Peano kernel theorem

In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano.^[1]

Let V [ a , b ] {\displaystyle {\mathcal {V}}[a,b]} be the space of all functions f {\displaystyle f} that are differentiable on ( a , b ) {\displaystyle (a,b)} that are of bounded variation on [ a , b ] {\displaystyle [a,b]} , and let L {\displaystyle L} be a linear functional on V [ a , b ] {\displaystyle {\mathcal {V}}[a,b]} . Assume that that L {\displaystyle L} annihilates all polynomials of degree ≤ ν {\displaystyle \leq \nu } , i.e. L p = 0 , ∀ p ∈ P ν [ x ] . {\displaystyle Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }[x].} Suppose further that for any bivariate function g ( x , θ ) {\displaystyle g(x,\theta )} with g ( x , ⋅ ) , g ( ⋅ , θ ) ∈ C ν + 1 [ a , b ] {\displaystyle g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]} , the following is valid: L ∫ a b g ( x , θ ) d θ = ∫ a b L g ( x , θ ) d θ , {\displaystyle L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,} and define the Peano kernel of L {\displaystyle L} as k ( θ ) = L [ ( x − θ ) + ν ] , θ ∈ [ a , b ] , {\displaystyle k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],} using the notation ( x − θ ) + ν = { ( x − θ ) ν , x ≥ θ , 0 , x ≤ θ . {\displaystyle (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}} The Peano kernel theorem^[1][2] states that, if k ∈ V [ a , b ] {\displaystyle k\in {\mathcal {V}}[a,b]} , then for every function f {\displaystyle f} that is ν + 1 {\textstyle \nu +1} times continuously differentiable, we have L f = 1 ν ! ∫ a b k ( θ ) f ( ν + 1 ) ( θ ) d θ . {\displaystyle Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .}

Several bounds on the value of L f {\displaystyle Lf} follow from this result: | L f | ≤ 1 ν ! ‖ k ‖ 1 ‖ f ( ν + 1 ) ‖ ∞ | L f | ≤ 1 ν ! ‖ k ‖ ∞ ‖ f ( ν + 1 ) ‖ 1 | L f | ≤ 1 ν ! ‖ k ‖ 2 ‖ f ( ν + 1 ) ‖ 2 {\displaystyle {\begin{aligned}|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{1}\|f^{(\nu +1)}\|_{\infty }\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{\infty }\|f^{(\nu +1)}\|_{1}\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{2}\|f^{(\nu +1)}\|_{2}\end{aligned}}}

where ‖ ⋅ ‖ 1 {\displaystyle \|\cdot \|_{1}} , ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} and ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }} are the taxicab, Euclidean and maximum norms respectively.^[2]

In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all f ∈ P ν {\displaystyle f\in \mathbb {P} _{\nu }} . The theorem above follows from the Taylor polynomial for f {\displaystyle f} with integral remainder:

f ( x ) = f ( a ) + ( x − a ) f ′ ( a ) + ( x − a ) 2 2 f ″ ( a ) + ⋯ ⋯ + ( x − a ) ν ν ! f ( ν ) ( a ) + 1 ν ! ∫ a x ( x − θ ) ν f ( ν + 1 ) ( θ ) d θ , {\displaystyle {\begin{aligned}f(x)=f(a)+{}&(x-a)f'(a)+{\frac {(x-a)^{2}}{2}}f''(a)+\cdots \\[6pt]&\cdots +{\frac {(x-a)^{\nu }}{\nu !}}f^{(\nu )}(a)+{\frac {1}{\nu !}}\int _{a}^{x}(x-\theta )^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}}

defining L ( f ) {\displaystyle L(f)} as the error of the approximation, using the linearity of L {\displaystyle L} together with exactness for f ∈ P ν {\displaystyle f\in \mathbb {P} _{\nu }} to annihilate all but the final term on the right-hand side, and using the ( ⋅ ) + {\displaystyle (\cdot )_{+}} notation to remove the x {\displaystyle x} -dependence from the integral limits.^[3]

• Divided differences