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Московский Доклад теорема о приближении тригонометрической суммы более короткой (ATS).

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??????????? ???????
??? 16 ?????? 1 (2015)
??????????????????????
??? 511
?????????? ??????: ??????? ?
??????????? ??????????????????
????? ????? ???????? (ATS)1
?. ?. ???????? (?. ??????)2
?????????
??????, ????????? ?? ???????? ?. ?. ?????? ? ?. ?. ???????? ???????? -??????????????? ?????????? ??????????? ???????????????? ???????????? ????? ?. ?. ?????????? 9 ?????? 2006 ?.
???????? ?????: ?????????????????? ?????, ???????????, ???????
????????, ??????? ???????????? ????????, ??????? ATS.
????????????: 15 ????????.
UDK 511
MOSCOW TALK: THE THEOREM ON
APPROXIMATION OF TRIGONOMETRIC SUM
BY A SHORT ONE (ATS)3
A. A. Karatsuba4
Abstract
The invited talk presented at the seminar of Prof. B. S. Kashin and Prof.
S. V. Konyagin at the Faculty of Mechanics and Mathematics of Moscow
Lomonosow University at the November 9, 2006.
Keywords: trigonometric sum, approximation, Voronoi? formula, the Poisson summation formula, the theorem ATS.
Bibliography: 15 titles.
1
?????? ??????????? ? ?????? ?. ?. ?????????.
???????? ?????????? ???????? (31.01.1937 ? 28.09.2008) ? ?????? ??????-?????????????? ????, ?????????, ?????????? ??????? ?????? ????? ??????????????? ????????? ??. ?. ?. ???????? ???, ????????? ??????????? ???????????????? ???????????? ?????
?. ?. ??????????.
3
The report was prepared for publication E. A. Karatsuba.
4
Anatoly Alekseevich Karatsuba (01.31.1937 ? 28.09.2008) ? Doctor of Physical and Mathematical Sciences, Professor, Head of Department of Number Theory Mathematics Institute. Russian
Academy of Sciences, professor of the Moscow State University named after M. V. Lomonosov.
2
?????????? ??????: ??????? ? ???????????. . .
7
8
?. ?. ????????
� ?????????????????? ?????? ?????????? ???????? ????? ???? S,
S=
?
?(x)e2?if (x) .
(1)
a<x?b
????? ? ? f ? ???????????? ???????, a ? 0, x ? ??????????? ?????, i2 = ?1,
e2?if (x) = cos (2?f (x)) + i sin (2?f (x)).
????? b?a ?????????? ?????? S; ???? a ? b ? ????? ?????, ?? b?a ? ??????????
????????? S.
? ?????????? ???????: ?(x) ? ?????????, f (x) ? ????.
� ??? ???????????? ???????? ?? ?(x) ? f (x) ????? S ????? ??????????
?????? ?????? S1 , ?????????? ????????? ? ??????? ????? ?????? b ? a, ?? ????
?
S = S1 + R, S1 =
?(x)e2?iF (x) , ? ? ? < b ? a.
(2)
?<x??
??????????? (2) ? ????? ???????? ATS (?Approximation of Trigonometric Sum?)
??? ???????? ATS.
� ???????????? ATS ? ??? ????, ? ????? ??? ???????????? ? [1] ? [2].
??????? 1. (ATS) ????? ???????????? ??????? ?(x) ? f (x) ????????????? ?? [a, b] ????????? ????????:
1) f (4) (x) ? ?(2) (x) ? ???????????;
2) ?????????? ????? H, U, A,
0 < H, 1 ? A ? U, 0 < b ? a ? U,
????? ???
1
A
? f (2) (x) ?
?(x) ? H,
1
A
,
1
f (3) (x) ? AU
,
f (4) (x) ? AU1 2 ;
?(1) (x) ? HU ?1 , ?(2) (x) ? HU ?2 .
?????, ????????? ????? xn ?? ?????????
f (1) (xn ) = n,
(3)
????? ?????
S=
?
?(x)e2?if (x) =
a<x6b
?
C(n)Z(n) + R,
?6n6?
???
? = f (1) (a),
? = f (1) (b),
(4)
?????????? ??????: ??????? ? ???????????. . .
{
C(n) =
{
T� =
1 ,
1
,
2
9
???? ? < n < ?;
???? n = ? ??? n = ?;
?
?(xn )
Z(n) = ei 4 ?
e2?i(f (xn )?nxn ) ;
(2)
f (xn )
(
)
A
+ Ta + Tb + ln (? ? ? + 2) ;
R?H
b?a
???? f (1) (�)- ????? ?????;
0,
? )
1
min ||f (1) (�)|| , A ,
(
???? ||f (1) (�)|| ?= 0.
????? ||x|| = min({x}, 1 ? {x}), ??????? ??????? ????? ????? x, y = {x},
??? ????????????? x, ???????????? ??????????
y = {x} = x ? [x],
??? [x] ???? ????? ????? x, ?? ???? ????? ????? ?????, ??? x ? 1 < [x] 6 x.
????? ????, ??? B ? +?, ??????
B?A?B
??????, ??? ?????????? ????????? C1 > 0 ? C2 > 0, ????? ???
C1 B ? |A| ? C2 B.
� ????????? ????????? ?? ?????? (4). ?????? ????? ?????:
? ? ? = f (1) (b) ? f (1) (a) = (b ? a)f (2) (?), a ? ? ? b.
?? ?? ??????? ???????,
f (2) (?) ?
1
, A ? 1,
A
?? ????
????
b?a
? b ? a.
A
????? S1 ? "??????".
???????, ????????? ? ???? S1 , ?????? ??????, ?????????? ?? ????????? ?
???? S.
???????? R ????? ?????? ?????. ????????, ???? ????? U = b ? a, 1 ? A ?
b ? a, ??????? |T� | ??????????, ?? ????
|T� | ?
?
A,
10
?. ?. ????????
?? ???????
(
R?H
?
A
+ A + ln
b?a
(
))
b?a
+2
.
A
????????? ?? ??????????? ?????? |S| :
|S| ? (b ? a)H.
????? ???????, R ????? ?????? ??????????? ?????? |S|, ????
?
(
)
A
A
1
b?a
+
+
ln
+ 2 ? 1,
(b ? a)2 b ? a b ? a
A
?? ???? ???
A ? (b ? a)2
(?? ??????? A ? b ? a).
?????? ??????????? ?????? |S1 |? ?????
H
H(b ? a)
|S1 | ? (? ? ?) ?
? ?
,
?1
A
A
? ??????????? ?????? |S1 | ????? ?????? R, ????
(
(
))
?
A
b?a
b?a
H
+ A + ln
+2
?H ? ,
b?a
A
A
?? ????
A
b?a
? ? =? A ? (b ? a)4/3 ;
b?a
A
?
b?a
A ? ? =? A ? (b ? a);
A
)
(
b?a
b?a
ln
+ 2 ? ? =? A ? (b ? a)2?? .
A
A
????, ???? A ? b ? a, ?? R ? (4) ?????, ?????? ??????, ?????? |S1 |.
?? ??????, ??? |S| ? R ??? A ? b ? a. ????? ????, ??????????
H(b ? a)
?
.
A
?????????????, ??????? (4) ???? ????????????? ??????, ????
|S1 | ?
H(b ? a)
?
,
A
?? ???? ??? ????? 1 ? A ? b ? a ??????? (4) ???? ????????????? ?????? |S|.
H(b ? a) ?
?????????? ??????: ??????? ? ???????????. . .
11
� ??????? ATS ????? ??????? ???????. ??????? ATS ??????? (??. [3])
??????? ???????? (??????? ????????) 1903 ?.
??????? 2. (??????? ????????) ?????
?
? (n) = x (ln x + 2? ? 1) + ?(x);
n?x
?????
1
N
( ?
( 1)
(( 2 ?1 )a )
( c ?1 )
x 4 ? ? (n)
?)
4
?(x) = ?
cos
4?
nx
?
+
O
x
+
O
T
x
+
O
xT
,
4
? 2 n=1 n3/4
???
T2
1
=
N
+
,
4? 2 x
2
a > 0, c > 1 ? ?????, ? ? ????????? ??????.
?????, ? 1914 ?. ????? ? ???????? ? ?????? ??????????????????? ????,
??????????????? ? ????????????? ? -????????? ([4]) ???????? ??????? ??????
ATS. ? 1917 ?. ?. ?. ?????????? ? ?????? ?? ??????? ???????? ????? ???????
????? ???????? ???? ?????????????? ????????????? (??. [5]) ????? ???????
??????? ?????? ATS, ????????? ? ?????? ??????? ? ???????? ?? ?????????
? ? ????????????. ???????, ??? ??? ?????? ? 1922 ?. ? ??????, ????????? ?
????????? ????????? (??. [6]), ??????? ??????? ATS, ? ??????? ???????? ?
?????? ?????????????????? ???? ??????? ????????? ???????? ?
( 1
)
3
R = O N ln N
??
(
)
R = O N 0,33 .
??????? ??????? ATS, ?????????????? ?. ?. ???????????? ? 1976 ?. ? ?????
??????? ???????? ?????? ?????????????????? ?????, ???. 22 (??. [7]):
??????? 3. ????
1
A
? f (2) (x) ?
?(x) ? H,
1
A
,
1
f (3) (x) ? AU
;
(1)
?1
? (x) ? HU , ?(2) (x) ? HU ?2 ,
?????? ??????? f (x) ? ?(x) ???????????? ??????????????, ??????? ???????
??????????, ????? ??????????? (4).
??????? ATS ??? ??? ??????? ????????????? ???:
12
?. ?. ????????
??????? 4. ????? f (x) ????????????, f (3) (x) ? ??????????? ?? [a, b],
f (x) ????????? ??????? ?? [a, b], f (1) (b) = ?, f (1) (a) = ?, ????? xn ???????????? ??????????
(1)
f (1) (xn ) = n,
? < n ? ?.
?????, ????? ????,
2??2 ? f (2) (x) ? A?2 ,
(3) f (x) ? A?3 .
?????
?
? e2?i(f (xn )?nxn )
?
+
(2) (x )|
|f
n
a<x6b
?<n6?
( 1)
(
)
1
?2
5
+O ?2
+ O (ln (2 + (b ? a)?2 )) + O (b ? a)(?2 ?3 ) .
e2?if (x) = e?i 4
?
??????, ??? ????? ??????????? ????????? ????? ??? ??? ???????:
????? 1. (????? ??? ??? ???????) ???? f (x) ????????????, f (1) (x) ????????? ?? (a, b) ? |f (1) (x)| ? ? < 1, ??
? b
?
2?if (x)
e
=
e2?if (x) dx + O (1) .
(5)
a
a<x6b
� ??? ???????????? ATS.
1) ??????? ??? ???????? ATS (? ???? ????????? ????? ??????) ????????
?????????:
?
a<x6b
?(x)e
2?if (x)
?
?
=
b
?(x)e2?i(f (x)?nx) dx + O (H ln (? ? ? + 2)) , (6)
???6n6?+?
a
??? ? ? ????? ????? ? ???????? 0 < ? < 1.
??????? (6) ???????? ???????? ???????????? ????????. ??????? ?? ?????
?????.
????? ?(x) ? 1. ??? a < n ? b, ??????????
?
Wn =
1
2
? 21
sin (2m + 1)?x 2?if (n+x)
e
dx,
sin ?x
??? m ? ???????? ?????????, ? Wn ????? ?????? ?
e2?if (n) .
????? ??????????? Vn ,
?????????? ??????: ??????? ? ???????????. . .
13
Wn = e2?if (n) + Vn .
?????
?
?
Wn =
a<n6b
e2?if (n) +
a<n6b
?
Vn ,
a<n6b
? ????? ????? ?????
? ?
a<n6b
1
2
? 12
m
?
e
2?ikx 2?if (n+x)
e
? ?
dx =
k=?m
a<n6b
=
m ?
?
b+ 12
1
2
? 12
m
?
e2?i(k(n+x)+f (n+x)) dx =
k=?m
e2?i(kx+f (x)) dx.
a? 12
k=?m
????????? ?? ????? ?????? ? (6) ? ?(x) ? 1. ? ????? ??????? ?(x) ?????????
? ??????? ??????? ???????????? ?????:
?
?
=?
2?if (x)
?(x)e
b
C(u)d?(u) + C(b)?(b),
a
a<x6b
???
C(u) =
?
e2?if (x) .
a<x6u
? C(u) ????????? ??, ??? ???????? ??? ?(x) ? 1.
2) ????????? ????
?
b
?(x)e2?i(f (x)?nx) dx
In =
a
????????? ??????? ???????????? ???? (????? ????????). ??? ???????? ATS
??????????? ???????:
?
b
2?i(f (c)
? ?(c)e
?(x)e2?if (x) dx = ei 4 ?
+
f (2) (c)
a
))
( (
(
(
? )
? )
A
(1)
?1
(1)
?1
+ min |f (a)| , A + min |f (b)| , A
,
+O H
U
???
f (1) (c) = 0.
???????? (7) ? In , ???????? ATS.
(7)
14
?. ?. ????????
� ?????????? ATS.
1) ???????? ??????? ? ????? ????????? ?????????? ATS ? ????????????
????????? ??????????????? ??? ?(s) :
?(s) =
( 1
)
? 1
? 1
( ?? )
?? ??1
2
+
F
(s)
+
O
x
+
O
t
y
,
ns
n1?s
n?x
n?y
??? 2?xy = t > 0; x ? 1, y ? 1; 0 < ? < 1; s = ? + it;
( )
1?s
? ? 2 ? 1?s
( s 2) .
F (s) =
? 2s
? ? 2
?
?
???? x = y, ?? x = y = 2?t , ? ?????? ??????? ???? ??????? ????? ? 2?t
?????????, ?? ???? ?(s) ?????? ???????????? ?????? "?????????" ???????
???????? ???? ???????.
??????? ??? z ? ?t ???????? ???????:
? 1
( ?? )
z 1?s
?(s) =
+
+
O
z
,
ns s ? 1
n?z
(8)
s = ? + it.
????? Res = ? > 1. ?????
?
N
M
?
?
?
1
1
1
=
+
lim
.
s
s
s
M ?+?
n
n
n
n=1
n=1
n=N +1
?(s) =
? ????????? ????? ???????? ??????? ???????????? ??????:
?
T =
n
?s
?
=
1
2
?s
x dx + ?(M )M
?s
? ?(N )N
?s
?
? {x}, ?(x) =
M
+s
N
N <n?M
?(x) =
M
?(x)x?s?1 dx,
N
?x
0
?(u)du, ?? ????
1
1
1
1
T =
M 1?s ?
N 1?s + M ?s ? N ?s + s
1?s
1?s
2
2
?
M
?(x)x?s?1 dx =
N
1
1
1
1
=
M 1?s +
N 1?s + M ?s ? N ?s + s(s + 1)
1?s
s?1
2
2
?
M
?(x)x?s?2 dx.
N
??? M ? +?, ??? ??? Res > 1,
T1 = lim T =
M ?+?
?
N <n
n
?s
1
1
=
N 1?s ? N ?s + s(s + 1)
s?1
2
?
?
N
?(x)x?s?2 dx =
?????????? ??????: ??????? ? ???????????. . .
=
15
(
)
1
1
N 1?s ? N ?s + O t2 N ?1 .
s?1
2
????,
N
?
( 2 ?1 )
1
1
1 ?s
1?s
?(s) =
+
N
?
N
+
O
tN
.
ns s ? 1
2
n=1
????? N > z ? ?t , ?????????? ?????
?
z<n?N
1 ?2?i t ln n
2?
e
n?
t
ln n
, f (1) (n) = n2?
? 12 , ?? ???? ? ????? ????????? ????? ??? ???
??? f (n) = t 2?
???????, ?? ???????
? N
? 1
( ?? ) N 1?s
(
)
du
z 1?s
=
+
O
z
=
?
+ O z ?? .
s
s
n
u
1?s 1?s
z
z<n?N
2 ?
???? ????? t2 N ?1 = z ?? , ?? ???? N
? = t z , ?? ??????? (8).
??? ????????, ????? x = y = 2?t . ????? z = ?t ? ?????????? ????? S,
S=
?
x<n? ?t
1 ?2?i t ln n
2? .
e
n?
S ??????????? ? ???? ????? ????????? ????
?
t ln n
n?? e?2?i 2? , ? = 0, 1, . . . , [ln t];
2? x<n?2?+1 x
???
Sa =
?
n?? e?2?i
t ln n
2?
.
a<n?2a
? Sa ????????? ATS ? f (x) =
?t ln x
,
2?
f (1) (x) =
?t
,
2?x
f (2) (x) =
t
,
2?x2
A=
a2
2?
t
? 1.
2) ? 1981 ?. ? ?????? ?? ?????????? ????? ????????? ?????? ?????-???????
??????, ???????? ?? ??????????? ??????? (??. [8]) ? ??????? ?????????
???????:
??????? 5. ?????
(
Z(t) = e
i?(t)
?
)
1
+ it ,
2
???
ei?(t)
( )
1?s
? ? 2 ? 1?s
( 2) ,
=
s
? ? 2 ? 2s
s=
1
+ it;
2
16
?. ?. ????????
k > 0; ????? Z (k) (t) ????? ???? ?? (T, T + Hk ), ????
2
1
Hk = T 6k+6 (ln T ) k+1 ,
? Z(t) ????? ???? ?? (T, T + H0 ), ????
5
H0 = T 32 ln T.
?????? (? ?. ??????, ??. [9]) ????
1
H0 = T 6 ln T.
5
1
?????????? 32
= 16 ? 96
??????? ????????, ???????? ??? ???????????????
5
5
?????????????????? ???? ATS. ?????? 32
??????? ?????????, ?????? 32
=
1
27
0, 1562500 ???????? ?? 0, 1559458, ? 6k+6 ?? 164k+168 (?. ????, ??. [10]).
3) ? ????????? ???? (????? 10?15 ???) ???????? ??????? ????????? ???????? ATS ? ??????: ????????? ??????, ?????? ????????? ? ?????? ????????
(??. [11]?[15]); ? ??? ????? ??? ????????? ????????????, ??????????? ????????????????.
?????? ???????????? ??????????
1. Karatsuba A. A. Approximation of exponential sums by shorter ones // Proc.
Indian. Acad. Sci. (Math. Sci.), vol. 97 (1-3), pp. 167?178 (1987).
2. ??????? ?. ?., ???????? ?. ?. ?????-??????? ??????. ?.: ?????????,
1994, 376 ?.
3. Voronoi, G. Sur un proble?me du calcul des fonctions asymptotiques. // Journal
fu?r die reine und angewandte Mathematik 126, s. 241?282 (1903).
4. Hardy, G. H., Littlewood, J. E. The trigonometrical series associated with the
elliptic ? -functions // Acta Math. 37, pp. 193?239 (1914).
5. ?????????? ?. ?., ? ??????? ???????? ????? ??????? ????? ???????? ????
?????????????? ???????????? // ????????? ?????. ?????. ??-??, ?. 16, ???.
10?38 (1917).
6. Van der Corput, J. G. Verscha?rfung der abscha?tzung beim teilerproblem //
Math. Ann., 87, pp. 39?65 (1922).
7. ?????????? ?. ?. ?????? ???????? ?????? ?????????????????? ????. ?.:
?????, 1976, 122 ?.
8. ???????? ?. ?. ? ?????????? ????? ????????? ?????? ?????-???????
??????, ???????? ?? ??????????? ?????? // ????? ????, ?. 157, ???.
49?63 (1981).
?????????? ??????: ??????? ? ???????????. . .
17
9. ?????, ?. ?? ????? ????? ? ?????? ?????-??????? ?????? // ??t? Arith.
31 (1976), 31?43; ??????????? ? ???????: Acta Arith. 31 (1976), ???. 31?
43; 31(1976), ???. 45?51; 35 (1979), ???. 403?404, Acta Arith. 40, ???. 97?107
(1981).
10. Ivic, A. Topics in Recent Zeta- Function Theory Publ. Math. d?Orsay, Universite de Paris-Sud, Orsay, 1983, 272 p.
11. Narozhny, N.B., Sanchez-Mondragon, J.J., and Eberly, J.H., Coherence versus
incoherence: collapse and revival in a single quantum model // Phys. Rev. A,
23, pp. 236?247 (1981).
12. Fleischhauer, M. and Schleich, W. P. 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).
13. Chassande - Mottin E. and Pai A., Best chirplet chain: Near-optimal detection
of gravitational wave chirps, Physical Review D73, 042003, 1 - 23 (2006).
14. Karatsuba E. A., Approximation of sums of oscillating summands in certain
physical problems, Journal of Math. Physics, 2004, Vol. 45, N 11, 4310?4321.
15. ???????? E. A. Approximation of exponential sums in the problem on the
oscillator motion caused by pushes. // ??????????? ???????. 2005. ?. 6, ???.
3(15), ?. 205?224.
REFERENCES
1. Karatsuba, A. A., 1987, "Approximation of exponential sums by shorter ones." ,
Proc. Indian. Acad. Sci. (Math. Sci.), Vol. 97 (1?3), pp. 167?178.
2. Voronin, S. M. & Karatsuba, A. A., 1994, "The Riemann zeta-function" ,
Moscow: Phys.-Math. Lit., 376 p. (Russian).
3. Voronoi, G., 1903, "Sur un proble?me du calcul des fonctions asymptotiques." ,
Journal fu?r die reine und angewandte Mathematik 126, s. 241?282.
4. Hardy, G. H. & Littlewood, J. E., 1914, "The trigonometrical series associated
with the elliptic ? -functions" , Acta Math. 37, pp. 193?239.
5. Vinogradov, I. M., 1917, "On the average value of the number of classes of
purely root form of the negative determinant" , Communications of Kharkhov
Mathematics Society 16, pp. 10?38.
6. Van der Corput, J. G., 1922, "Verscha?rfung der abscha?tzung beim teilerproblem" , Math. Ann., 87, pp. 39?65.
18
?. ?. ????????
7. Vinogradov, I. M., 1976, "Special Variants of the Method of Trigonometric
Sums" , Moscow: Nauka, pp. 1?122.
8. Karatsuba, A. A., 1981, "On the distance between adjacent zeros of the Riemann
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????????? 10.02.2015
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