On dual Euler-Simpson formulae (Q5952929)

A simple quadrature rule associated with Simpson's rule is the Three Point Rule, called here the Dual Simpson Rule, NEWLINE\[NEWLINE\int_a^bf= D(a,b; f) + {{7(b-a)^3}\over{23040}}f^{(4)}( \eta),NEWLINE\]NEWLINE where NEWLINE\[NEWLINED(a,b; f)= {{b-a}\over{3}} \Biggl(2f\biggl({{3a+b}\over{4}}\biggr)- f\biggl({{a+b}\over{2}}\biggr) +f \biggl({{a+3b}\over{4}}\biggr)\Biggr),NEWLINE\]NEWLINE the formula being valid if \(f\in{\mathcal C}^{(4)}(ab)\). The basic idea in this paper is to introduce a term from the Euler interpolation formula, and obtain results under weaker conditions; NEWLINE\[NEWLINE\int_a^bf= D(a,b; f-T_m) + R_m(a,b)NEWLINE\]NEWLINE where NEWLINE\[NEWLINET_m(x) = \sum_{k=1}^m {{(b-a)^{k-1}}\over{k!}}B_k \biggl({{x-a}\over{b-a}}\biggr) \bigl(f^{k-1}(b)-f^{k-1}(a)\bigr),NEWLINE\]NEWLINE \(B_k\) being the Bernoulli polynomial of degree \(k\), NEWLINE\[NEWLINER_m(a,b)= {{(b-a)^n}\over{3 n!}}\int_a^b F_m\biggl({{t-a}\over{b-a}}\biggr) df^{(n-1)}(t)NEWLINE\]NEWLINE where if \(m=n\), NEWLINE\[NEWLINEF_n(t) = G_n(t)= 2B_n(-t+1/4)- B_n(-t+ 1/2) + 3B_n(-t+3/4),NEWLINE\]NEWLINE and if \(m=n-1\), NEWLINE\[NEWLINEF_{n-1}(t)= G_n(t) - B_n(t);NEWLINE\]NEWLINE here the Bernoulli polynomials are extended to \(\mathbb R\) by periodicity. These results hold if \(f^{(n-1)}\) is continuous and of bounded variation. Inequalities are obtained for the two functions \( G_n, F_{n-1}\) which are used to estimate the remainder terms when \(f^{(n-1)}\) is Lipschitz continuous of bounded variation.