Melnik, R.V.N. and Melnik, K.N.
Problems of numerical integration with fast oscillatory integrands constitute an intrinsic part of a wide class of challenging applied problems in digital signal processing, image recognition, complex systems control, quantum mechanics, and many other applied areas. In this paper we propose effective algorithms that guarantee optimal-by-order solution of the problem on computing integrals with fast oscillatory functions in interpolational classes that are typical in applications. The proposed algorithms provide an effective tool for numerical integration of highly oscillatory functions, especially in the case when a priori information about the problem is given inaccurately. Such a consideration puts us closer to situations which are typical in the solution of practical problems.
The developed formulae alow simple computational implementation which is an important feature for the design of software application packages. The importance of the design and implementation of optimal algorithms (and algorithms close to optimal with respect to their computational characteristics) is comparable with the importance of the design and implementation of new computer hardware. The algorithms and methods proposed in this paper offer a flexible choice of options depending on practical aspects of the problems to be solved and permit the application of these techniques in a wide variety of circumstances.
Key words: integration of fast oscillatory functions; digital signal processing and image recognition; limit functions method; Chebyshev radius; asymptotically optimal integration formulae; optimal-by-order and optimal-by-accuracy integration formulae; error estimation of numerical integration in interpolational classes; inaccurate a priori information.
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