ATC theorem

ATS theorem - a theorem on the approximation of a shorter trigonometric sum.

In some areas of mathematics and mathematical physics, sums of the form

S=\sum _{a<k\leq b}\varphi (k)e^{2\pi if(k)}~~~(1).

{\ displaystyle S = \ sum _ {a <k \ leq b} \ varphi (k) e ^ {2 \ pi if (k)} ~~~ (1).}

S = \ sum_ {a <k \ le b} \ varphi (k) e ^ {2 \ pi i f (k)} ~~~ (1).

Here $\varphi (x)$ ${\ displaystyle \ varphi (x)}$ $\ varphi (x)$ and $f(x)$ ${\ displaystyle f (x)}$ $f (x)$ - real functions of the real argument, $i^{2}=-1.$ ${\ displaystyle i ^ {2} = - 1.}$ $i ^ 2 = -1.$

Such sums appear, for example, in number theory in the analysis of the Riemann zeta function , in solving problems related to the distribution of integer points in various regions on the plane and in space, in studying Fourier series , in solving differential equations such as the wave equation , the heat equation etc.

Introductory remarks

We call the length of the sum $S$ ${\ displaystyle S}$ $S$ number $b-a$ ${\ displaystyle ba}$ $b-a$ (for whole $a$ ${\ displaystyle a}$ $a$ and $b$ ${\ displaystyle b}$ $b$ it's just the number of terms in $S$ ${\ displaystyle S}$ $S$ )

We will use the following notation:

At $B>0,B\to +\infty$ ${\ displaystyle B> 0, B \ to + \ infty}$ $B > 0, B \to +\infty$ or $B\to 0$ ${\ displaystyle B \ to 0}$ $B \to 0$ record $1\ll {\frac {A}{B}}\ll 1$ ${\ displaystyle 1 \ ll {\ frac {A} {B}} \ ll 1}$ $1 \ll \frac{A}{B} \ll 1$ means there are constants $C_{1}>0$ ${\ displaystyle C_ {1}> 0}$ $C_1 > 0$ and $C_{2}>0,$ ${\ displaystyle C_ {2}> 0,}$ $C_2 > 0,$ such that
$C_{1}\leq {\frac {|A|}{B}}\leq C_{2}$ ${\ displaystyle C_ {1} \ leq {\ frac {| A |} {B}} \ leq C_ {2}}$ $C_1 \leq\frac{|A|}{B} \leq C_2$ .
For material $\alpha$ ${\ displaystyle \ alpha}$ $\alpha$ record $||\alpha ||$ ${\ displaystyle || \ alpha ||}$ $||\alpha||$ means what
$||\alpha ||=\min(\{\alpha \},1-\{\alpha \})$ ${\ displaystyle || \ alpha || = \ min (\ {\ alpha \}, 1 - \ {\ alpha \})}$ $||\alpha|| = \min(\{\alpha\},1- \{\alpha\})$ ,
Where $\{\alpha \}$ ${\ displaystyle \ {\ alpha \}}$ $\{\alpha\}$ - fraction $\alpha .$ ${\ displaystyle \ alpha.}$ $\alpha.$

We formulate the main theorem on replacing the trigonometric (sometimes also called exponential) sum by a shorter one.

ATS Theorem

Let real functions $f(x)$ ${\ displaystyle f (x)}$ and $\varphi (x)$ ${\ displaystyle \ varphi (x)}$ satisfy on the segment $[a,b]$ ${\ displaystyle [a, b]}$ the following conditions:

$f''(x)$ ${\ displaystyle f '' (x)}$ and $\varphi ''(x)$ ${\ displaystyle \ varphi '' (x)}$ are continuous;
there are numbers $H$ ${\ displaystyle H}$ , $U$ ${\ displaystyle U}$ and $V$ ${\ displaystyle V}$ such that
$H>0,~~1\ll U\ll V,~~0<b-a\leq V$ ${\ displaystyle H> 0, ~~ 1 \ ll U \ ll V, ~~ 0 <ba \ leq V}$
${\begin{array}{rc}{\frac {1}{U}}\ll f''(x)\ll {\frac {1}{U}}\ ,&\varphi (x)\ll H,\\f'''(x)\ll {\frac {1}{UV}}\ ,&\varphi '(x)\ll {\frac {H}{V}},\\f''''(x)\ll {\frac {1}{UV^{2}}}\ ,&\varphi ''(x)\ll {\frac {H}{V^{2}}}.\\\end{array}}$ ${\ displaystyle {\ begin {array} {rc} {\ frac {1} {U}} \ ll f '' (x) \ ll {\ frac {1} {U}} \, & \ varphi (x) \ ll H, \\ f '' '(x) \ ll {\ frac {1} {UV}} \, & \ varphi' (x) \ ll {\ frac {H} {V}}, \\ f '' '' (x) \ ll {\ frac {1} {UV ^ {2}}} \, & \ varphi '' (x) \ ll {\ frac {H} {V ^ {2}}}. \\\ end {array}}}$

Then, determining the numbers $x_{\mu }$ ${\ displaystyle x _ {\ mu}}$ from the equation

f'(x_{\mu })=\mu ,

{\ displaystyle f '(x _ {\ mu}) = \ mu,}

we have

\sum _{a<\mu \leq b}\varphi (\mu )e^{2\pi if(\mu )}=\sum _{f'(a)\leq \mu \leq f'(b)}C(\mu )Z(\mu )+R,

{\ displaystyle \ sum _ {a <\ mu \ leq b} \ varphi (\ mu) e ^ {2 \ pi if (\ mu)} = \ sum _ {f '(a) \ leq \ mu \ leq f '(b)} C (\ mu) Z (\ mu) + R,}

Where

R=O\left({\frac {HU}{b-a}}+HT_{a}+HT_{b}+H\log \left(f'(b)-f'(a)+2\right)\right);

{\ displaystyle R = O \ left ({\ frac {HU} {ba}} + HT_ {a} + HT_ {b} + H \ log \ left (f '(b) -f' (a) +2 \ right) \ right);}

T_{j}=\left\{{\begin{array}{rc}0\ ,&\ {{\textit {i}}f}\ \ f'(j)\ {{\textit {i}}s\ an\ integer};\\\min \left({\frac {1}{||f'(j)||}},{\sqrt {U}}\right)\ ,&\ {{\textit {i}}f}\ \ ||f'(j)||\neq 0;\\\end{array}}\right.

{\ displaystyle T_ {j} = \ left \ {{\ begin {array} {rc} 0 \, & \ {{\ textit {i}} f} \ \ f '(j) \ {{\ textit {i }} s \ an \ integer}; \\\ min \ left ({\ frac {1} {|| f '(j) ||}}, {\ sqrt {U}} \ right) \, & \ { {\ textit {i}} f} \ \ || f '(j) || \ neq 0; \\\ end {array}} \ right.}

j=a,b;

{\ displaystyle j = a, b;}

C(\mu )=\left\{{\begin{array}{rc}1\ \ ,&\ \ {{\textit {i}}f}\ \ f'(a)<\mu <f'(b);\\{\frac {1}{2}}\ \ ,&\ \ {{\textit {i}}f}\ \ \mu =f'(a)\ \ {{\textit {o}}r}\ \ \mu =f'(b);\\\end{array}}\right.

{\ displaystyle C (\ mu) = \ left \ {{\ begin {array} {rc} 1 \ \, & \ \ {{\ textit {i}} f} \ \ f '(a) <\ mu < f '(b); \\ {\ frac {1} {2}} \ \, & \ \ {{\ textit {i}} f} \ \ \ mu = f' (a) \ \ {{\ textit {o}} r} \ \ \ mu = f '(b); \\\ end {array}} \ right.}

Z(\mu )={\frac {1+i}{\sqrt {2}}}{\frac {\varphi (x_{\mu })}{\sqrt {f''(x_{\mu })}}}e^{2\pi i(f(x_{\mu })-\mu x_{\mu })}\ .

{\ displaystyle Z (\ mu) = {\ frac {1 + i} {\ sqrt {2}}} {\ frac {\ varphi (x _ {\ mu})} {\ sqrt {f '' (x _ {\ mu})}}} e ^ {2 \ pi i (f (x _ {\ mu}) - \ mu x _ {\ mu})} \.}

Van der Corpute Lemma

The simplest version of the formulated theorem is a statement called the van der Corpute lemma in the literature.

Let be $f(x)$ ${\ displaystyle f (x)}$ Is a real differentiable function on the interval $a<x\leq b$ ${\ displaystyle a <x \ leq b}$ , in addition, within this interval its derivative $f'(x)$ ${\ displaystyle f '(x)}$ is a monotonous and constant function, and when $\delta =const$ ${\ displaystyle \ delta = const}$ , $0<\delta <1$ ${\ displaystyle 0 <\ delta <1}$ satisfies the inequality

|f'(x)|\leq \delta .

{\ displaystyle | f '(x) | \ leq \ delta.}

Then

\sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}dx+\theta \left(3+{\frac {2\delta }{1-\delta }}\right),

{\ displaystyle \ sum _ {a <k \ leq b} e ^ {2 \ pi if (k)} = \ int _ {a} ^ {b} e ^ {2 \ pi if (x)} dx + \ theta \ left (3 + {\ frac {2 \ delta} {1- \ delta}} \ right),}

Where $|\theta |\leq 1.$ ${\ displaystyle | \ theta | \ leq 1.}$

If the parameters $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ are integers, then the last expression can be replaced by this:

\sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}dx+{\frac {1}{2}}e^{2\pi if(b)}-{\frac {1}{2}}e^{2\pi if(a)}+\theta {\frac {2\delta }{1-\delta }},

{\ displaystyle \ sum _ {a <k \ leq b} e ^ {2 \ pi if (k)} = \ int _ {a} ^ {b} e ^ {2 \ pi if (x)} dx + {\ frac {1} {2}} e ^ {2 \ pi if (b)} - {\ frac {1} {2}} e ^ {2 \ pi if (a)} + \ theta {\ frac {2 \ delta} {1- \ delta}},}

Where $|\theta |\leq 1$ ${\ displaystyle | \ theta | \ leq 1}$ .

Application

For applications of automatic telephone exchanges in physics, see ^[1] , ^[2] , see also ^[3] , ^[4] .

History

The problem of approximating a trigonometric series by any suitable function was already considered by Euler and Poisson .

Under certain conditions on $\varphi (x)$ ${\ displaystyle \ varphi (x)}$ and $f(x)$ ${\ displaystyle f (x)}$ the amount $S$ ${\ displaystyle S}$ can be replaced with good accuracy by another sum $S_{1},$ ${\ displaystyle S_ {1},}$

S_{1}=\sum _{\alpha <k\leq \beta }\Phi (k)e^{2\pi iF(k)},

{\ displaystyle S_ {1} = \ sum _ {\ alpha <k \ leq \ beta} \ Phi (k) e ^ {2 \ pi iF (k)},}

whose length $\beta -\alpha$ ${\ displaystyle \ beta - \ alpha}$ much less than $b-a.$ ${\ displaystyle ba.}$ The first relations of the form

S=S_{1}+R

{\ displaystyle S = S_ {1} + R}

Where $R$ ${\ displaystyle R}$ - residual member, with specific functions $\varphi (x)$ ${\ displaystyle \ varphi (x)}$ and $f(x),$ ${\ displaystyle f (x),}$ were obtained by G. Hardy and J. Littlewood ^[5] , ^[6] , ^[7] when deriving the functional equation for the Riemann zeta function $\zeta (s)$ ${\ displaystyle \ zeta (s)}$ and I. Vinogradov ^[8] , when considering the number of integer points in areas on the plane. In general, the theorem was proved by J. Van der Corpute ^[9] , ^[10] (recent results related to the Van der Corpute theorem can be found in ^[11] ).

In each of the above work on the function $\varphi (x)$ ${\ displaystyle \ varphi (x)}$ and $f(x)$ ${\ displaystyle f (x)}$ some restrictions were imposed. With restrictions convenient for applications, the theorem was proved by A. A. Karatsuba in ^[12] (see also ^[13] , ^[14] ).

Notes

↑ EA Karatsuba Approximation of sums of oscillating summands in certain physical problems, - JMP 45:11 , pp. 4310-4321 (2004).
↑ EA Karatsuba On an approach to the study of the Jaynes-Cummings sum in quantum optics, - Numerical Algorithms, Vol. 45, No.1-4, pp. 127-137 (2007).
↑ E. Chassande-Mottin, A. Pai Best chirplet chain: near-optimal detection of gravitational wave chirps, - Phys. Rev. D 73: 4 , 042003, pp. 1-23 (2006).
↑ M. Fleischhauer, WP Schleich Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model, - Phys. Rev. A 47: 3 , pp. 4258-4269 (1993).
↑ GH Hardy and JE Littlewood The trigonometrical series associated with the elliptic θ-functions, - Acta Math. 37 , pp. 193–239 (1914).
↑ GH Hardy and JE Littlewood Contributions to the theory of Riemann Zeta-Function and the theory of the distribution of primes, - Acta Math. 41 , pp. 119-196 (1918).
↑ GH Hardy and JE Littlewood The zeros of Riemann's zeta-function on the critical line, - Math. Z., 10 , pp. 283-317 (1921).
↑ I. M. Vinogradov On the average value of the number of classes of purely root forms of the negative determinant, - Communication. Kharkiv. Mat. Islands, vol. 16, No. 1/2, pp. 10–38 (1918).
↑ JG Van der Corput Zahlentheoretische Abschätzungen, - Math. Ann. 84 , pp. 53-79 (1921).
↑ JG Van der Corput Verschärfung der abschätzung beim teilerproblem, - Math. Ann., 87 , pp. 39-65 (1922).
↑ HL Montgomery Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, - Am. Math. Soc., 1994.
↑ AA Karatsuba Approximation of exponential sums by shorter ones, - Proc. Indian. Acad. Sci. (Math. Sci.) 97: 1-3 , pp. 167-178 (1987).
↑ S. M. Voronin, A. A. Karatsuba The Riemann Zeta Function, - M .: Fizmatlit, 1994.
↑ A. A. Karatsuba, M. A. Korolev A theorem on the approximation of a trigonometric sum shorter, - Izvestiya RAS. Series of Mathematics, vol. 71, No. 2, p. 123-150 (2007).

[1] EA Karatsuba Approximation of sums of oscillating summands in certain physical problems, - JMP 45:11 , pp. 4310-4321 (2004).

[2] EA Karatsuba On an approach to the study of the Jaynes-Cummings sum in quantum optics, - Numerical Algorithms, Vol. 45, No.1-4, pp. 127-137 (2007).

[3] E. Chassande-Mottin, A. Pai Best chirplet chain: near-optimal detection of gravitational wave chirps, - Phys. Rev. D 73: 4 , 042003, pp. 1-23 (2006).

[4] M. Fleischhauer, WP Schleich Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model, - Phys. Rev. A 47: 3 , pp. 4258-4269 (1993).

[5] GH Hardy and JE Littlewood The trigonometrical series associated with the elliptic θ-functions, - Acta Math. 37 , pp. 193–239 (1914).

[6] GH Hardy and JE Littlewood Contributions to the theory of Riemann Zeta-Function and the theory of the distribution of primes, - Acta Math. 41 , pp. 119-196 (1918).

[7] GH Hardy and JE Littlewood The zeros of Riemann's zeta-function on the critical line, - Math. Z., 10 , pp. 283-317 (1921).

[8] I. M. Vinogradov On the average value of the number of classes of purely root forms of the negative determinant, - Communication. Kharkiv. Mat. Islands, vol. 16, No. 1/2, pp. 10–38 (1918).

[9] JG Van der Corput Zahlentheoretische Abschätzungen, - Math. Ann. 84 , pp. 53-79 (1921).

[10] JG Van der Corput Verschärfung der abschätzung beim teilerproblem, - Math. Ann., 87 , pp. 39-65 (1922).

[11] HL Montgomery Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, - Am. Math. Soc., 1994.

[12] AA Karatsuba Approximation of exponential sums by shorter ones, - Proc. Indian. Acad. Sci. (Math. Sci.) 97: 1-3 , pp. 167-178 (1987).

[13] S. M. Voronin, A. A. Karatsuba The Riemann Zeta Function, - M .: Fizmatlit, 1994.

[14] A. A. Karatsuba, M. A. Korolev A theorem on the approximation of a trigonometric sum shorter, - Izvestiya RAS. Series of Mathematics, vol. 71, No. 2, p. 123-150 (2007).