ATS theorem
In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful. History of the problem. In some fields of mathematics and mathematical physics, sums of the form are under study. Here formula_2 and formula_3 are real valued functions of a real argument, and formula_4 Such sums appear, for example, in number theory in the analysis of the Riemann zeta function, in the solution of problems connected with integer points in the domains  on plane and in space, in the study of the Fourier series, and in the solution of such differential equations as the wave equation, the potential equation, the heat conductivity equation. The problem of approximation of the series (1) by a suitable function was studied already by Euler and Poisson. We shall define the length of the sum formula_5 to be the number formula_6 (for the integers formula_7 and formula_8 this is the number of the summands in formula_5). Under certain conditions on formula_2 and formula_3 the sum formula_5 can be substituted with good accuracy by another sum formula_13 where  the length formula_15 is far less than formula_16 First relations of the form where formula_18 formula_19 are the sums (1) and (2) respectively, formula_20 is a remainder term, with concrete functions formula_2 and formula_22 were obtained by G. H. Hardy and J. E. Littlewood, when they deduced approximate functional equation for the Riemann zeta function formula_23 and by I. M. Vinogradov, in the study of the amounts of integer points in the domains on plane. In general form the theorem was proved by J. Van der Corput, (on the recent results connected with the Van der Corput theorem one can  read at In every one of the above-mentioned works, some restrictions on the functions formula_2 and formula_3 were imposed. With convenient (for applications) restrictions on formula_2 and formula_22 the theorem was proved by A. A. Karatsuba in (see also,). Certain notations. . "For" formula_28 "or" formula_29 "the record" . "For a real number" formula_34 "the record" formula_35 "means that" ATS theorem. "Let the real functions" "ƒ"("x") "and" formula_2 "satisfy on the segment" ["a", "b"] "the following conditions:" 1) formula_40 "and" formula_41 "are continuous;" 2) "there exist numbers" formula_42 formula_43 "and" formula_44 "such that" "Then, if we define the numbers  formula_47 from the equation" "we have" "where" formula_52 The most simple variant of the formulated theorem is the statement, which is called in the literature the Van der Corput lemma. Van der Corput lemma. "Let" formula_3 "be a real differentiable function in the interval" formula_56 "moreover, inside of this interval, its derivative" formula_57 "is a monotonic and a sign-preserving function, and for the constant" formula_58 "such that" formula_59 "satisfies the inequality" formula_60 "Then" "where" formula_62 Remark. If the parameters formula_7 and formula_64 are integers, then it is possible to substitute the last relation by the following ones: where formula_62  On the applications of ATS to the problems of physics see,; see also.