset{{[a,b]}^2}{\int \int }{\mathcal{K}}_r(x,t){\tilde{\mathcal{K}}}_s(y,\tau )D^{r,s}f(t,\tau )dtd\tau ,$$

(12)

where

$${\mathcal{K}}_r(x,t)={\displaystyle \frac{1}{(r-1)!}}({(x-t)}_{+}^{r-1}-\sum _{\mu =1}^{m_1}{\ell }_{\mu }\left(x\right){(x_{\mu }-t)}_{+}^{r-1}),$$

$${\tilde{\mathcal{K}}}_s(y,\tau )={\displaystyle \frac{1}{(s-1)!}}({(y-\tau )}_{+}^{s-1}-\sum _{\nu =1}^{m_2}{\tilde{\ell }}_{\nu }\left(y\right){(y_{\nu }-\tau )}_{+}^{s-1}).$$

Integrating (12) over ${[a,b]}^2$ we obtain the approximation

$$I[f]\approx I[Bf]=:C[f].$$

$C[f]$ is called blending cubature formula and is of the form

$$C[f]=\sum _{\mu =1}^{m_1}{b}_{\mu }\underset{a}{\overset{b}{\int }}f(x_{\mu },t)dt+\sum _{\nu =1}^{m_2}{\tilde{b}}_{\nu }\underset{a}{\overset{b}{\int }}f(t,y_{\nu })dt-\sum _{\mu =1}^{m_1}\sum _{\nu =1}^{m_2}{b}_{\mu }{\tilde{b}}_{\nu }f(x_{\mu },y_{\nu }),$$

(13)

where $Q^{\prime }[g]={\sum }_{\mu =1}^{m_1}{b}_{\mu }g\left(x_{\mu }\right)$ and $Q^{″}[g]={\sum }_{\nu =1}^{m_2}{\tilde{b}}_{\nu }g\left(x_{\nu }\right)$ are the interpolatory quadrature formulae with sets of nodes X and Y, respectively. The remainder of this blending cubature formula admits the integral representation

$$R[C;f]=I[f]-C[f]=\underset{{[a,b]}^2}{\int \int }{K}_r(Q^{\prime },t)K_s(Q^{″};\tau )D^{r,s}f(t,\tau )dtd\tau .$$

(14)

Now, assuming that the integrand belongs to a certain Sobolev class of bivariate functions, one can derive error estimates using Hölder’s inequality.

The presence of $m_1+m_2$ univariate integrals in the blending cubature formula (13) is a drawback, as the precise values of these integrals may be not accessible. A sequence of blending cubature formulae involving only these $m_1+m_2$ univariate integrals and approximating $I[f]$ with increasing accuracy is constructed through the scheme

$$I[f-Bf]\approx C_n[f-Bf],$$

where $\left\{C_n\right\}$ is a sequence of cubature formulae of standard type, i.e., using only point evaluations. We denote the resulting cubature formulae by $S_n$,

$$S_n[f]=I[Bf]+C_n[f]-C_n[Bf].$$

(15)

A reasonable choice for $C_n$ is to be a product cubature formula generated by a quadrature formula

$$Q_n[g]=\sum _{j=0}^n{a}_{j,n}g\left(x_{j,n}\right)\approx \underset{a}{\overset{b}{\int }}g\left(t\right)dt,$$

(16)

i.e.,

$$C_n[f]=\sum _{i=0}^n\sum _{j=0}^n{a}_{i,n}{a}_{j,n}f(x_{i,n},x_{j,n}).$$

With such a choice of $C_n$, (12) and (13) imply an integral representation of the remainder of $S_n$,

$$R[S_n,f]=I[f]-S_n[f]=\underset{{[a,b]}^2}{\int \int }{K}_{r,s}(S_n;t,\tau )D^{r,s}f(t,\tau )dtd\tau$$

(17)

with

$$K_{r,s}(S_n;t,\tau )=K_r(Q^{\prime };t)K_s(Q^{″};\tau )-Q_n[{\mathcal{K}}_r(·,t)]Q_n[{\tilde{\mathcal{K}}}_s(·,\tau )].$$

Further representations of $S_n$ and its Peano kernel $K_{r,s}(S_n;·)$ are given in the following theorem:

Theorem 3

([8], Theorem 3.1). Assume that in the blending cubature formula $S_n$ defined by (15), $C_n$ is the product cubature formula generated by a quadrature formula $Q_n$ with $ADP\left(Q_n\right)\ge max\{ADP\left(Q^{\prime }\right),ADP\left(Q^{″}\right)\}$, where $Q^{\prime }$ and $Q^{″}$ are the interpolatory quadrature formulae with sets of nodes X and Y, respectively. Then for $(t,\tau )\in {[a,b]}^2$ the Peano kernel $K_{r,s}(S_n;t,\tau )$ has the following representations:

$$K_{r,s}(S_n;t,\tau )=K_r(Q^{\prime };t)K_s(Q_n;\tau )+K_r(Q_n;t)Q_n[{\tilde{\mathcal{K}}}_s(·,\tau )],$$

(18)

$$K_{r,s}(S_n;t,\tau )=K_s(Q^{″};\tau )K_r(Q_n;t)+K_s(Q_n;\tau )Q_n[{\mathcal{K}}_r(·,t)],$$

(19)

$$\begin{array}{cc}\hfill K_{r,s}(S_n;t,\tau )=& K_r(Q^{\prime };t)K_s(Q_n;\tau )+K_s(Q^{″};\tau ))K_r(Q_n;t)\hfill \\ \hfill & -K_r(Q_n;t))K_s(Q_n;\tau ).\hfill \end{array}$$

(20)

The cubature formula $S_n$ admits the representation

$$S_n[f]=C_n[f]+Q^{\prime }\left[R[Q_n;f({(·)}_{Q^{\prime }},{(·)}_R)]\right]+Q^{″}\left[R[Q_n;f({(·)}_R,{(·)}_{Q^{″}})]\right].$$

(21)

See also [10] (Theorems 1 and 3).

In view of the integral representation (17), we may have definite blending cubature formulae.

Definition 2.

The blending cubature formula $S_n$ is called positive (negative) definite of order $(r,s)$ if $K_{r,s}(t,\tau )\ge 0$ (resp. $K_{r,s}(t,\tau )\le 0$) in ${[a,b]}^2$.

Definite cubature formulae of order $(r,s)$ provide one-sided approximation to $I[f]$ for integrands from the class

$${\mathcal{C}}^{r,s}[a,b]=\{f\in C^{r,s}\left({[a,b]}^2\right):D^{r,s}f\mathrm{does}\mathrm{not}\mathrm{change}\mathrm{sign}\mathrm{in}{(a,b)}^2\}.$$

Set

$$\begin{array}{ccc}& & {\mathcal{C}}_{+}^{r,s}[a,b]=\{f\in {\mathcal{C}}^{r,s}[a,b]:D^{r,s}f\ge 0\mathrm{in}{[a,b]}^2\},\hfill \\ & & {\mathcal{C}}_{-}^{r,s}[a,b]=\{f\in {\mathcal{C}}^{r,s}[a,b]:-f\in {\mathcal{C}}_{+}^{r,s}[a,b]\}.\hfill \end{array}$$

If $S_n$ is, say, positive definite of order $(r,s)$, then $R[S_n;f]\ge 0$ for $f\in {\mathcal{C}}_{+}^{r,s}[a,b]$ and the reversed inequality holds when $f\in {\mathcal{C}}_{-}^{r,s}[a,b]$. Various sufficient conditions for definiteness of blending cubature formulae are given in [10] (Theorem 5).

We conclude this section with a general observation about definite blending cubature formulae.

Theorem 4.

Let $(S^{\prime },S^{″})$ be a pair of positive (negative) definite blending cubature formulae of order $(r,s)$. Assume that, for some $c>0$, the cubature formula

$$\widehat{S}:=(c+1)S^{\prime }-c{S}^{″}$$

is negative (positive) definite of order $(r,s)$. Then the following inequalities hold true whenever $f$ belongs to ${\mathcal{C}}^{r,s}[a,b]$:

(i)

${\displaystyle |R[S^{\prime };f]|\le \frac{c}{c+1}|R[S^{″};f]|;}$

(ii)

${\displaystyle |R[S^{\prime };f]|\le c|S^{\prime }[f]-S^{″}[f]|;}$

(iii)

${\displaystyle |R[S^{″};f]|\le (c+1)|S^{\prime }[f]-S^{″}[f]|.}$

Proof. 

Let us consider, e.g., the case when $S^{\prime }$ and $S^{″}$ are negative definite and $\widehat{S}$ is positive definite, of order $(r,s)$. Without loss of generality we may assume that $f\in {\mathcal{C}}_{+}^{r,s}[a,b]$. Then $R[S^{\prime };f]\le 0$, $R[S^{″};f]\le 0$, and $R[\widehat{S};f]\ge 0$, therefore

$$0\le R[\widehat{S};f]=(c+1)R[S^{\prime };f]-cR[S^{″};f],$$

and hence

$$-R[S^{\prime };f]\le -{\displaystyle \frac{c}{c+1}}R[S^{″};f],$$

which, in this case, is claim (i) of Theorem 4. Claim (iii) follows from

$$\begin{array}{cc}\hfill |S^{\prime }[f]-S^{″}[f]|& =|R[S^{″};f]-R[S^{\prime };f]|\ge |R[S^{″};f]|-|R[S^{\prime };f]|\hfill \\ & \ge |R[S^{″};f]|-{\displaystyle \frac{c}{c+1}}|R[S^{″};f]|={\displaystyle \frac{1}{c+1}}|R[S^{″};f]|.\hfill \end{array}$$

Part (ii) is a consequence of (iii) and (i). The proof of the case when $S^{\prime }$ and $S^{″}$ are positive definite and $\widehat{S}$ is negative definite of order $(r,s)$ is analogous, and we omit it. □

3. Proofs

3.1. Proof of Theorem 1

The cubature formula $S_n^{-}$ is obtained through the Scheme (15), where $Bf$ is the Hermite blending interpolant for f at the blending grid $G(X,Y)$ formed by $X=Y=\left({\displaystyle \frac{a+b}{2}},{\displaystyle \frac{a+b}{2}}\right)$, and $C_n[f]=C_n^{Tr}[f]$. In this case $Q^{\prime }=Q^{″}=Q^{Mi}$ is the midpoint quadrature formula

$$Q^{Mi}[g]=(b-a)g\left({\displaystyle \frac{a+b}{2}}\right),$$

and (21) implies the representation (1) of $S_n^{-}$. According to (20), the Peano kernel $K_{2,2}(S_n^{-};t,\tau )$ is equal to

$$K_2(Q^{Mi};t)K_2(Q_n^{Tr};\tau )+K_2(Q^{Mi};\tau )K_2(Q_n^{Tr};t)-K_2(Q_n^{Tr};t)K_2(Q_n^{Tr};\tau ).$$

(22)

Since $Q_n^{Tr}$ and $Q^{Mi}$ are, respectively, negative definite and positive definite quadrature formulae of order two, (22) implies $K_{2,2}(S_n^{-};t,\tau )\le 0$ in ${[a,b]}^2$, i.e., $S_n^{-}$ is a negative definite cubature formula of order $(2,2)$. By the mean value theorem, there is a point $P\in {[a,b]}^2$ such that

$$R[S_n^{-};f]=D^{2,2}f\left(P\right)\underset{{[a,b]}^2}{\int \int }{K}_{2,2}(S_n^{-};t,\tau )dtd\tau .$$

From (22) and

$$\underset{a}{\overset{b}{\int }}{K}_2(Q^{Mi};t)dt={\displaystyle \frac{{(b-a)}^3}{24}},\underset{a}{\overset{b}{\int }}{K}_2(Q_n^{Tr};t)dt=-{\displaystyle \frac{{(b-a)}^3}{12n^2}},$$

we find

$$\underset{{[a,b]}^2}{\int \int }{K}_{2,2}(S_n^{-};t,\tau )dtd\tau =-{\displaystyle \frac{{(b-a)}^6}{144n^2}}\left(1+{\displaystyle \frac{1}{n^2}}\right),$$

thus proving part (i) of Theorem A.

Theorem 1 follows from Theorem 4, applied with $r=s=2$, $S^{\prime }=S_{2n}^{-}$, $S^{″}=S_n^{-}$, and $c=1$. We shall prove that the cubature formula

$$\widehat{S}=(c+1)S_{2n}^{-}-c{S}_n^{-}$$

is positive definite of order $(2,2)$ for every $c\ge 1$, or, equivalently, that

$$\phi (t,\tau ):=(c+1)K_{2,2}(S_{2n}^{-};t,\tau )-c{K}_{2,2}(S_n^{-};t,\tau )\ge 0,(t,\tau )\in {[a,b]}^2$$

(23)

when $c\ge 1$. Moreover, we shall show that $c=1$ is the smallest constant with this property. We assume in what follows that $c\ge 1$, and set

$$t_i={\tau }_i=a+ih,i=0,1,\dots ,2n,h={\displaystyle \frac{b-a}{2n}}.$$

(24)

We split ${[a,b]}^2$ in the following way:

$${[a,b]}^2=\bigcup _{k=0}^{n-1}\bigcup _{\ell =0}^{n-1}{\Delta }_{k,\ell },{\Delta }_{k,\ell }=\{(t,\tau ):t_{2k}\le t\le t_{2k+2},{\tau }_{2\ell }\le \tau \le {\tau }_{2\ell +2}\},$$

with a further decomposition ${\displaystyle {\Delta }_{k,\ell }=\bigcup _{i=1}^4{\Delta }_{k,\ell }^i}$, where

$$\begin{array}{ccc}& & {\Delta }_{k,\ell }^1=\{(t,\tau ):t_{2k}\le t\le t_{2k+1},{\tau }_{2\ell }\le \tau \le {\tau }_{2\ell +1}\},\hfill \\ & & {\Delta }_{k,\ell }^2=\{(t,\tau ):t_{2k+1}\le t\le t_{2k+2},{\tau }_{2\ell }\le \tau \le {\tau }_{2\ell +1}\},\hfill \\ & & {\Delta }_{k,\ell }^3=\{(t,\tau ):t_{2k}\le t\le t_{2k+1},{\tau }_{2\ell +1}\le \tau \le {\tau }_{2\ell +2}\},\hfill \\ & & {\Delta }_{k,\ell }^4=\{(t,\tau ):t_{2k+1}\le t\le t_{2k+2},{\tau }_{2\ell +1}\le \tau \le {\tau }_{2\ell +2}\}.\hfill \end{array}$$

The Peano kernels $K_2(Q^{Mi};·)$, $K_2(Q_n^{Tr};·)$ and $K_2(Q_{2n}^{Tr};·)$, which according to (22) and (23) occur in $\phi$, admit the representations (see (10)):

$$\begin{array}{cc}\hfill & K_2(Q^{Mi};t)={\displaystyle \frac{{(t-a)}^2}{2}}-(b-a){\left(t-{\displaystyle \frac{a+b}{2}}\right)}_{+},t\in [a,b],\hfill \\ \hfill & K_2(Q_n^{Tr};t)={\displaystyle \frac{1}{2}}(t-t_{2k})(t-t_{2k+2}),t\in \left[t_{2k},t_{2k+2}\right],0\le k\le n-1,\hfill \\ \hfill & K_2(Q_{2n}^{Tr};t)={\displaystyle \frac{1}{2}}(t-t_i)(t-t_{i+1}),t\in \left[t_i,t_{i+1}\right],0\le i\le 2n-1.\hfill \end{array}$$

(25)

Since either of these Peano kernels is an even function with respect ${\displaystyle \frac{a+b}{2}}$, i.e., $K_2(·;{\displaystyle \frac{a+b}{2}}-t)=K_2(·;{\displaystyle \frac{a+b}{2}}+t)$ for $a\le t\le {\displaystyle \frac{a+b}{2}}$, then $\phi (t,\tau )$ possesses the same property in each of variables t and $\tau$. Therefore it suffices to verify (25) only for $a\le t,\tau \le {\displaystyle \frac{a+b}{2}}$.

Assume first that $(t,\tau )\in {\Delta }_{0,0}^1$, then using (22) and (25) we find

$$\begin{array}{cc}\hfill \phi (t,\tau )={\displaystyle \frac{(t-a)(\tau -a)}{4}}[& (c+1)\left[(t-a)(\tau -t_1)+(\tau -a)(t-t_1)-(t-t_1)(\tau -t_1)\right]\hfill \\ \hfill & -c\left[(t-a)(\tau -t_2)+(\tau -a)(t-t_2)-(t-t_2)(\tau -t_2)\right]],\hfill \end{array}$$

and substituting $t_2=t_1+h$, get

$$\phi (t,\tau )={\displaystyle \frac{(t-a)(\tau -a)}{4}}\left[(t-a)(\tau -t_1)+(\tau -a)(t-t_1)-(t-t_1)(\tau -t_1)+3c{h}^2\right].$$

Since the expression in the brackets is a bilinear function, its minimal value in ${\Delta }_{0,0}^1$ is attained at a vertex of ${\Delta }_{0,0}^1$. Hence, this expression is non-negative when $c\ge 1/3$, and consequently $\phi (t,\tau )\ge 0$ for $(t,\tau )\in {\Delta }_{0,0}^1$.

Next, we compare the values of $\phi (t,\tau )$ in ${\Delta }_{k,\ell }^1$ and ${\Delta }_{k+1,\ell }^1$, assuming that $n\ge 2$ and $\ell \le {\displaystyle \frac{n-1}{2}}$, $k\le {\displaystyle \frac{n-2}{2}}$. By using (25), for $(t,\tau )\in {\Delta }_{k,\ell }^1$ we obtain

$$\begin{array}{cc}\hfill 4\phi (t,\tau )=& (c+1)[{(t-a)}^2(\tau -t_{2\ell })(\tau -t_{2\ell +1})+{(\tau -a)}^2(t-t_{2k})(t-t_{2k+1})\hfill \\ \hfill & -(t-t_{2k})(t-t_{2k+1})(\tau -t_{2\ell })(\tau -t_{2\ell +1})]\hfill \\ \hfill & -c[{(t-a)}^2(\tau -t_{2\ell })(\tau -t_{2\ell +2})+{(\tau -a)}^2(t-t_{2k})(t-t_{2k+2})\hfill \\ \hfill & -(t-t_{2k})(t-t_{2k+2})(\tau -t_{2\ell })(\tau -t_{2\ell +2})].\hfill \end{array}$$

With $t=t_{2k}+uh$, $\tau =t_{2\ell }+vh$, $(u,v)\in {[0,1]}^2$, this expression simplifies to

$$\begin{array}{cc}\hfill {\psi }_{k,\ell }(u,v)=h^4[& {(2k+u)}^{2}v(v-1+c)+{(2\ell +v)}^{2}u(u-1+c)\hfill \\ \hfill & -uv(1-u)(1-v)+cuv(3-u-v))].\hfill \end{array}$$

(26)

Analogously, by substituting $t=t_{2k+2}+uh$, $\tau =t_{2\ell }+vh$ with $(u,v)\in {[0,1]}^2$ we find that $4\phi (t,\tau )={\psi }_{k+1,\ell }(u,v)$ for $(t,\tau )\in {\Delta }_{k+1,\ell }^1$. Since

$${\psi }_{k+1,\ell }(u,v)-{\psi }_{k,\ell }(u,v)=4h^4(2k+1+u)v(c-1+v)\ge 0,(u,v)\in {[0,1]}^2,$$

the following implication holds true for $0\le k\le {\displaystyle \frac{n-2}{2}}$ and $0\le \ell \le {\displaystyle \frac{n-1}{2}}$:

$$\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k+1,\ell }^1.$$

(27)

Next, we compare the values of $\phi$ on ${\Delta }_{k,\ell }^1$ and ${\Delta }_{k,\ell }^2$, where $0\le k,\ell \le {\displaystyle \frac{n-1}{2}}$. For $(t,\tau )\in {\Delta }_{k,\ell }^2$ we have

$$\begin{array}{cc}\hfill 4\phi (t,\tau )=& (c+1)[{(t-a)}^2(\tau -t_{2\ell })(\tau -t_{2\ell +1})+{(\tau -a)}^2(t-t_{2k+1})(t-t_{2k+2})\hfill \\ \hfill & -(t-t_{2k+1})(t-t_{2k+2})(\tau -t_{2\ell })(\tau -t_{2\ell +1})]\hfill \\ \hfill & -c[{(t-a)}^2(\tau -t_{2\ell })(\tau -t_{2\ell +2})+{(\tau -a)}^2(t-t_{2k})(t-t_{2k+2})\hfill \\ \hfill & -(t-t_{2k})(t-t_{2k+2})(\tau -t_{2\ell })(\tau -t_{2\ell +2})].\hfill \end{array}$$

With $t=t_{2k+2}-uh$, $\tau =t_{2\ell }+vh$, where $(u,v)\in {[0,1]}^2$, the above expression becomes

$$\begin{array}{cc}\hfill {\tilde{\psi }}_{k,\ell }(u,v)=h^4[& {(2k+2-u)}^{2}v(v-1+c)+{(2\ell +v)}^{2}u(u-1+c)\hfill \\ \hfill & -uv(1-u)(1-v)+cuv(3-u-v))].\hfill \end{array}$$

Since ${\tilde{\psi }}_{k,\ell }(u,v)-{\psi }_{k,\ell }(u,v)=4h^4(2k+1-u)(1-u)v(v-1+c)\ge 0$ for $(u,v)\in {[0,1]}^2$, we have the following implication when $0\le k,\ell \le {\displaystyle \frac{n-1}{2}}$:

$$\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell }^2.$$

(28)

We apply the same procedure to deduce similar conclusions in the “vertical” direction. Thus, for $0\le k,\ell \le {\displaystyle \frac{n-1}{2}}$ we have the implication

$$\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell }^3,$$

(29)

and for $k\le {\displaystyle \frac{n-1}{2}}$, $\ell \le {\displaystyle \frac{n-2}{2}}$,

$$\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\ge 0,(t,\tau )\in {\Delta }_{k,\ell +1}^1.$$

(30)

Since we already showed that $\phi (t,\tau )\ge 0$ in ${\Delta }_{0,0}^1$, from (27)–(30) we conclude that $\phi (t,\tau )\ge 0$ for all $(t,\tau )$ satisfying $a\le t,\tau \le {\displaystyle \frac{a+b}{2}}$. By symmetry, it follows that $\phi (t,\tau )\ge 0$ in ${[a,b]}^2$.

Thus, we showed that the cubature formula

$$\widehat{S}=(c+1)S_{2n}^{-}-c{S}_n^{-}$$

is positive definite of order $(2,2)$ whenever $c\ge 1$. Theorem 1 follows from Theorem 4, applied with $r=s=2$, $S^{\prime }=S_{2n}^{-}$, $S^{″}=S_n^{-}$, and $c=1$. □

Remark 3.

To demonstrate that $\widehat{S}$ is not positive definite when $c<1$, we show that $K_{2,2}(\widehat{S};t,\tau )$ assumes negative values on the segment $t=\tau$ in ${\Delta }_{k,k}^1$, $1\le k\le {\displaystyle \frac{n-1}{2}}$. In view of (26), the trace of $K_{2,2}(\widehat{S};·)$ on this segment is the univariate function

$$g(u)=h^4\left[2{(2k+u)}^{2}u(u-1+c)-u^2{(1-u)}^2+c{u}^2(3-2u)\right],u\in [0,1].$$

From $g\left(0\right)=0$ and $g^{\prime }\left(0\right)=8(c-1)h^4{k}^2$ it follows that $g\left(u\right)$ assumes negative values in $(0,1)$ when $c<1$.

3.2. Proof of Theorem 2

$S_n^{+}$ is built according to (15), where $Bf$ is the Lagrange blending interpolant for f at the blending grid $G(X,Y)$ with $X=Y=\{a,b\}$, and $C_n[f]=C_n^{Tr}[f]$. In this case $Q^{\prime }=Q^{″}=Q^{Tr}$, and the representation (2) of $S_n^{+}$ follows from (21).

From (19) we have

$$K_{2,2}(S_n^{+};t,\tau )=K_2(Q^{Tr};\tau )K_2(Q_n^{Tr};t)+K_2(Q_n^{Tr};\tau )Q_n^{Tr}[{\mathcal{K}}_2(·,t)],$$

(31)

where

$${\mathcal{K}}_2(x,t)={(x-t)}_{+}-{\displaystyle \frac{x-a}{b-a}}(b-t).$$

Since $Q_n^{Tr}$ (in particular, $Q^{Tr}$) is negative definite of order two and ${\mathcal{K}}_2(x,t)\le 0$ for $t,\tau \in [a,b]$, it follows from (31) that $K_{2,2}(S_n^{+};t,\tau )\ge 0$ in ${[a,b]}^2$, i.e., $S_n^{+}$ is a positive definite cubature formula of order $(2,2)$. According to (20),

$$\begin{array}{cc}\hfill K_{2,2}(S_n^{+};t,\tau )=& K_2(Q^{Tr};t)K_2(Q_n^{Tr};\tau )+K_2(Q^{Tr};\tau )K_2(Q_n^{Tr};t)\hfill \\ \hfill & -K_2(Q_n^{Tr};t)K_2(Q_n^{Tr};\tau ).\hfill \end{array}$$

(32)

Claim (ii) of Theorem A follows from the mean value theorem, applied to the integral representation of the remainder $R[S_n^{+};f]$, taking into account (32).

The proof of Theorem 2 is based on Theorem 4 applied with $r=s=2$, $S^{\prime }=S_{2n}^{+}$ and $S^{″}=S_n^{+}$. We shall find condition on the constant $c>0$ so that the cubature formula $\widehat{S}=(c+1)S_{2n}^{+}-c{S}_{2n}^{+}$ is negative definite of order $(2,2)$, i.e.,

$$\phi (t,\tau ):=(c+1)K_{2,2}(S_{2n}^{+};t,\tau )-c{K}_{2,2}(S_n^{+};t,\tau )\le 0,(t,\tau )\in {[a,b]}^2$$

(33)

The verification of (33) goes along the same lines as that of (23): with ${\left\{t_i\right\}}_{i=0}^{2n}$ defined in (24) we decompose ${[a,b]}^2$ in the same way, and study $\phi (t,\tau )$ on the different sub-domains. By symmetry, we may restrict our considerations to the region $a\le t,\tau \le {\displaystyle \frac{a+b}{2}}$. Assume first that $(t,\tau )\in {\Delta }_{0,0}^1$. Using (25) and (32) and

$$K_2(Q^{Tr};t)={\displaystyle \frac{1}{2}}(t-a)(t-b),$$

we find that for $(t,\tau )\in {\Delta }_{0,0}^1$ the function $\phi (t,\tau )$ takes the form

$$\begin{array}{cc}\hfill \phi (t,\tau )={\displaystyle \frac{(t-a)(\tau -a)}{4}}[& (c+1)\left[(t-b)(\tau -t_1)+(\tau -b)(t-t_1)-(t-t_1)(\tau -t_1)\right]\hfill \\ \hfill & -c\left[(t-b)(\tau -t_2)+(\tau -b)(t-t_2)-(t-t_2)(\tau -t_2)\right]].\hfill \end{array}$$

Replacing $t_2=t_1+h$ and using that $b-a=2nh$, after simplification we get

$$\phi (t,\tau )={\displaystyle \frac{(t-a)(\tau -a)}{4}}\left[t\tau -b(t+\tau )+t_1(2b-t_1)-c(4n-3)h^2\right].$$

The expression in the brackets is a bilinear function in t and $\tau$ and attains its extreme values at the vertices of ${\Delta }_{0,0}^1$. Its maximum in ${\Delta }_{0,0}^1$ is attained at $(t,\tau )=(t_0,t_0)=(a,a)$ and equals $\left(4n-1-(4n-3)c\right)h^2$. Hence, the inequality

$$c\ge {\displaystyle \frac{4n-1}{4n-3}}$$

(34)

is a necessary and sufficient condition for $\phi (t,\tau )\le 0$ in ${\Delta }_{0,0}^1$.

The rest of the proof of (33) follows the scheme of the proof of (23): under the assumption (34) we establish the implications

$$\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k+1,\ell }^1$$

(35)

for $0\le k\le {\displaystyle \frac{n-2}{2}}$ and $0\le \ell \le {\displaystyle \frac{n-1}{2}}$;

$$\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell }^2$$

(36)

for $0\le k,\ell \le {\displaystyle \frac{n-1}{2}}$;

$$\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell +1}^1$$

(37)

for $0\le k\le {\displaystyle \frac{n-1}{2}}$ and $0\le \ell \le {\displaystyle \frac{n-2}{2}}$, and

$$\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell }^1⟹\phi (t,\tau )\le 0,(t,\tau )\in {\Delta }_{k,\ell }^3$$

(38)

for $0\le k,\ell \le {\displaystyle \frac{n-1}{2}}$.

Let us prove (35). For $(t,\tau )\in {\Delta }_{k,\ell }^1$ we have

$$\begin{array}{cc}\hfill 4\phi (t,\tau )=& (c+1)[(t-a)(t-b)(\tau -t_{2\ell })(\tau -t_{2\ell +1})+(\tau -a)(\tau -b)(t-t_{2k})(t-t_{2k+1})\hfill \\ \hfill & -(t-t_{2k})(t-t_{2k+1})(\tau -t_{2\ell })(\tau -t_{2\ell +1})]\hfill \\ \hfill & -c[(t-a))t-b)(\tau -t_{2\ell })(\tau -t_{2\ell +2})+(\tau -a)(\tau -b)(t-t_{2k})(t-t_{2k+2})\hfill \\ \hfill & -(t-t_{2k})(t-t_{2k+2})(\tau -t_{2\ell })(\tau -t_{2\ell +2})].\hfill \end{array}$$

With $t=t_{2k}+uh$, $\tau =t_{2\ell }+vh$, $(u,v)\in {[0,1]}^2$, we have $4\phi (t,\tau )=h^4{\psi }_{k,\ell }(u,v)$, where

$$\begin{array}{cc}\hfill {\psi }_{k,\ell }(u,v)=& (2k+u)(2k+u-2n)v(v-1+c)+(2\ell +v)(2\ell +v-2n)u(u-1+c)\hfill \\ \hfill & -uv(1-u)(1-v)+cuv(3-u-v)).\hfill \end{array}$$

For $(t,\tau )\in {\Delta }_{k+1,\ell }^1$, we substitute $t=t_{2k+2}+uh$, $\tau =t_{2\ell }+vh$, $(u,v)\in {[0,1]}^2$, and obtain $4\phi (t,\tau )={\psi }_{k+1,\ell }(u,v)$. Now (35) follows from

$${\psi }_{k+1,\ell }(u,v)-{\psi }_{k,\ell }(u,v)=4v(v-1+c)(2k+1+u-n)\le 0.$$

The proof of implications (36)–(38) is similar, therefore we skip it. Since $\phi (t,\tau )\le 0$ in ${\Delta }_{0,0}^1$ for every c satisfying (34), it follows from (35)–(38) that $\phi (t,\tau )\le 0$ whenever $a\le t,\tau \le {\displaystyle \frac{a+b}{2}}$. By symmetry, $\phi (t,\tau )\le 0$ for every $(t,\tau )\in {[a,b]}^2$. Hence, the cubature formula

$$\widehat{S}=(c+1)S_{2n}^{+}-c{S}_{2n}^{+}$$

is negative definite of order $(2,2)$ whenever c satisfies (34). Theorem 2 now follows from Theorem 4 applied with $r=s=2$, $S^{\prime }=S_{2n}^{+}$, $S^{″}=S_n^{+}$ and $c={\displaystyle \frac{4n-1}{4n-3}}$. Let us point out that this value of c is the best possible, i.e., the smallest one for which $\widehat{S}$ is negative definite cubature formula of order $(2,2)$.

4. Numerical Examples and Concluding Remarks

We illustrate theoretical results in Theorems 1 and 2 by considering double integrals over ${[0,1]}^2$ of $f(x,y)=e^{xy}$, $g(x,y)=sin\left(xy\right)$ and $h(x,y)=x^2{y}^{2}ln(1+x^2+y^2)$. The numerical values $I[f]\approx 1.317902151$ and $I[g]\approx 0.239811742$ are found from

$$\underset{{[0,1]}^2}{\int \int }{e}^{xy}dxdy=\underset{0}{\overset{1}{\int }}{\displaystyle \frac{e^x-1}{x}}dx,\underset{{[0,1]}^2}{\int \int }sin\left(xy\right)dxdy=\underset{0}{\overset{1}{\int }}{\displaystyle \frac{1-cosx}{x}}dx$$

and appropriate truncation of the power series of the integrands in the univariate integrals. The numerical value $I[h]=0.085922803$ is found from

$$I[h]={\displaystyle \frac{1}{9}}ln3+{\displaystyle \frac{88}{135}}-{\displaystyle \frac{4\sqrt{2}}{9}}arctan{\displaystyle \frac{\sqrt{2}}{2}}-{\displaystyle \frac{2}{3}}\underset{0}{\overset{1}{\int }}{x}^2{(x^2+1)}^{3/2}arctan{\displaystyle \frac{1}{\sqrt{x^2+1}}}dx$$

and tight two-sided estimation of the univariate integral. Since the integrand has positive second and fourth derivative in $[0,1]$, the two-sided estimation is obtained by usage of compound midpoint and Simpson quadrature formulae. To ensure reliability of the results presented in

Table 1. Remainders and error bounds of $S_n^{-}[f]$ and $S_n^{+}[f]$.

n	${\displaystyle \mathit{R}[{\mathit{S}}_{\mathit{n}}^{-};\mathit{f}]}$	${\displaystyle \frac{1}{2}|{\mathit{S}}_{2\mathit{n}}^{-}[\mathit{f}]-{\mathit{S}}_{\mathit{n}}^{-}[\mathit{f}]|}$	${\displaystyle \mathit{R}[{\mathit{S}}_{\mathit{n}}^{+};\mathit{f}]}$	${\displaystyle \frac{4\mathit{n}-1}{4\mathit{n}-3}|{\mathit{S}}_{2\mathit{n}}^{+}[\mathit{f}]-{\mathit{S}}_{\mathit{n}}^{+}[\mathit{f}]|}$
4	$-1.947\times {10}^{-3}$		$3.615\times {10}^{-3}$
		$7.411\times {10}^{-4}$		$3.101\times {10}^{-3}$
8	$-4.648\times {10}^{-4}$		$9.274\times {10}^{-4}$
		$1.750\times {10}^{-4}$		$7.419\times {10}^{-4}$
16	$-1.148\times {10}^{-4}$		$2.333\times {10}^{-4}$
		$4.310\times {10}^{-5}$		$1.806\times {10}^{-4}$
32	$-2.862\times {10}^{-5}$		$5.842\times {10}^{-5}$
		$1.073\times {10}^{-5}$		$4.451\times {10}^{-5}$
64	$-7.149\times {10}^{-6}$		$1.461\times {10}^{-5}$
		$2.681\times {10}^{-6}$		$1.104\times {10}^{-5}$
128	$-1.787\times {10}^{-6}$		$3.653\times {10}^{-6}$

,

Table 2. Remainders and error bounds of $S_n^{-}[g]$ and $S_n^{+}[g]$.

n	${\displaystyle \mathit{R}[{\mathit{S}}_{\mathit{n}}^{-};\mathit{g}]}$	${\displaystyle \frac{1}{2}|{\mathit{S}}_{2\mathit{n}}^{-}[\mathit{g}]-{\mathit{S}}_{\mathit{n}}^{-}[\mathit{g}]|}$	${\displaystyle \mathit{R}[{\mathit{S}}_{\mathit{n}}^{+};\mathit{g}]}$	${\displaystyle \frac{4\mathit{n}-1}{4\mathit{n}-3}|{\mathit{S}}_{2\mathit{n}}^{+}[\mathit{g}]-{\mathit{S}}_{\mathit{n}}^{+}[\mathit{g}]|}$
4	$6.300\times {10}^{-4}$		$-1.129\times {10}^{-3}$
		$2.397\times {10}^{-4}$		$9.697\times {10}^{-4}$
8	$1.507\times {10}^{-4}$		$-2.886\times {10}^{-4}$
		$5.674\times {10}^{-5}$		$2.309\times {10}^{-4}$
16	$3.726\times {10}^{-5}$		$-7.254\times {10}^{-5}$
		$1.399\times {10}^{-5}$		$5.616\times {10}^{-5}$
32	$9.289\times {10}^{-6}$		$-1.816\times {10}^{-5}$
		$3.484\times {10}^{-6}$		$1.384\times {10}^{-5}$
64	$2.321\times {10}^{-6}$		$-4.541\times {10}^{-6}$
		$8.703\times {10}^{-7}$		$3.433\times {10}^{-6}$
128	$5.801\times {10}^{-7}$		$-1.135\times {10}^{-6}$

and

Table 3. Remainders and error bounds of $S_n^{-}[h]$ and $S_n^{+}[h]$.

n	${\displaystyle \mathit{R}[{\mathit{S}}_{\mathit{n}}^{-};\mathit{h}]}$	${\displaystyle \frac{1}{2}|{\mathit{S}}_{2\mathit{n}}^{-}[\mathit{h}]-{\mathit{S}}_{\mathit{n}}^{-}[\mathit{h}]|}$	${\displaystyle \mathit{R}[{\mathit{S}}_{\mathit{n}}^{+};\mathit{h}]}$	${\displaystyle \frac{4\mathit{n}-1}{4\mathit{n}-3}|{\mathit{S}}_{2\mathit{n}}^{+}[\mathit{h}]-{\mathit{S}}_{\mathit{n}}^{+}[\mathit{h}]|}$
4	$-2.935\times {10}^{-3}$		$5.431\times {10}^{-3}$
		$1.117\times {10}^{-3}$		$4.659\times {10}^{-3}$
8	$-7.010\times {10}^{-4}$		$1.393\times {10}^{-3}$
		$2.639\times {10}^{-4}$		$1.114\times {10}^{-3}$
16	$-1.732\times {10}^{-4}$		$3.504\times {10}^{-4}$
		$6.501\times {10}^{-5}$		$2.712\times {10}^{-4}$
32	$-4.317\times {10}^{-5}$		$8.773\times {10}^{-5}$
		$1.619\times {10}^{-5}$		$6.684\times {10}^{-5}$
64	$-1.078\times {10}^{-5}$		$2.194\times {10}^{-5}$
		$4.045\times {10}^{-6}$		$1.658\times {10}^{-5}$
128	$-2.696\times {10}^{-6}$		$5.486\times {10}^{-6}$

, high precision calculations are carried out with Wolfram Mathematica software.

Table 1, Table 2 and Table 3 represent the remainders and a posteriori error estimates of cubature formulae $S_n^{-}$ and $S_n^{+}$ applied to $f(x,y)$, $g(x,y)$ and $h(x,y)$, respectively. The remainder signs ($R[S_n^{-};f]<0$, $R[S_n^{+};f]>0$, $R[S_n^{-};g]>0$, $R[S_n^{+};g]<0$, $R[S_n^{-};h]<0$, $R[S_n^{+};h]>0$) are in agreement with the fact that $D^{2,2}f>0$, $D^{2,2}g<0$, $D^{2,2}h>0$ in ${[0,1]}^2$. In the three tables it is observed that a posteriori error estimates exceed the true error magnitude by a factor ranging between $1.5$ and $1.6$ for $S_n^{-}$ and between $3.02$ and $3.36$ for $S_n^{+}$. Of course, the mean cubature formula $(S_n^{+}+S_n^{-})/2$ furnishes another reasonable approximation to I, and for integrands from ${\mathcal{C}}^{2,2}[0,1]$ it has error magnitude not exceeding $|R[S_n^{+};·]+R[S_n^{-};·]|/2$.

Remark 4.

One may wonder if similar results may be established for the modified composite midpoint rules, constructed through the Scheme (15) with blending interpolation on the same blending grids. Unfortunately, we cannot prove the analogues of Theorems 1 and 2 (monotonicity and a posteriori error estimates). Theorem 4 is not applicable in this situation, due to the fact that the associated Peano kernels $K_{2,2}(S_n;·)$ and $K_{2,2}(S_{2n};·)$ coincide on some subregions of ${[a,b]}^2$.

5. Conclusions

We have proved error bounds and a posteriori error estimates for some appropriately modified trapezium product cubature formulae, provided the integrand f belongs to the class of functions $f\in {\mathcal{C}}^{2,2}[a,b]$ described by a single derivative $D^{2,2}f$. Besides the standard data of point evaluations, the modified cubature formulae involve two or four univariate integrals on some traces of the integrand. Practical implementation of our results is limited because of the assumptions on the integrands and the necessity to know the true values of some univariate integrals. On the other hand, a cubature formula C which involves only point evaluations cannot have error representation of the form

$$R[C;f]=\underset{{[a,b]}^2}{\int \int }{K}_{r,s}(t,\tau )D^{r,s}f(t,\tau )dtd\tau .$$

Indeed, such a representation would mean that C evaluates to the exact value every integral of a function f from the linear space of blending functions $B^{r,s}\left({[a,b]}^2\right)$, which is impossible, since $B^{r,s}\left({[a,b]}^2\right)$ is of infinite dimension. This is the difference with the univariate case (where the corresponding set is the linear space of algebraic polynomials of finite degree). Our results in this paper contribute mainly to the theory of numerical integration, showing a way for extending useful properties of a quadrature formula to its product-type counterpart.