# The order of convergence of an optimal quadrature formula with derivative in the space $W_2^{(2,1)}$

## TL;DR

This paper develops an optimal quadrature formula with derivatives in the space W_2^{(2,1)}, deriving explicit coefficients, proving exactness for specific functions, and demonstrating its lower error compared to Euler-Maclaurin in the same space.

## Contribution

The paper extends the trapezoidal rule to the space W_2^{(2,1)} by deriving explicit optimal coefficients and establishing its accuracy and error bounds.

## Key findings

- Explicit formulas for quadrature coefficients are obtained.
- The quadrature formula is exact for span{1, x, e^x, e^{-x}}.
- The error is shown to be less than Euler-Maclaurin in W_2^{(2,1)}.

## Abstract

The present work is devoted to extension of the trapezoidal rule in the space $W_2^{(2,1)}$. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of a integrand. Using the discrete analog of the operator $\frac{d^2}{dx^{2}}-1$ the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set $\mathbf{F}=\mathrm{span}\{1,x,e^{x},e^{-x}\}$. Finally, in the space $W_2^{(2,1)}$ the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space $L_2^{(2)}$.