# ATS theorem

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.

Contents 1 History of the problem 2 Certain notations 3 ATS theorem 4 Van der Corput lemma 5 Remark 6 Notes History of the problem In some fields of mathematics and mathematical physics, sums of the form {displaystyle S=sum _{a0,Bto +infty ,} or {displaystyle Bto 0,} the record {displaystyle 1ll {frac {A}{B}}ll 1} means that there are the constants {displaystyle C_{1}>0} and {displaystyle C_{2}>0,} such that {displaystyle C_{1}leq {frac {|A|}{B}}leq C_{2}.} . For a real number {displaystyle alpha ,} the record {displaystyle |alpha |} means that {displaystyle |alpha |=min({alpha },1-{alpha }),} where {displaystyle {alpha }} is the fractional part of {displaystyle alpha .} ATS theorem Let the real functions ƒ(x) and {displaystyle varphi (x)} satisfy on the segment [a, b] the following conditions: 1) {displaystyle f''''(x)} and {displaystyle varphi ''(x)} are continuous; 2) there exist numbers {displaystyle H,} {displaystyle U} and {displaystyle V} such that {displaystyle H>0,qquad 1ll Ull V,qquad 0

Si quieres conocer otros artículos parecidos a ATS theorem puedes visitar la categoría Theorems in analysis.

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