Finding The Norm View of The Optimal Formula of The Quadratury Error Function in Gilbert Space
Abstract
The implementation of quadrature formulas serves as an essential tool for numerical integration because they convert definite integrals into discrete summation approximations. Numerical analysis depends on the Euler-Maclaurin quadrature formula for wide scientific use since it enables computational error determination and correction work. The application of quadrature formulas within Hilbert space needs optimal configurations because they reduce approximation errors. These mathematical formulas have been applied in numerous spaces by previous researchers although additional development is required to elevate both computational speed and accuracy levels. Existing research on optimal quadrature formulas fails to establish the specific norm of error functional within Hilbert space using an extended Euler-Maclaurin formula. The current need exists to develop an optimal formulation that reduces errors successfully yet keeps mathematical precision intact. The study constructs an optimal quadrature formula in Hilbert space through determination of the error functional norm using Riesz’s theorem and extremum function theory. The study establishes these optimized coefficients to achieve precision improvements in numerical quadrature computations. A new improved quadrature formula emerges from this research which satisfies Sard’s problem definitions. Through explicit and mathematical derivations the norm of the error functional becomes clear as it demonstrates achieved minimum error bounds. The research introduces a mathematically founded quadrature formula specialized for Hilbert spaces to achieve error reduction in integration methods. The implementation of Riesz’s theorem together with extremum function theory delivers a new method to construct error functionals. New mathematical discoveries from the study enhance numerical integration techniques thus improving the precision of computational mathematics and physical as well as engineering applications' quadrature formulas. The proposed integration formula provides substantial improvements for minimizing errors so it enhances theoretical computational and applied research results.
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