MULTIDIMENSIONAL INTEGRATION VIA DIMENSION REDUCTION AND GENERATORS

P. CERONE

Abstract. An iterative approach is used to represent multidimensional integrals in terms of lower dimensional integrals and function evaluations. The procedure is quite general utilising one dimensional identities as the generator to procure multidimensional identities. Bounds are obtained from the identities. Both weighted and unweighted integrals are considered.

1. Introduction

We firstly present one-dimensional identities which may be used as generators for higher dimensional results.

For f : [a, b] → ℝ we define the Ostrowski and Trapezoidal functionals by

(1.1)  S (f; c, x, d) := f (x) − M (f; c, d)

and

(1.2)  T (f; c, x, d) := (\frac{x − c}{d − c}) f (c) + (\frac{d − x}{d − c}) f (d) − M (f; c, d) ,

respectively, where

(1.3)  M (f; c, d) := \frac{1}{d − c} ∫_{c}^{d} f (u) du, the integral mean.

We note that taking x = \frac{a+b}{2} and multiplication by (b − a) in (1.1) and (1.2) recapture the traditional midpoint and trapezoidal rules for the evaluation of the integrals. With this in mind, the most common task is to obtain bounds on the above functionals. This task is perhaps best accomplished from identities involving the functionals. The following identities may be easily shown to hold for f of bounded variation, by an integration by parts argument of the Riemann-Stieltjes integrals and so

(1.4) S (f; c, x, d) = ∫_{c}^{d} p (x, t, c, d) df (t) , p (x, t, c, d) = \{ \begin{matrix}\frac{t − c}{d − c}, & t ∈ [c, x] \\ \frac{t − d}{d − c}, & t ∈ (x, d] \end{matrix}

and

(1.5) T (f; c, x, d) = ∫_{c}^{d} q (x, t, c, d) df (t) , q (x, t, c, d) = \frac{t − x}{d − c}, x, t ∈ [c, d] .

1991 Mathematics Subject Classification. Primary 26D15, 26D10, 26D99, 41A55.

Key words and phrases. Multidimensional integrals, Dimension Reduction, Recursive method, Ostrowski, trapezoidal and three point generators, a priori bounds.

The book [13] is devoted to Ostrowski type results involving (1.1) and numerous generalisations. See also [1], [14], [17] and [19].

Further, define the three point functional 𝔗 (f; a, α, x, β, b) which involves the difference between the integral mean and, a weighted combination of a function evaluated at the end points and an interior point. Namely, for a ≤ α < x < β ≤ b,

(1.6) 𝔗 (f; a, α, x, β, b)

:= (\frac{α − a}{b − a}) f (a) + (\frac{β − α}{b − a}) f (x) + (\frac{b − β}{b − a} ) f (b) − M (f; a, b) .

Cerone and Dragomir [8] showed that for f of bounded variation, the identity

(1.7) 𝔗 (f; a, α, x, β, b) = ∫_{a}^{b} r (x, t) df (t) , r (x, t) = \{ \begin{matrix}\frac{t − α}{b − a}, & t ∈ [a, x] \\ \frac{t − β}{b − a} , & t ∈ (x, b] \end{matrix}

is valid. They effectively demonstrated that the Ostrowski functional and the trapezoid functional could be recaptured as particular instances. Specifically, from (1.6) and (1.7)

S (f; a, x, b) = 𝔗 (f; a, a, x, b, b)

and

T (f; a, x, b) = 𝔗 (f; a, x, x, x, b) ,

where S (f; a, x, b) and T (f; a, x, b) are defined by (1.1) and (1.2) and satisfy identities (1.4) and (1.5) respectively.

It should be noted at this stage that

(b − a) 𝔗 (f; a, \frac{5a + b}{6}, \frac{a + b}{2}, \frac{a + 5b}{6}, b)

= \frac{b − a}{6} [f (a) + 4f (\frac{a + b}{2}) + f (b)] − ∫_{a}^{b} f (x) dx

is the Simpson functional.

Further, if f (t) is assumed to be absolutely continuous for t over its respective interval, then df (t) = f′ (t) dt and the Riemann-Stieltjes integrals in (1.4), (1.5) and (1.7) are equivalent to Riemann integrals.

Pachpatte [20] obtains a trapezoidal type result for double integrals that involves single integrals and functional evaluation. He also investigated triple integrals, however, the techniques would be challenging to express in general dimensions.

In the current work, the Ostrowski (1.4), the generalised trapezoidal (1.5) and three point identities (1.6) – (1.7) for absolutely continuous functions are used as generators to produce identities involving multidimensional integrals in terms of lower dimensional integrals and function evaluations. These are used to procure bounds for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{p} [I^{n}] , 1 ≤ p ≤ ∞, where I^{n} = [a_{1}, b_{1}] × ⋯ × [a_{n}, b_{n}] . Here for h : I^{n} → ℝ we mean by h ∈ L_{p} [I^{n}] , the Lebesgue norms, that is,

(1.8) ∥h∥_{p} := (∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |h (t_{1}, t_{2}, …, t_{n})|^{p} dt_{1} … dt_{n})^{\frac{1}{p}} ,

1 ≤ p < ∞, for h ∈ L_{p} [I^{n}] , and

(1.9)  ∥h∥_{∞} := ess sup_{t_{∼}∈[a_{∼}, b_{∼}]} |h (t_{1}, t_{2}, …, t_{n})| , for h ∈ L_{∞} [I^{n}] .

In the current work, weighted generators are used to obtain identities involving multidimensional integrals. The identities allow a priori bounds on the error. Ostrowski, Trapezoid and three-point generators are utilised to procure multidimensional results. The results of Cerone [5] and [6] are recaptured if the weights are taken to be identically one.

2. Weighted Multidimensional Ostrowski Identities and Bounds from an Iterative Approach

The following theorem uses an iterative approach to extend a weighted Ostrowski functional identity to multidimensions. Firstly, we will require some notation.

Let I^{n} = ∏_{i=1}^{n} [a_{i}, b_{i}] = [a_{1}, b_{1}] × [a_{2}, b_{2}] × ⋯ × [a_{n}, b_{n}] . Further, let f : I^{n} → ℝ and define operators F_{i} (f) and λ_{i,w_{i}} (f) by

(2.1)  F_{i} (f) := f (t_{1}, …, t_{i−1}, x_{i}, t_{i+1}, …, t_{n}) where x_{i} ∈ [a_{i}, b_{i}]

and

(2.2)  λ_{i,w_{i}} (f) := \frac{1}{W_{i}} ∫_{a_{i}}^{b_{i}} w_{i} (t_{i}) f (t_{1}, …, t_{i−1}, t_{i}, t_{i+1}, …, t_{n}) dt_{i},

where w_{i} (t_{i}) are positive weight functions for t_{i} ∈ [a_{i}, b_{i}] , i = 1, 2, …, n satisfying

(2.3)  W_{i} = ∫_{a_{i}}^{b_{i}} w_{i} (t_{i}) dt_{i} > 0.

That is, F_{i} (f) evaluates f (·) in the ith variable at x_{i} ∈ [a_{i}, b_{i}] and λ_{i,w_{i}} (f) is the weighted integral mean of f (·) in the ith variable. Assuming that f (·) is absolutely continuous in the ith variable t_{i} ∈ [a_{i}, b_{i}] , we have

(2.4)  L_{i,w_{i}} (f) = \frac{1}{W_{i}} ∫_{a_{i}}^{b_{i}} P_{i} (x_{i}, t_{i}) \frac{∂f}{∂t_{i}} dt_{i} = (F_{i} − λ_{i,w_{i}}) (f) ,

for i = 1, 2, …, n, where

(2.5)  \frac{P_{i} (x_{i}, t_{i})}{W_{i}} = \{ \begin{matrix}^{\frac{∫_{a_{i}}^{t_{i}} w_{i}(s)ds}{W_{i}}}, & t_{i} ∈ [a_{i}, x_{i}] \\ ^{\frac{− ∫_{t_{i}}^{b_{i}} w_{i}(s)ds}{W_{i}}}, & t_{i} ∈ (x_{i}, b_{i}] . \end{matrix}

Thus (2.4) – (2.5) is ostensibly equivalent to a weighted Montgomery identity which reduces to (1.4) for w_{i} (t_{i}) ≡ 1 for f (t_{1}, …, t_{i−1}, t_{i,}t_{i+1}, …, t_{n}) absolutely continuous for t_{i} ∈ [a_{i}, b_{i}] .

Theorem 1. Let f : I^{n} → ℝ be absolutely continuous in such a manner that the partial derivatives of order one with respect to every variable exist. Then

(2.6) E_{n} (f) = f (x_{1}, x_{2}, …, x_{n})

− ∑_{i=1}^{n} \frac{1}{W_{i}} ∫_{a_{i}}^{b_{i}} w_{i} (t_{i}) f (x_{1}, x_{2}, …, x_{i−1}, t_{i}, x_{i+1}, …, x_{n}) dt_{i}

+ ∑_{i<j}^{n} \frac{1}{W_{i}W_{j}} ∫_{a_{j}}^{b_{j}} ∫_{a_{i}}^{b_{i}} w_{i} (t_{i}) w_{j} (t_{j})

× f (x_{1}, …, x_{i−1}, t_{i}, x_{i+1}, …, t_{j}, …, x_{n}) dt_{i}dt_{j}

− ⋯ − \frac{(−1)^{n}}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} w_{i} (t_{i}) f (t_{1}, …, t_{n}) dt_{1} … dt_{n}

:= τ_{n} (a_{∼}, x_{∼}, b_{∼}) ,

where

(2.7)  E_{n} (f) = \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} P_{i} (x_{i}, t_{i}) \frac{∂^{n}f}{∂t_{n} … ∂t_{1}} dt_{1} … dt_{n},

(2.8)  W^{∗} = ∏_{i=1}^{n} W_{i},

with W_{i} given by (2.3) and P_{i} (x_{i}, t_{i}) is given by (2.5).

Proof. Define E_{r} (f) by

(2.9)  E_{r} (f) = (∏_{i=1}^{r} L_{i,w_{i}}) (f)

then from the left identity in (2.4), E_{n} (f) is as given by (2.7). Further,

(2.10)  E_{r} (f) = L_{r,w_{r}} (E_{r−1} (f)) , for r = 1, 2, …, n

where E_{0} (f) = f.

Now, from (2.9),

E_{1} (f) = L_{1,w_{1}} (f) = (F_{1} − λ_{1,w_{1}}) (f) ,

which is the weighted Montgomery identity for t_{1}, x_{1} ∈ [a_{1}, b_{1}]

(2.11)  E_{1} (f) = \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} P_{1} (x_{1}, t_{1}) \frac{∂f (t_{1}, t_{2}, …, t_{n})}{∂t_{1}}dt_{1}

= f (x_{1}, t_{2}, …, t_{n}) − \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, t_{2}, …, t_{n}) dt_{1}.

Further,

E_{2} (f) = L_{2,w_{2}} (E_{1} (f)) = (F_{2} − λ_{2,w_{2}}) (E_{1} (f))

= F_{2} (E_{1} (f)) − λ_{2,w_{2}} (E_{1} (f))

= f (x_{1}, x_{2}, t_{3}, …, t_{n}) − \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, x_{2}, t_{3}, …, t_{n}) dt_{1}

− \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) [f (x_{1}, t_{2}, …, t_{n})

− \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, t_{2}, …, t_{n}) dt_{1}] dt_{2}

= f (x_{1}, x_{2}, t_{3}, …, t_{n}) − \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, x_{2}, t_{3}, …, t_{n}) dt_{1}

− \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) f (t_{1}, t_{2}, t_{3}, …, t_{n}) dt_{2}

+ \frac{1}{W_{1}W_{2}} ∫_{a_{2}}^{b_{2}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) w_{2} (t_{2}) f (t_{1}, t_{2}, …, t_{n}) dt_{1}dt_{2}

and continuing in this manner until r = n gives the result as stated in (2.6).

Remark 1. The result given by (2.6) may be utilised to approximate the weighted n− dimensional integral in terms of lower dimensional integrals and a function evaluation f (x_{1}, x_{2}, …, x_{n}) where x_{i} ∈ [a_{i}, b_{i}] , i = 1, 2, …, n. Specifically, there are \binom{n}{0} function evaluations, \binom{n}{1} single integral evaluations in each of the axes, \binom{n}{2} double integral evaluations and so on, and, of course, \binom{n}{n} n−dimensional integral evaluations. This results from the fact that from (2.9) and (2.1) – (2.4)

(2.12)  E_{n} (f) = (∏_{i=1}^{n} L_{i,w_{i}}) (f) = (∏_{i=1}^{n} (F_{i} − λ_{i,w_{i}})) (f) .

The above procedure of utilising a one-dimensional identity as the generator to recursively obtain a multidimensional identity which is quite general, may be extended to utilising other one-dimensional identities.

Theorem 2. Let f : I^{n} → ℝ be absolutely continuous in a manner that the partial derivatives of order one with respect to every variable exist. Then

(2.13) W^{∗} |τ_{n} (a_{∼}, x_{∼}, b_{∼})|

≤ \{ \begin{matrix}∏_{i=1}^{n} (∫_{a_{i}}^{b_{i}} |x_{i} − t_{i}| w_{i} (t_{i}) dt_{i}) ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{∞} , & \frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∈ L_{∞} [I^{n}] ; \\ (∏_{i=1}^{n} P_{i} (q))^{\frac{1}{q}} ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{p} , & \frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∈ L_{p} [I^{n}] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ ∏_{i=1}^{n} θ_{i} ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{1} , & \frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∈ L_{1} [I^{n}] , \end{matrix}

where τ_{n} (a_{∼}, x_{∼}, b_{∼}) is as defined by (2.6),

(2.14)  P_{i} (q) = ∫_{a_{i}}^{x_{i}} (∫_{a_{i}}^{t_{i}} w_{i} (s) ds)^{q} dt_{i} + ∫_{x_{i}}^{b_{i}} (∫_{t_{i}}^{b_{i}} w_{i} (s) ds)^{q} dt_{i},
(2.15)  θ_{i} = \frac{1}{2} ∫_{a_{i}}^{b_{i}} w_{i} (s) ds + \frac{1}{2} |∫_{a_{i}}^{x_{i}} w_{i} (s) ds − ∫_{x_{i}}^{b_{i}} w_{i} (s) ds| .

Proof. From (2.6) and (2.7) we have

(2.16) |τ_{n} (a_{∼}, x_{∼}, b_{∼})| = |E_{n} (f)|

≤ \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} P_{i} (x_{i}, t_{i}) \frac{∂^{n}f}{∂t_{n} … ∂t_{1}} | dt_{1} … dt_{n}.

Now, for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{∞} [I^{n}] , we have

(2.17) W^{∗} · |E_{n} (f)| ≤ ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} P_{i} (x_{i}, t_{i})| dt_{1} … dt_{n} ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{∞} ,

where

(2.18) ∏_{i=1}^{n} ∫_{a_{i}}^{b_{i}} |P_{i} (x_{i}, t_{i})| dt_{i} = ∏_{i=1}^{n} [∫_{a_{i}}^{x_{i}} ∫_{a_{i}}^{t_{i}} w_{i} (s) dsdt_{i} + ∫_{x_{i}}^{b_{i}} ∫_{t_{i}}^{b_{i}} w_{i} (s) dsdt_{i}]

= ∏_{i=1}^{n} ∫_{a_{i}}^{b_{i}} |x_{i} − t_{i}| w_{i} (t_{i}) dt_{i}.

Hence, combining (2.17) and (2.18) gives the first inequality of (2.13). Further, using the Hölder inequality, we have for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{p} [I^{n}] , 1 ≤ p < ∞,

W^{∗} · |E_{n} (f)| ≤ (∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} P_{i} (x_{i}, t_{i})|^{q} dt_{1} … dt_{n}) ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{p} ,

where

∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} P_{i} (x_{i}, t_{i})|^{q} dt_{1} … dt_{n}

= ∏_{i=1}^{n} ∫_{a_{i}}^{b_{i}} |P_{i} (x_{i}, t_{i})|^{q} dt_{i}

= ∏_{i=1}^{n} [∫_{a_{i}}^{x_{i}} (∫_{a_{i}}^{t_{i}} w_{i} (s) ds)^{q} dt_{i} + ∫_{x_{i}}^{b_{i}} (∫_{t_{i}}^{b_{i}} w_{i} (s) ds)^{q} dt_{i}]

and so the second inequality is valid on noting (2.13).

The final inequality in (2.13) is obtained from (2.16) for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{1} [I^{n}] , giving

W^{∗} · |E_{n} (f)| ≤ sup_{t_{∼}∈[a_{∼}, b_{∼}]} |∏_{i=1}^{n} P_{i} (x_{i}, t_{i})| ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} | dt_{1} … dt_{n}

= ∏_{i=1}^{n} sup_{t_{i}∈[a_{i},b_{i}]} |P_{i} (x_{i}, t_{i})| ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{1}

= ∏_{i=1}^{n} max \{∫_{a_{i}}^{x_{i}} w_{i} (s) ds, ∫_{x_{i}}^{b_{i}} w_{i} (s) ds\} ∥\frac{∂^{n}f}{∂t_{n} … ∂t_{1}} ∥_{1} .

On noting that max \{X, Y \} = \frac{X+Y}{2} + \frac{|X−Y |}{2} readily produces the stated result.

Remark 2. The expression for τ_{n} (a_{∼}, x_{∼}, b_{∼}) may be written in a less explicit form which is perhaps more appealing. Namely,

(2.19)  τ_{n} (a_{∼}, x_{∼}, b_{∼}) = f (x_{1}, x_{2}, …, x_{n}) + ∑_{k=1}^{n−1} (−1)^{k} ∑_{k} M_{k} + (−1)^{n} M_{n},

where M_{k} represents the integral means in k variables with the remainder being evaluated at their respective interior point and ∑_{k} M_{k} is a sum over all \binom{n}{k} , k−dimensional integral means. Here

M_{n} = \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} w_{i} (t_{i}) f (t_{1}, …, t_{n}) dt_{1} … dt_{n}

and

∑_{1} M_{1} = \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, x_{2}, …, x_{n}) dt_{1}

+ \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) f (x_{1}, t_{2}, x_{3}, …, x_{n}) dt_{2}

+ ⋯ + \frac{1}{W_{n}} ∫_{a_{n}}^{b_{n}} w_{n} (t_{n}) f (x_{1}, x_{2}, …, x_{n−1}, t_{n}) dt_{n}.

It should be noted that (2.19) may be written as

(2.20)  τ_{n} (a_{∼}, x_{∼}, b_{∼}) = ∑_{k=0}^{n} (−1)^{k} ∑_{k} M_{k}

if we define the degenerate 0th integral mean M_{0} = f (x_{1}, x_{2}, …, x_{n}) .

Remark 3. The following result was developed by Cerone [5] using an iterative approach from an unweighted Montgomery identity as generator.

The following theorem bounds for τ_{n} (a_{∼}, x_{∼}, b_{∼}) were obtained where

(2.21) τ_{n} (a_{∼}, x_{∼}, b_{∼})

= f (x_{1}, x_{2}, …, x_{n}) − ∑_{i=1}^{n} \frac{1}{d_{i}} ∫_{a_{i}}^{b_{i}} f (x_{1}, x_{2}, …, x_{i−1}, t_{i}, x_{i+1}, …, x_{n}) dt_{i}

+ ∑_{i<j}^{n} \frac{1}{d_{j}d_{i}} ∫_{a_{j}}^{b_{j}} ∫_{a_{i}}^{b_{i}} f (x_{1}, …, x_{i−1}, t_{i}, x_{i+1}, …, x_{j−1}, t_{j}, x_{j+1}, …, x_{n}) dt_{i}dt_{j}

− · · · · · · · · · − \frac{(−1)^{n}}{D_{n}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} f (t_{1}, …, t_{n}) dt_{1} … dt_{n}

where D_{n} = ∏_{i=1}^{n} d_{i}, d_{i} = b_{i} − a_{i} and z_{∼} = (z_{1}, z_{2}, …, z_{n}) .

Theorem 3. Let f : I^{n} → ℝ be absolutely continuous in a manner that the partial derivatives of order one with respect to every variable exist. Then

(2.22) D_{n} · |τ_{n} (a_{∼}, x_{∼}, b_{∼})|

≤ \{ \begin{matrix}∏_{i=1}^{n} P_{i} (1) ∥\frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∥_{∞} , & \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{∞} [I^{n}] ; \\ ( ∏_{i=1}^{n} P_{i} (q))^{\frac{1}{q}} ∥\frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∥_{p} , & \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{p} [I^{n}] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ ∏_{i=1}^{n} θ_{i} ∥\frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∥_{1} , & \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{1} [I^{n}] , \end{matrix}

where τ_{n} (a_{∼}, x_{∼}, b_{∼}) is as defined by (2.21)

(2.23)  (q + 1)P_{i} (q) = (x_{i} − a_{i})^{q+1} + (b_{i} − x_{i})^{q+1},

(2.24)  θ_{i} = \frac{b_{i} − a_{i}}{2} + |x_{i} − \frac{a_{i} + b_{i}}{2} | .

We note that, not suprisingly, more explicit expressions for the bounds are possible in Theorem 3 (for w_{i} (t_{i}) = 1, i = 1, 2, …, n) than those of Theorem 2.

Further, the tightest bounds from Theorem 3 occur when we choose to sample at the mid-points of the respective intervals, namely, for x_{i} = \frac{a_{i}+b_{i}}{2} . This is not the case for the weighted results depicted in Theorem 2. If we let

m (c, d) = ∫_{c}^{d} w (s) ds and M (c, d) = ∫_{c}^{d} sw (s) ds.

then we observe that, for example,

∫_{a}^{b} |x − t| w (t) dt = ∫_{a}^{x} (x − t) w (t) dt + ∫_{t}^{b} (t − x) w (t) dt
= x [m (a, x) − m (x, b)] + M (x, b) − M (a, x) .

Thus we see that the bound is simplified, although not necessarily globally minimised, at the median x = x^{∗}, where m (a, x^{∗}) = m (x^{∗}, b) .

3. Multidimensional Trapezoidal Identities and Bounds

The work of Section 2 used the weighted Ostrowski functional, which satisfies a weighted Montgomery identity, as a generator for extension to higher dimensions. We may prove, in an equivalent manner, utilising the generalised weighted trapezoidal identity as the generator of a higher dimensional result. We will restrict the current work to absolutely continuous functions so that the Riemann integral identity will be used. Let f : I^{n} → ℝ and define the operator

(3.1) G_{i} (f) := \frac{A_{i}}{W_{i}} f (t_{1}, …, t_{i−1}, a_{i}, t_{i+1}, …, t_{n})

+ \frac{B_{i}}{W_{i}} f (t_{1}, …, t_{i−1}, b_{i}, t_{i+1}, …, t_{n}) ,

where

(3.2)  A_{i} = ∫_{a_{i}}^{x_{i}} w_{i} (s) ds, B_{i} = ∫_{x_{i}}^{b_{i}} w_{i} (s) ds, W_{i} = A_{i} + B_{i}.

Here W_{i}G_{i} (f) represents the generalised weighted trapezoid in the i^{th} variable giving the standard trapezoid when x_{i} = \frac{a_{i}+b_{i}}{2} .

Now, for f (·) absolutely continuous in the i^{th} variable t_{i} ∈ [a_{i}, b_{i}] we have

(3.3) 𝔐_{i,w_{i}} (f) = \frac{1}{W_{i}} ∫_{a_{i}}^{b_{i}} Q_{i} (x_{i}, t_{i}) \frac{∂f}{∂t_{i}} dt_{i} = (G_{i} − λ_{i,w_{i}}) (f) , i = 1, 2, …, n,

where G_{i} (f) and λ_{i,w_{i}} (f) are as given by (3.1) – (3.2) and (2.2) respectively and,

(3.4)  \frac{Q_{i} (x_{i}, t_{i})}{W_{i}} = \frac{∫_{x_{i}}^{t_{i}} w_{i} (s) ds}{∫_{a_{i}}^{b_{i}} w_{i} (s) ds}, x_{i}, t_{i} ∈ [a_{i}, b_{i}] .

Let c^{(0)} = (c_{1}, c_{2}, …, c_{n}) , where c_{i} = a_{i} or b_{i} in the i^{th} position for i = 1, 2, …, n. Also, let σ_{0} (c^{(0)}) be the set of all such vectors which consists of 2^{n} possibilities. Further, let

(3.5)  χ_{k} = ∏_{j=1}^{n}^{(k)} \frac{C_{j}}{W_{j}} , k = 0, 1, …, n,

where C_{j} = A_{j} or B_{j} with the exception that k of the C_{j} = W_{j} and so \frac{C_{j}}{W_{j}} = 1.

In a similar fashion, let c^{(k)} be a vector taking on the fixed values a_{i} or b_{i} in the i^{th} position except for k of the positions which are variable, t_{•}. Let M_{k} be k−dimensional weighted integral means for f (c^{(k)}) . Here c^{(k)} ∈ σ_{k} (c^{(k)}) the set of all such elements, of which there are \binom{n}{k} 2^{n−k}.

With the above notation in place, the following theorem holds.

Theorem 4. Let f : I^{n} → ℝ be absolutely continuous and be such that all partial derivatives of order one in each of the variables exist. Then

(3.6)  R_{n} (f) = ∑_{0} χ_{0}f (c^{(0)}) − ∑_{1} χ_{1}M_{1} + ∑_{2} χ_{2}M_{2}
− ⋯ − (−1)^{n−1} ∑_{n−1} χ_{n−1}M_{n−1} + (−1)^{n} M_{n}

:= ρ_{n} (a_{∼}, x_{∼}, b_{∼}) ,

where, χ_{k} is as defined in (3.5), M_{k} is the k−dimensional integral mean for f (c^{(k)}) , specifically,

M_{n} = \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} w_{i} (t_{i}) f (t_{1}, t_{2}, …, t_{n}) dt_{1} … dt_{n}

and ∑_{k} is a sum involving each of the elements of σ_{k} (c^{(k)}) of which there are \binom{n}{k} 2^{n−k} terms.

Further,

(3.7)  R_{n} (f) = \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} Q_{i} (x_{i}, t_{i}) \frac{∂^{n}f}{∂t_{n} ⋯ ∂t_{1}} dt_{1} ⋯ dt_{n},

and Q_{i} (x_{i}, t_{i}) is given by (3.4), W^{∗} by (2.8). Here, c_{i} is equal to either a_{i} or b_{i} in which case C_{i} = A_{i} or B_{i}.

Proof. Let R_{r} (f) be defined by

(3.8)  R_{r} (f) := (∏_{i=1}^{r} 𝔐_{i,w_{i}}) (f) ,

then from the left identity in (3.3), R_{n} (f) is as given by (3.7). Now,

(3.9)  R_{r} (f) = 𝔐_{r,w_{r}} (R_{r−1} (f)) , for r = 1, 2, …, n,

where R_{0} (f) = f.

Thus, from (3.8)

R_{1} (f) = 𝔐_{1,w_{1}} (f) = (G_{1} − λ_{1,w_{1}}) (f) ,

which is the generalised trapezoidal identity for t_{1}, x_{1} ∈ [a_{1}, b_{1}]

(3.10)  R_{1} (f) = \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} Q_{1} (x_{1}, t_{1}) \frac{∂f}{∂t_{1}} (t_{1}, t_{2}, …, t_{n}) dt_{1}

= \frac{A_{1}}{W_{1}} f (a_{1}, t_{2}, …, t_{n}) + \frac{B_{1}}{W_{1}} f (b_{1}, t_{2}, …, t_{n})

− \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, t_{2}, …, t_{n}) dt_{1}

contains three entities; two function evaluations and one integral. Further,

R_{2} (f) = 𝔐_{2,w_{2}} (R_{1} (f)) = (G_{2} − λ_{2,w_{2}}) (R_{1} (f))

= G_{2} (R_{1} (f)) − λ_{2,w_{2}} (R_{1} (f))

= \frac{A_{2}}{W_{2}} R_{1} (f)|_{t_{2}=a_{2}} + \frac{B_{2}}{W_{2}} R_{1} (f)|_{t_{2}=b_{2}} − \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) R_{1} (f) dt_{2}

contains nine entities. Thus,

(3.11) R_{2} (f) = \frac{A_{2}}{W_{2}} \{\frac{A_{1}}{W_{1}} f (a_{1}, a_{2}, t_{3}, …, t_{n}) + \frac{B_{1}}{W_{1}} f (b_{1}, a_{2}, t_{3}, …, t_{n})

− \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, a_{2}, t_{3}, …, t_{n}) dt_{1}\}

+ \frac{B_{2}}{W_{2}} \{\frac{A_{1}}{W_{1}} f (a_{1}, b_{2}, t_{3}, …, t_{n}) + \frac{B_{1}}{W_{1}} f (b_{1}, b_{2}, t_{3}, …, t_{n})

− \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, b_{2}, t_{3}, …, t_{n}) dt_{1}\}

− \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) \{\frac{A_{1}}{W_{1}} f (a_{1}, t_{2}, …, t_{n}) + \frac{B_{1}}{W_{1}} f (b_{1}, t_{2}, t_{3}, …, t_{n})

− \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, t_{2}, …, t_{n}) dt_{1}\} dt_{2}

= \frac{A_{2}}{W_{2}} \frac{A_{1}}{W_{1}} f (a_{1}, a_{2}, t_{3}, …, t_{n}) + \frac{A_{2}}{W_{2}} \frac{B_{1}}{W_{1}} f (b_{1}, a_{2}, t_{3}, …, t_{n})

+ \frac{B_{2}}{W_{2}} \frac{A_{1}}{W_{1}} f (a_{1}, b_{2}, t_{3}, …, t_{n}) + \frac{B_{2}}{W_{2}} \frac{B_{1}}{W_{1}} f (b_{1}, b_{2}, t_{3}, …, t_{n})

− \frac{A_{2}}{W_{2}} · \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, a_{2}, t_{3}, …, t_{n}) dt_{1}

− \frac{B_{2}}{W_{2}} · \frac{1}{W_{1}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) f (t_{1}, b_{2}, t_{3}, …, t_{n}) dt_{1}

− \frac{A_{1}}{W_{1}} · \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) f (a_{1}, t_{2}, …, t_{n}) dt_{2}

− \frac{B_{1}}{W_{1}} · \frac{1}{W_{2}} ∫_{a_{2}}^{b_{2}} w_{2} (t_{2}) f (b_{1}, t_{2}, t_{3}, …, t_{n}) dt_{2}

+ \frac{1}{W_{2}W_{1}} ∫_{a_{2}}^{b_{2}} ∫_{a_{1}}^{b_{1}} w_{1} (t_{1}) w_{2} (t_{2}) f (t_{1}, t_{2}, …, t_{n}) dt_{1}dt_{2}.

From the 3^{2} entities of R_{2} (f) there are 2^{2} function evaluations, 2×2 single integrals and one double integral

R_{3} (f) = 𝔐_{3,w_{3}} (R_{2} (f)) = (G_{3} − λ_{3,w_{3}}) (R_{2} (f)) = G_{3} (R_{2} (f)) − λ_{3,w_{3}} (R_{2} (f))
= \frac{A_{3}}{W_{3}} R_{2} (f)|_{t_{3}=a_{3}} + \frac{B_{3}}{W_{3}} R_{2} (f)|_{t_{3}=b_{3}} − \frac{1}{W_{3}} ∫_{a_{3}}^{b_{3}} w_{3} (f_{3}) R_{2} (f) dt_{3}.

This will produce 3^{2} entities with \binom{3}{0} 2^{3} function evaluations, \binom{3}{1} 2^{2} single integrals, \binom{3}{2} 2^{1} double integrals and \binom{3}{0} 2^{0} triple integrals. The 2 occurs since evaluation is at either the a_{i} or the b_{i}.

Continuing in this manner we obtain the result as stated where there are 3^{n} entities for R_{n} (f), with \binom{n}{0} 2^{n} function evaluations only, \binom{n}{1} 2^{n−1} single integrals,

\binom{n}{2} 2^{n−2} double integrals, ... , \binom{n}{n−1} 2, (n − 1)^{th} integrals and one n-dimensional integral.

The following theorem gives bounds for ρ_{n} (a_{∼}, x_{∼}, b_{∼}) , as defined in (3.6) for

\frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{p} [I^{n}], 1 ≤ p ≤ ∞ with the usual the Lebesgue norms. Following the proof of Theorem 2 closely, it may be shown that

Theorem 5. Let f : I^{n} → ℝ be absolutely continuous in a manner such that all partial derivatives of order one with respect to every variable exists. Then

(3.12) W^{∗} · |ρ_{n} (a_{∼}, x_{∼}, b_{∼})|

≤ \{ \begin{matrix}∏_{i=1}^{n} (∫_{a_{i}}^{b_{i}} |x_{i} − t_{i}| w_{i} (t_{i}) dt_{i}) ∥\frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∥_{∞} , & \frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∈ L_{∞} [I^{n}] ; \\ ( ∏_{i=1}^{n} Q_{i} (q))^{\frac{1}{q}} ∥\frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∥_{p} , & \frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∈ L_{p} [I^{n}] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1 \\ ∏_{i=1}^{n} ϕ_{i} ∥\frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∥_{1} , & \frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∈ L_{1} [I^{n}] ; \end{matrix}

where ρ_{n} (a_{∼}, x_{∼}, b_{∼}) is as defined by (2.13),

(3.13) Q_{i} (q) = ∏_{i=1}^{n} [∫_{a_{i}}^{x_{i}} (∫_{a_{i}}^{t_{i}} w_{i} (t_{i}) dt_{i})^{q} dt_{i} + ∫_{x_{i}}^{b_{i}} (∫_{t_{i}}^{b_{i}} w_{i} (t_{i}) dt_{i})^{q} dt_{i}]

(3.14)  θ_{i} = \frac{1}{2} ∫_{a_{i}}^{b_{i}} w_{i} (s) ds + \frac{1}{2} |∫_{a_{i}}^{x_{i}} w_{i} (s) ds − ∫_{x_{i}}^{b_{i}} w_{i} (s) ds| .

Remark 4. The bound for the unweighted result for |ρ_{n} (a_{∼}, x_{∼}, b_{∼})| may be shown

to be in agreement with that for |τ_{n} (a_{∼}, x_{∼}, b_{∼})| given in Theorem 3. It was shown in Cerone [7] that the bounds in terms of the Lebesgue norms on |S (f; c, x, d)| and |T (f; c, x, d)| , as defined in (1.1) and (1.2), are the same. This result carries across to weighted functionals which when used as generators would produce the same bounds since the kernels P_{i} (x_{i}, t_{i}) from (2.5) and Q_{i} (x_{i}, t_{i}) from (3.5) produce the same norms.

4. Three Point Identities and their Bounds

It may be noticed from (1.4) – (1.5) that the three point functional can be expressed in terms of a combination of Trapezoidal functionals, see [12] and [14] for treatment of three point weighted quadrature rules.

In particular, for f (·) absolutely continuous and using the results of Section 3 for weighted trapezoidal functionals as generators, we have the identity

(4.1)  Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼}) := ρ_{n} (a_{∼}, α_{∼}, x_{∼}) + ρ_{n} (x_{∼}, β_{∼}, b_{∼})

= ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} r_{i} (x_{i}, t_{i}) \frac{∂^{n}f}{∂t_{n} ⋯ ∂t_{1}} dt_{1} ⋯ dt_{n},

where

(4.2)  \frac{r_{i} (x_{i}, t_{i})}{W_{i}} = \{ \begin{matrix}\frac{∫_{α_{i}}^{t_{i}} w_{i} (t_{i}) dt_{i}}{W_{i}} , & t_{i} ∈ [a_{i}, x_{i}] \\ \frac{∫_{β_{i}}^{t_{i}} w_{i} (t_{i}) dt_{i}}{W_{i}} , & t_{i} ∈ (x_{i}, b_{i}] \end{matrix} .

It is important to obtain an identity for the three point rule since the bounds are tighter than using the bounds of the two trapezoidal rules as this would entail using

the triangle inequality. We notice that Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼}) in (4.1) is not expressed

explicitly. This may be accomplished by returning to (3.6) or else we may use the generator methodology utilised to obtain the results in Section 2 and 3.

Let f : I^{n} → ℝ and define the operator

(4.3) H_{i} (f) := \frac{ν_{i}^{(a)}}{W_{i}} f (t_{1}, …, t_{i−1}, a_{i}, t_{i+1}, …, t_{n})

+ \frac{ν_{i}^{(x)}}{W_{i}} f (t_{1}, …, t_{i−1}, x_{i}, t_{i+1}, …, t_{n})

+ \frac{ν_{i}^{(b)}}{W_{i}} f (t_{1}, …, t_{i−1}, b_{i}, t_{i+1}, …, t_{n}) ,

where,

(4.4)  ν_{i}^{(a)} = ∫_{a_{i}}^{α_{i}} w_{i} (t_{i}) dt_{i}, ν_{i}^{(x)} = ∫\begin{matrix}β & _{i} \\ α & i \end{matrix} w_{i} (t_{i}) dt_{i},

ν_{i}^{(b)} = ∫_{β_{i}}^{b_{i}} w_{i} (t_{i}) dt_{i}, W_{i} = ∫_{a_{i}}^{b_{i}} w_{i} (t_{i}) dt_{i}.

Then, for f (·) absolutely continuous in the i^{th} variable t_{i} ∈ [a_{i}, b_{i}] we have

(4.5)  𝔑_{i} (t) = \frac{1}{W_{i}} ∫_{a_{i}}^{b_{i}} r_{i} (x_{i}, t_{i}) \frac{∂f}{∂t_{i}} dt_{i} = (H_{i} − λ_{i,w_{i}}) (f) , i = 1, 2, …, n.

where H_{i} (f) and λ_{i,w_{i}} (f) are as given by (4.3) – (4.4) and (2.2) respectively.

If we now follow the work of the previous section and let c^{(0)} = (c_{1}, c_{2}, …, c_{n}), where now c_{i} = a_{i}, x_{i} or b_{i} in the i^{th} partition for i = 1, 2, …, n. Then σ_{0} (c^{(0)}) which is the set of all such vectors consists of 3^{n} possibilities. Further, let χ_{n} be as in (3.5) where now, C_{j} = ν_{j}^{(a)} or ν_{j}^{(x)} or ν_{j}^{(b)} are as defined by (4.4) with the exception that k of the C_{j} = W_{j} and so \frac{C_{j}}{W_{j}} = 1.

Further, c^{(k)} is a vector taking on fixed values of either a_{i}, x_{i} or b_{i} in the i^{th} position except for k of the positions which are variable, t_{•}. Let M_{k} be a k−dimensional weighted integral means for f (c^{(k)}) , then the following theorem holds.

14 P. CERONE

Theorem 6. Let f : I^{n} → ℝ be absolutely continuous and be such that all partial derivatives of order one in each of the variables exist. Then

(4.6)  B_{n} (f) = ∑_{0} χ_{0}f (c^{(0)}) − ∑_{1} χ_{1}M_{1} + ∑_{2} χ_{2}M_{2}
− ⋯ − (−1)^{n−1} ∑_{n−1} χ_{n−1}M_{n−1} + (−1)^{n} M_{n}

:= Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼}) ,

where

(4.7)  χ_{k} = ∏_{j=1}^{n}^{(k)} \frac{C_{j}}{W_{j}} , k = 0, 1, …, n,

with C_{j} = ν_{j}^{(a)}, ν_{j}^{(x)} or ν_{j}^{(b)} as defined in (4.4) except for k of the C_{j} = W_{j} giving \frac{C_{j}}{W_{j}} = 1, M_{k} is the k−dimensional weighted integral mean of f (c^{(k)}) and specifically,

(4.8) M_{n} = \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} ∏_{i=1}^{n} w_{i} (t_{i}) f (c^{(n)}) dt_{1} … dt_{n}, c^{(n)} = (t_{1}, t_{2}, …, t_{n}) .

Finally, ∑_{k} is a sum over all \binom{n}{k} 3^{n−k} terms and B_{n} (f) is as defined in (4.1). The following theorem gives bounds for the Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼}) as given in either

(4.6) or (4.1). Theorem 7. Let the conditions of Theorem 6 continue to hold. Then,

(4.9) W^{∗} |Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼})|

≤ \{ \begin{matrix}∏_{i=1}^{n} S_{i} (1) ∥\frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∥_{∞} , & \frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∈ L_{∞} [I^{n}] ; \\ ( ∏_{i=1}^{n} S_{i} (q))^{\frac{1}{q}} ∥\frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∥_{p} , & \frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∈ L_{∞} [I^{n}] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1 \\ ∏_{i=1}^{n} ζ_{i} ∥\frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∥_{1} , & \frac{∂^{n}f}{∂t_{n}···∂t_{1}} ∈ L_{1} [I^{n}] ; \end{matrix}

where ∥h∥_{p}, 1 ≤ p < ∞ and ∥h∥_{∞} are defined by (1.8) and (1.9), Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼}) is defined by (4.1) or, explicitly, by (4.6),

(4.10)  S_{i} (q) = ∫_{a_{i}}^{b_{i}} |r_{i} (x_{i}, t_{i})|^{q} dt_{i} and ζ_{i} = sup_{t_{i}∈[a_{i},b_{i}]} |r_{i} (x_{i}, t_{i})| .

Proof. (Sketch) From (4.1) – (4.2) and (4.6) we have

(4.11) Ψ_{n} (a_{∼}, α_{∼}, x_{∼}, β_{∼}, b_{∼}) = |B_{n} (f)|

≤ \frac{1}{W^{∗}} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} r_{i} (x_{i}, t_{i}) \frac{∂^{n}f}{∂t_{n} … ∂t_{1}} | dt_{1} … dt_{n}.

Now, for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{∞} [I^{n}] , then

(4.12)  W^{∗} |B_{n} (f)| ≤ ∥\frac{∂^{n}f}{∂t_{n} ⋯ ∂t_{1}} ∥_{∞} ∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} r_{i} (x_{i}, t_{i})| dt_{1} … dt_{n},

where ∥h∥_{∞} is defined by (1.9).

Moreover, using the Hölder inequality for multiple integrals, we have, from (4.11) for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{p} [I^{n}] , 1 < p < ∞,

(4.13) W^{∗} |B_{n} (f)| ≤ ∥\frac{∂^{n}f}{∂t_{n} ⋯ ∂t_{1}} ∥_{p} (∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} r_{i} (x_{i}, t_{i})|^{q} dt_{1} … dt_{n})^{\frac{1}{q}} ,

where

∫_{a_{n}}^{b_{n}} ⋯ ∫_{a_{1}}^{b_{1}} |∏_{i=1}^{n} r_{i} (x_{i}, t_{i})|^{q} dt_{1} … dt_{n} = ∏_{i=1}^{n} ∫_{a_{i}}^{b_{i}} |r_{i} (x_{i}, t_{i})|^{q} dt_{i}.

Finally, for \frac{∂^{n}f}{∂t_{n}...∂t_{1}} ∈ L_{1} [I^{n}] , we have from (4.11)

W^{∗} |B_{n} (f)| ≤ sup_{t_{∼}∈[a_{∼}, b_{∼}]} |∏_{i=1}^{n} r_{i} (x_{i}, t_{i})| ∥\frac{∂^{n}f}{∂t_{n} ⋯ ∂t_{1}} ∥_{1}

= ∏_{i=1}^{n} sup_{t_{i}∈[a_{i},b_{i}]} |r_{i} (x_{i}, t_{i})| ∥\frac{∂^{n}f}{∂t_{n} ⋯ ∂t_{1}} ∥_{1} ,

giving the third bound.

Remark 5. The bounds obtained above in Theorem 7 are the product of the bounds for the one dimensional integral results. These were studied extensively in Cerone and Dragomir [9]. It should further be noted that the three point results of the current section recaptures the generalised trapezoidal results of the previous section if we take α_{i} = β_{i} = x_{i}. In addition, the Ostrowski type results of Cerone [5] are recaptured if we take α_{i} = a_{i} and β_{i} = b_{i} and the weight functions w_{i} (t_{i}) ≡ 1, i = 1, …, n. The results of Cerone [6] would be recaptured if w_{i} (t_{i}) ≡ 1 from Theorem 6 and the results in (4.10) may be simplified further.

5. Concluding Remarks

Weighted rules of Ostrowski, Trapezoidal and Three-point type have been investigated in the current work as generators for multidimensional integration. This results in product form weight functions in the multidimensional integral. The procedure developed in [5] and [6] may also be used to include higher order formulae involving the behaviour of higher derivatives for its bounds. Multidimensional results based on an m branched Peano kernel producing function evaluations at m+1 points are also possible using the methodology. Finally, we are not restricted to using the same identity in each of the directions but may use different ones as long as we are able to justify this.

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[3] N.S. BARNETT and S.S. DRAGOMIR, A trapezoid type inequality for double integrals, RGMIA Research Report Collection, 3(4) (2000), Article 5. [ONLINE

http://rgmia.vu.edu.au/v3n4.html]

[4] N.S. BARNETT and S.S. DRAGOMIR, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27(1) (2001), 1-10.

[5] P. CERONE, Approximate multidimensional integration through dimension reduction via the Ostrowski functional, Nonlinear Funct. Anal. & Applics., 8(3) (2003), 313-333.

[6] P. CERONE, Multidimensional integration via trapezoidal and three point generators, J. Korean Math. Soc., 40(2) (2000), 251-272.

[7] P. CERONE, On relationships between Ostrowski, trapezoidal and Chebychev identities and inequalities, Soochow J. Math., 28(3) (2002), 311-328.

[8] P. CERONE and S.S. DRAGOMIR, On some inequalities arising from Montgomery’s identity, J. of Computational Analysis and Applications, 5(4) (2003), 341-367.

[9] P. CERONE and S.S. DRAGOMIR, Three point quadrature rules involving, at most, a first derivative, RGMIA Res. Rep. Coll., 2(4) (1999), Article 8. [ONLINE]

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[21] B.G. PACHPATTE, On an inequality of Ostrowski type in three independent variables, J. Math. Anal. & Applics., 249 (2000), 583-591.

School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City MC, Victoria 8001, Australia.

E-mail address: pietro.cerone@vu.edu.au
URL: http://rgmia.vu.edu.au/cerone