Exponentially weighted optimal quadrature formulas with complex exponential weights in the periodic Sobolev space ^ W (2,1,0) 2 (0,1]
This paper is devoted to the construction of optimal quadrature formulas for the approximate
evaluation of integrals of periodic functions in the Sobolev space
^
W
(2,1,0)
2
(0,1]. The quadrature
formulas involve a complex exponential weight function e 2πiωx .The coefficients of the formulas are
obtained by minimizing the norm of the corresponding error functional in the conjugate space. Using
Fourier analysis and extremal function methods, explicit expressions for the optimal coefficients are
derived. These results extend the classical theory of quadrature formulas to exponentially weighted
and oscillatory cases, yielding efficient schemes for the numerical integration of periodic functions.
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