Optimal quadrature evaluation of integrals of rapidly oscillating functions in Lipschitz interpolation class in case of
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OPTIMAL QUADRATURE EVALUATION OF INTEGRALS OF RAPIDLY OSCILLATING FUNCTIONS IN LIPSCHITZ INTERPOLATION CLASS IN CASE OF STRONG OSCILLATION V. K. Zadiraka,a S. S. Melnikova,a and L. V. Lutsa
UDC 519.64: 517.443: 519.254-37
An accuracy-optimal quadrature formula is derived to evaluate the Fourier transform of compact functions from an interpolation Lipschitz class. The case of strong oscillation of the integrand is considered. The optimality is substantiated based on the boundary function method, namely, constructing the Chebyshev center and Chebyshev radius in the uncertainty domain of the problem solution. Keywords: rapidly oscillating functions, accuracy-optimal algorithm.
In solving problems of classes such as statistical processing of experimental data, digital filtration, pattern recognition, modeling of optical systems and synthesized holograms, boundary-value problems for partial differential equations, etc., the need arises to evaluate integrals [1] 1
1
I 1 ( w ) = ò f ( x )sin wxdx, I 2 ( w ) = ò f ( x )cos wxdx, 0
(1)
0
where f ( x ) Î F (F is a given class of functions), w is an arbitrary real number, | w| ³ 2p, and the information about f ( x ) is available at no more than N points. In the present study, which continues and develops the results obtained in [2], we will find optimal estimates [3] and derive accuracy-optimal quadrature formulas to evaluate the Fourier transform of compact functions (1) on the assumption that f ( x ) Î C L, N , C L, N is an interpolation class of functions that satisfy the Lipschitz property (| f ( x1 ) - f ( x 2 )| £ L| x1 - x 2 | , x1 , x 2 Î [ 0, 1]) and are defined by 2N fixed values { f i }0N -1 and {x i }0N -1 . Such a way of presenting the initial information can be used to narrow the class F onto the class F N . It brings closer to the real situation that arises in solving a specific problem [3]. The paper considers the case of strong oscillation of the integrand: | w| ³ 2p and N £ | w| . Denote by R = R ( f , A , w ) the result of the approximate evaluation of I ( w ) by the quadrature formula À. Let us consider the following characteristics: d( f , A , { f i }0N -1 , w ) = r( I , R ) , d( F N , A , w ) = sup d( f , A , { f i }0N -1 , w ) ,
(2)
f ÎFN
d = d( F N , w ) = inf d( F N , A , w ) , A
where r( I , R ) is the numerical integration error: r( I , R ) = | I ( w ) - R | . The quadrature formula A * on which the optimal estimate d is attained will be called accuracy-optimal. If d( F N , A *, N , w ) £ d + h for the quadrature formula A * , then we will call A * accuracy optimal up to h. If h = o[ d], O[ d] , a
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 105–112, March–April 2010. Original article submitted May 19, 2009. 264
1060-0396/10/4602-0264
©
2010 Springer Science+Business Media, Inc.
then we will call A * asymptotically optimal or order-optimal, respectively. To derive and substantiate accuracy-optimal and close
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