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О приближении непрерывных функций двух переменных.

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??? 517.51
??????? ?????. ???. 1, 2007, ???. 2
?. ?. ?????????
? ??????????? ??????????? ???????
???? ??????????
1. ????????
?????? ????????, ?????? ? ???????? ?????????? ???? ? ???????? ????? ?? ??????
??????-??????????? ?????. ????????????? ?????? ?????? ????????? ?? ?????? ????????, ?? ? ?????????????? ????????????. ????????, ?????????? ???????? ?????? ?
?????? ?????????? ??? ????????? ????????, ?????????? ? ??????????? ???????????
?????????, ??? ???????? ?????? ?????? ????????????? ? ???????????? ????????????,
?????????? ? ?????????? ??????? ???????????????? ?????????, ? ?. ?.
? ????????? ????? ?????? ????????? ? ????? ??????????? ??????? ??? ?????? ?????????? ???????? ???????????? ????? ????????? ??????????. ???? ??? ??????????? ? ?????????. ?????? ????, ????? ?????????? ??????? ???????? ??????? f (t, ? ) ???????????? ????????? ?????? ????????????? ????? cmn ??????????
??????? f (t, ? ) ?? ?????? ?????????????? ??????? ??????? ?mn (t, ? ) = xm (t)yn (? ),
m, n = 0, 1, . . . :
?
X
f (t, ? ) =
cmn ?mn (t, ? ),
m,n=0
??? ????????? ???????????? ????? ???
(
(?mn , ?kl ) =
0, ????(m, n) 6= (k, l),
1, ????(m, n) = (k, l).
(1)
???? ??????? f (t, ? ) ???????? ?????? ?????????? ?????????? f (ti , ?j ), i, j =
1, . . . , N , ?? ??? ?????????? ????????????? cmn ??????? ???????????? ????? ?????????????? ??????? ?mn (t, ? ), ??? ??????? ????????? ???????????? ????? ??? ?????:
(?mn , ?kl ) =
N X
N
X
?ij ?mn (ti , ?j )?kl (ti , ?j ).
i=1 j=1
??? ??? ??????? ??????????????? (1) ?? ????????? ????????????? ? ???? ????? ?????
????? ??? ?????????????? ??????????? ?????????? ??????????, ?? ??? ??????????
?????? ???????????? ??? ?????? ??????????.
????? H n ? ???????????? ?????????????? ??????????? pn = pn (x) ??????? ?? ???? n, C[?1,1] ? ???????????? ??????????? ?? [?1, 1] ???????, ? = {x0 , x1 , . . . , xn , . . .} ?
????? ? ?????????? ?????????, ????????? ?? ????????? ??? ???????????? ????? ????????? ????? ?????????????? ??? R. ? ?????? [1] ? ??????? ???????????? ??????? ??????
?1
???? ?????????? ??????? ??????????? ????? {Pi?,? (x)}N
i=0 (?, ? > ?1, N = 2, 3, . . . ),
????????????? ?? ????? ?N = {x1 , x2 , . . . , xN } ???????????? ?????????? ????????????
X
(f, g) =
х(x)f (x)g(x) (х(xj ) = хj ),
x??N
c
58
?. ?. ?????????, 2007
N
X
?,?
хj Pm
(xj )Pn?,? (xj ) = h?,?
n ?mn ,
(2)
j=1
??? xj ? ?N ? ???? ?????????? ????? PN?,? (x), хj ? ????? ??????????? ??? ???? ????2?+?+1
и ?(n+?+1)?(n+?+1)
???????? ??????? ??????, h?,?
= 2n+?+?+1
n
?(n+1)?(n+?+?+1) , ?mn ? ?????? ?????????.
???????
?1/2 ?,?
Pbn?,? (x) = {h?,?
Pn (x),
n }
??? ???????????? ??????? f (x) ? C[?1,1] ????? ?????????? ?????????? ??????? ????1
?? ??????????? ??????? n 6 N ? 1 ?? ????????????????? ??????? {Pbn?,? (x)}N
n=0 :
?,?
?,?
Sn,N
(f ) = Sn,N
(f, x) =
n
X
k=0
?,? b ?,?
fbk,N
Pk (x),
(3)
N
P
?,?
??? fbk,N
=
хj f (xj )Pbk?,? (xj ) ? ?????????? ???????????? ???????????.
j=1
????? ?????? C[?1,1]2 ? ???????????? ??????????? ?? ???????? [?1, 1]2 = [?1, 1] О
[?1, 1] ??????? f (x, y) ? ?????? kf k =
max 2 |f (x, y)|. ???????, ??? ??????? ?????(x,y)?[?1,1]
r
?,?
?,?
r
?????? ???? ?????????? {??,?
mn (x, y)}m,n=0 = {Pm (x)Pn (y)}m,n=0 (r = m+n 6 N ?1)
N
???????? ????????????? ?? ????? ?N ОN = {(xi , yj )}i,j=1 ???????????? ?????????? ????????????
N X
N
X
(?, ?) =
хi хj ?(xi , yj )?(xi , yj ),
(4)
i=1 j=1
??? xi , yj ? ???? ?????????? ????? PN?,? (x). ?????????????, ???????? (2) ? (4), ?????
N X
N
X
?,?
хi хj ??,?
mn (xi , yj )?kl (xi , yj ) =
i=1 j=1
N
X
?,?
хi Pm
(xi )Pk?,? (xi )
i=1
?,?
= h?,?
m ?mk hn ?nl =
(
N
X
хj Pn?,? (yj )Pl?,? (yj ) =
j=1
0,
???? (m, n) 6= (k, l),
?,? 2
(hn ) , ???? (m, n) = (k, l).
???????
b ?,? (x, y) = {h?,? }?1/2 {h?,? }?1/2 ??,? (x, y),
?
mn
m
n
mn
????????? ??? ???????????? ??????? f (x, y) ? C[?1,1]2 ?????????? ??????? ?????
??????????? ?????????????? ????
?,?
?,?
Sm,n,N
(f ) = Sm,n,N
(f ; x, y) =
X
k6m,l6n
?,? b ?,?
fbk,l,N
?kl (x, y) =
m X
n
X
k=0 l=0
?,? b ?,?
fbk,l,N
?kl (x, y),
(5)
N P
N
P
?,?
b ?,? (xi , yj ) ? ?????????? ???????????? ??????????
??? fbk,l,N
=
хi хj f (xi , yj )?
kl
i=1 j=1
??????? f (x, y) ? ??????? ??? ???????????.
59
????????? (5) ? ????????? ????:
?
?
m X
n
N X
N
X
X
?,?
?,?
b (xi , yj )f (xi , yj )? ?
b ?,? (x, y) =
?
Sm,n,N (f ; x, y) =
хi хj ?
kl
kl
k=0 l=0
=
+
Pb0?,? (x) ?
?
Pb1?,? (x) ?
иии +
=
i=1 j=1
?
N
X
i=1
N
X
i=1
?
?,?
Pbm
(x) ?
N X
N
X
хi Pb0?,? (xi )
хi Pb0?,? (xi )
N
X
i=1
N
X
j=1
N
X
j=1
хi Pb0?,? (xi )
?
хj f (xi , yj )Kn?,? (yj , y)? +
?
хj f (xi , yj )Kn?,? (yj , y)? + . . .
N
X
j=1
?
хj f (xi , yj )Kn?,? (yj , y)?
=
?,?
f (xi , yj )хi хj Km
(xi , x)Kn?,? (yj , y),
i=1 j=1
??? Kk?,? (u, v) =
k
P
i=0
Pbi?,? (u)Pbi?,? (v).
?????? ????????????, ???????? ? [2], ????????? ?????
pmn (x, y) =
m?1
k
XX
akl xk?l y l +
k=0 l=0
n
X
aml xm?l y l
l=0
?????????????? ????????? ???? ?????????? ??????? (m, n). ????? p?mn (x, y) ? ????????? ?????????? ??????????? ??????? f (x, y) ? ???????????? C[?1,1]2 ? Emn (f ) =
max 2 |f (x, y) ? p?mn (x, y)| ? ????????? ??????????? ??????? f (x, y) ??????????-
(x,y)?[?1,1]
????? ???????????? ???? ?????????? ??????? (m, n). ?? (5) ???????? ??????, ???
?,?
?????????? ??????? ????? ??????????? ?????????????? ???? Sm,n,N
(f ; x, y) ?? ???,?
?????? ??????????????? ?????????? pmn (x, y) ??????? (m, n), ?. ?. Sm,n,N (pmn ; x, y) ?
pmn (x, y). ???????
?,?
?,?
|f (x, y) ? Sm,n,N
(f ; x, y)| 6 |f (x, y) ? p?mn (x, y)| + |p?mn (x, y) ? Sm,n,N
(f ; x, y)| =
?,?
= |f (x, y) ? p?mn (x, y)| + |Sm,n,N
(p?mn ? f ; x, y)| 6
6 Emn (f ) + Emn (f )L?,?
m,n,N (x, y) =
(6)
= (1 + L?,?
m,n,N (x, y))Emn (f ),
???
L?,?
m,n,N (x, y) =
N
X
i=1
?,?
хi |Km
(xi , x)| и
N
X
j=1
?,?
хj |Kn?,? (yj , y)| = L?,?
m,N (x)Ln,N (y)
(7)
? ??????? ?????? ?????????? ??????? ???? ??????????? ?????????????? ????
?,?
Sm,n,N
(f ; x, y), L?,?
m,N (x) ? ??????? ?????? ?????????? ??????? ???? ???????????
?,?
Sm,N (f, x), ???????????? ?????????? (3). ??????? ?????? ???????? ????????? ????????????? ???????????? ?????? ?????? ???? ??? ???????????? ???????? ??????????
60
????? ????? ??????????? ???????. ?????? ??????? ?????? ????????? ????????????? ??????????? ??????? ??????????? ?????????? ????? ????? ?? ?????????????
??????????? ?? ???? ????????? (???????) ???????????????.
??????? ?????, ??? ?? ??????????? ????????? ??????
?,?
?,?
kSm,n,N
k = sup kSm,n,N
(f )k =
kf k61
max
(x,y)?[?1,1]2
L?,?
m,n,N (x, y),
(8)
??????????? (6) ????????? ? ????
?,?
?,?
|f (x, y) ? Sm,n,N
(f ; x, y)| 6 (1 + kSm,n,N
k)Emn (f ).
(9)
? ????????? ?????? ?? ??????????? ?? ??????? ?????????? ????????????? ?????,?
??? ???? Sm,n,N
(f ; x, y) ? ???????????? ??????? f (x, y) ? C[?1,1]2 , ??????? ? ????
?,?
???????, ??? ????? ?? (9), ???????? ? ???????? ????????? ???????? ?????? kSm,n,N
k?
?,?
???? ????????? Sm,n,N : C[?1,1]2 ? C[?1,1]2 . ???????, ??? ??? ??????? ????? ????? ??????? ????????? ????? ?????????? ??-???????, ? ?????????, ???????? ???????????????? ???????? ????????? ???? ??????????: ????????????? (??????????), ???????????
? ??????????? (??. [3]). ???, ? ?????????????????? ?????????
?????? ??? ??????????P
cх,? ei(хx+?y) ?????????
??? ??????? ???? ????? Sm,n
(f ) = Sm,n
(f ; x, y) =
|х|6m,|?|6n
?????? ????????????? ?????????
kSm,n
k=
sup
|f (x,y)|61
kSm,n
(f )k = 16? ?4 ln m ln n + O(ln m + ln n),
??????? ???????? ?????????? ?????? ????????? ?????? ?????????? ???????? ??????.
?
?
?????? ??????????? ??????? ???? ??????? ????? ????? Sm,n
(f ) = Sm,n
(f ; x, y) =
P
i(хx+?y)
cх,? e
?????????????? ? ??????? [4, 5]. ? ?????????, ? [4] ???? ????|х|/m+|?|/n61
????, ???
?
kSm,m
k = 16? ?4 ln2 m + O(ln m),
? ? [5] ???????? ????? ????? ???????????
?
kSm,n
k = 32? ?4 ln m ln n ? 16? ?4 ln2 m + O(ln n)
??? m, n = 1, 2, . . . , l = n/m. ??????? ??? ??????????, ? ??????? [6, 7] ???? ????????
?????? ???????? ?????? ??????? ???? ?? ??????????? ????????????? ?????????????? ? Rm .
??? ?????? ????????????? ??????? ???? ??????? ????? ????? ?? ?????????????
??????? ??????????? ????? Pn?,? (x) (?(u) = (1 ? u)? (1 + u)? )
Sm,n
(f )
=
Sm,n
(f ; x, y)
=
Z1 Z1
?,?
?(s)?(t)Km
(x, s)Kn?,? (t, y)dsdt,
?1 ?1
?????? kSm,n
k ???????? ????????, ????????? ??????????? ??? ?????????? ???????
R1
?(t)|Kn?,? (x, t)|dt, ?????????? ? ??????? [8, 9].
?????? L?,?
n (x) =
?1
61
? ????????? ?????? ?? ?????? ?????????? ?????? ??????? ?????? L?,?
n,N (x) ????????,?
??? ??????? ???? ??????????? Sn,N
(f, x) (??????? 1) ??????????? ???????????????
?,?
??????????? ??? ???????? ?????? kSm,n,N
k (??????? 2) ?, ??? ?????????, ????????
?,?
????????? ???????????????? ???????? ????????????? ???? Sm,n,N
(f ; x, y).
2. ??????????????? ???????????
???????? ??? ?????????????? ????????? ????????? ???????????.
????? 1. ????? ??????? f (x) ?????????? ? ?????????????? ?? ??????????
[a1 , b1 ] ? {tj }m
j=0 ? ?????, ????? ??? a1 < t0 < t1 < . . . < tm < b1 . ????? ?tj = tj+1 ?tj
? [a2 , b2 ] ? [a1 , b1 ]. ?????, ????
1) f (x) ????????? ?????????? ?? [a2 , b2 ], ??
X
f (tj )?tj 6
a2 6tj 6b2
Zb2
f (x) dx + f (b2 )?? ;
(10)
a2
2) f (x) ????????? ??????? ?? [a2 , b2 ], ??
X
f (tj )?tj 6
a2 6tj 6b2
?
Zb2
f (x) dx + f (a2 )?? ,
(11)
a2
??? ? = max ?tj .
j
????? 2 [10, ?.15.3]. ???? xj = cos ?j (0 < ?j < ?) ? ???? ?????????? ?????
PN?,? (x), ?1/2 < ?, ? < 1/2, ?? ??? ????? ??????????? хj ??????????? ?????????
??????:
?
хj 6 (sin ?j )2?+1 (0 < ?j 6 ? ? ?),
(12)
N
хj 6
?
(sin ?j )2?+1
N
(? 6 ?j < ?),
(13)
??? ? ? ? = ?(?) ? ????????????? ????????????? ?????, j = 1, 2, . . . , N .
??? ??????????? ????????? ???????? ??????????? ????? [10]. ??? ???????? ??????
?? ??????? ?? ? ???? ?????????.
??????????? ????????? ?????????:
Pn?,? (x) = (?1)n Pn?,? (?x),
(14)
Pn?+1,? (x) =
?,?
(n + ? + 1)Pn?,? (x) ? (n + 1)Pn+1
(x)
2
и
,
2n + ? + ? + 2
1?x
(15)
Pn?,?+1 (x) =
?,?
(n + ? + 1)Pn?,? (x) + (n + 1)Pn+1
(x)
2
и
.
2n + ? + ? + 2
1+x
(16)
??? ?1 6 x 6 1, n > 1 ??????????? ????????? ?????? (??., ????., [9]):
|Pn?,? (x)|
62
c(?, ?)
6
n1/2
???1/2 ???1/2
?
?
1
1
1?x+
1+x+
.
n
n
(17)
????? ? ????? ????? ck , c(?, ?, . . . , ?) ???????????? ????????????? ??????????, ????????? ???? ?? ????????? ??????????.
???? xj = cos?j ? ???? ?????????? ????? PN?,? (x), ?1/2 6 ?, ? 6 1/2, ?????????????? ? ????????? ???????:
1 > x1 > x2 > . . . > xN > ?1,
0 < ?1 < ?2 < . . . < ?N < ?,
?? (??. [10])
2j ? 1
2j
? 6 ?j 6
?
2N + 1
2N + 1
??????
??j = ?j+1 ? ?j >
??j 6
(j = 1, 2, . . . , N ).
(18)
?
,
2N + 1
(19)
3?
.
2N + 1
(20)
??? ?1 6 u, v 6 1, u 6= v ????? ????? ????????? ????????? [11]:
Kn?,? (u, v)
=
n
X
k=0
=
Pbk?,? (u)Pbk?,? (v) =
n
X
?1 ?,?
{h?,?
Pk (u)Pk?,? (v) =
k }
k=0
?,?
?,?
1
?(n + 2)?(n + ? + ? + 2) Pn+1 (u)Pn?,? (v) ? Pn?,? (u)Pn+1 (v)
и
и
.
2?+? (2n + ? + ? + 2) ?(n + ? + 1)?(n + ? + 1)
u?v
(21)
3. ??????????? ??????????? ???????
?,?
???? ?????????? ??????? Sm,n,N
(f ; x, y)
????????? ???????? ???????????, ????????? ? ????????????????? ?????????? ????,?
??????? ??????? ???? ??????????? ?????????????? ???? Sm,n,N
(f ; x, y), ?????? ?
?,?
???????? ??????????????? ??????? ??????? ?????? Ln,N (x) ?????????? ??????? ????
?,?
??????????? Sn,N
(f, x), ???????????? ?????????? (3). ???? ????? ???????? (??????? 1), ??? ??????? ??????
L?,?
n,N (x)
=
N
X
j=1
n
X
?,?
?,?
хj Pbk (x)Pbk (xj )
(22)
k=0
?? ??????? [?1 + ?, 1 ? ?], ? > 0, ????? ???????
O(ln n), ? ?? ???????? [?1, ?1 + ?] ?
[1 ? ?, 1], ??????????????, ??????? O n?+1/2 ? O n?+1/2 .
???????, ??? ???????, ????????? ? ??????? ??????? ?????? ???????????? ????
?????, ? ?????? ????? ??????????????? ? ??????? [8?17] ? ??. ? ?????????, ? ?????? [8] ??? ??????? ?????? L?,?
n (x) ???????????? ???? ??????????? ??? x ? [?1, 1],
?, ? > ?1/2 ???? ???????? ???????????
h
i ? h
i
?,?
?(?)
L?,?
(1 + x)?(?) + 1 + n |Pn?,? (x)| + |Pn+1
(x)| ,
n (x) ? ln n(1 ? x)
63
? ??? ??????, ??? ????????? ???? ????????? ?????????
( ????? ????? ??????????1/2 ??? ? 6= 1/2
???? ???????????, ?????????? ?? ? ? ?, ??? ?(?) =
. ?? ?????
0 ??? ? = 1/2
q+1/2
?????????? ?????, ??? ??????? ????? L?,?
, ???
n (x) ?? ??????? [?1, 1] ???? O n
q = max{?, ?}, ? ?? ????? ??????? [?1 + ?, 1 ? ?], ? > 0 ??????? ????? O(ln n).
? ?????? [9] ??????? ?????? ???????????? ???? ??????????? ??????????? ???
x ? [?1, 1] ? ??????, ????? ?? ??????? ???? ???? ?? ????? ? ? ? ??????????? ?????????
(?1, ?1/2), ? ?????????,
2
1 ? x2
(?, ? ? (?1, ?1/2)),
L?,?
n (x) = O(1) 1 + ln 1 + n
"
???1/2 #
?
1
?,?
2
Ln?1 (x) = O(1) ln 1+n (1?x) +
1 + x+
(? ? (?1, ?1/2), ? > ?1/2),
n
??? O(1) ??????? ?? ? ? ?.
??????? ?????, ??? ??? x ? [?1, 1], ? = ? = ?1/2 (??., ????????, [11], [13])
? 1 ,? 12
Ln 2
(x) = O(ln n).
????? ????? ?????????
??????? 1. ????? ?1/2 < ?, ? < 1/2, n 6 N ? 1, n > 1, ????? ??? ???? x ? [?1, 1]
h
i
?,?
1/2
?,?
L?,?
(x)
=
O(1)
ln
n
+
n
|P
(x)|
+
|P
(x)|
+
1
,
(23)
n
n+1
n,N
??? O(1) ??????? ?? ? ? ?.
??????????????. ?????? ???????? L?,?
n,N (x) ??? x ? [?1, 1]. ?????????? ????????? ??????: 1) x ? [0, 1], 2) x ? [?1, 0].
1. x ? [0, 1]. ??????? ???? ?????????? ????? PN?,? (x) ? ????????? ??????? ?1 <
xN < xN ?1 < ... < x1 < 1 ? ??????? ?????? x = cos ?, xj = cos ?j . ? ?????? ?????? (19)
?? (22) ???????, ???
n
N
2N + 1 X X b?,?
?,?
?,?
хj Pk (cos ?)Pbk (cos ?j ) ??j .
(24)
Ln,N (cos ?) 6
?
j=1
k=0
3?
1
1
1
??????? ?1 = 5 , ? , ?2 = ? + n1 , 3?
5 , ?3 = ? ? n , ? + n , ?4 = 0, ? ? n .
????? ???????? L?,?
n,N (cos ?) ???????? ?? ????????? ?????:
?
?
X
X
X
X
2N + 1 ?
? = U1 + U2 + U3 + U4 .
L?,?
+
+
+
(25)
n,N (cos ?) 6
?
?j ??1
?j ??2
?j ??3
?j ??4
???? ????????, ??? ? 6 n1 , ?? ????? U3 ??????? ?? ?????????? 0, ? + n1 , ? ?????
U4 ????????????? ?? ????.
??????????? ????????? (21). ??? ????? ???????? ??? ?????????????? ?????????
??????????? 1. [11] ??? ????????????? p ????? ????? ?????????
?(m + p)
1
p
= m 1+O
,
m ? ?,
(26)
?(m)
m
?????????? ?? ?????? ????????? ??????? ?????????.
64
? ???? ????? ??????????? (?1/2 < ?, ? < 1/2)
?,?
Kn (u, v) = (n + 1)(n + ? + ? + 1) и ?(n + 1)?(n + ? + ? + 1) О
2?+? (2n + ? + ? + 2) ?(n + ? + 1)?(n + ? + 1)
P ?,? (u)P ?,? (v) ? P ?,? (u)P ?,? (v) n+1
n
n
n+1
О
6
u?v
P ?,? (u)P ?,? (v) ? P ?,? (u)P ?,? (v) 3
n+1
n
n
n+1
6 c1 (n + 1) .
5
u?v
(27)
???????? (27), ?????? ?? ???? Ui (i = 1, 2, 4) ?????? ???:
Ui 6
О
X
?j ??i
3
c1 (n + 1)(2N + 1)О
5?
P ?,? (cos ?)P ?,? (cos ? ) ? P ?,? (cos ?)P ?,? (cos ? ) j
j n+1
n
n
n+1
хj ??j .
cos ? ? cos ?j
(28)
??????????? ????????? ? ?????? ????? ??????? (28) ? ??????? ????????? (15):
?+?
?,?
?,?
?,?
?,?
Pn+1 (cos ?)Pn (cos ?j ) ? Pn (cos ?)Pn+1 (cos ?j ) = 1 +
О
2n + 2
О (1 ? cos ?j )Pn?+1,? (cos ?j )Pn?,? (cos ?) ? (1 ? cos ?)Pn?+1,? (cos ?)Pn?,? (cos ?j ) .
?????
?+?
3
Ui 6
c1 (n + 1) 1 +
(2N + 1)О
5?
2n + 2
?
?+1,?
X
P
(cos ?j ) О ?|Pn?,? (cos ?)|
хj (1 ? cos ?j ) n
??j +
cos ? ? cos ?j ?j ??i
?
?,?
X
P (cos ?j ) + (1 ? cos ?)|Pn?+1,? (cos ?)|
хj n
??j ? .
cos ? ? cos ?j (29)
?j ??i
??? ?????????????? ? ????? 2 ????? ??????? ? = 2?/5. ??????? ?? ????????? ?1
????? ???????????? ??????? (12), ? ?? ?????????? ?2 , ?3 , ?4 ? ??????? (13).
??? ?????????? ???????? U1 ????? ?????????, ??? (sin ?j )2?+1 = (1 ?
cos ?j )?+1/2 (1+cos ?j )?+1/2 ? ??? ?j ? ?1 ????? (1?cos ?j )??/2+?+1/4 6 2, cos ??cos ?j >
? cos 3?/5 > 3/10, (1 + cos ?j )?/2+1/4 6 1. ???????? ?? ???????? (13), (17), ?? (29) ???????
8
2N + 1
U1 6 c(?, ?)c1 ?n1/2
О
N
h?
i X
?,?
О |Pn?,? (cos ?)| + |Pn+1
(cos ?)|
(1 + cos ?j )?/2+1/4 ??j 6
(30)
?j ??1
i
48
?,?
c(?, ?)c1 ?n1/2 |Pn?,? (cos ?)| + |Pn+1
(cos ?)| .
5
?????? ???????? U2 . ?????????????? ??????? ?????????
6
h
65
??????????? 2. ???? ?1/2 < ? < 1/2, n 6 N ? 1, ??
X ?j??1/2
3?
3?
??1/2
??j 6 ?
ln
n+
,
?j ? ?
5
2
(31)
X ?j?+1/2
3?
3?
4
3?
?+1/2
??j 6
+1 +?
ln
n+
.
?j ? ?
2 5(2? + 1)
5
2
(32)
?j ??2
?j ??2
1
??????????????. ????????? ??????? g1 (?) = ???
? g2 (?) = (? ? ?)??1/2 ????????? ??????? ?? ?????????? (?, ?], ? ?????? ?????? (11), (20) ?????
X
??1/2
?j
?j ??2
?j ? ?
??j 6 ???1/2
X
?j ??2
?
1
?
??j 6 ???1/2 ?
?j ? ?
3?/5
Z
1
?+ n
3?
3?
6 ???1/2 ln
n+
,
5
2
?
d?
3? ?
+n
?6
???
2N
X ?j?+1/2
X (?j ? ?)?+1/2 + ??+1/2
??j 6
??j =
?j ? ?
?j ? ?
?j ??2
?j ??2
X
X
1
??j 6
?j ? ?
?j ??2
?j ??2
?
?
3?/5
3?/5
Z
Z
3? ?
3? ???1/2
d?
?
6
(? ? ?)??1/2 d? +
n
+ ??+1/2 ?
+
?6
2
???
2
=
(?j ? ?)??1/2 ??j + ??+1/2
1
?+ n
1
?+ n
1
3? 3?
3?
3?
?+1/2
6
и
ln
+
+?
n+
6
2
5
2
? + 12 5
4
3?
3?
3?
?+1/2
6
+1 +?
ln
n+
.
2 5(2? + 1)
5
2
??????????? 2 ????????.
??? ?????????? Ui (i = 2, 3, 4) ????? ???????????? ?????????? (sin ?j )2?+1 = (1 ?
cos ?j )?+1/2 (1+cos ?j )?+1/2 , ? ??? ?????? U2 ?????, ??? 1?cos ? 6 ?2 ? ??? ?j ? ?2 ?????
? ?+1/2
?+3/2
(1 + cos ?j )???/2+1/4 6 2, (1 ? cos ?j )?/2+3/4 6 2?j
, (1 ? cos ?j )?/2+1/4 6 2?j
.
??????? ?????, ??? ??? ? ? (0, ?/2], ?j ? (?, 3?/5] ??????????? ???????????
cos ? ? cos ?j = 2 sin
=
66
?j ? ?
?j + ?
2 ?j ? ? 2 ?j + ?
sin
>2 и
и и
=
2
2
?
2
?
2
2
2
?j2 ? ?2 = 2 (?j ? ?)(?j + ?).
2
?
?
(33)
???????? (12), (15), (31)?(33), ????? ?? (29)
?
X (1 ? cos ?j )?/2+3/4
36
c(?, ?)c1 ?n1/2 ?|Pn?,? (cos ?)|
??j +
U2 6
5?
cos ? ? cos ?j
?j ??2
?
X (1 ? cos ?j )?/2+1/4
+ (1 ? cos ?)|Pn?+1,? (cos ?)|
??j ? 6
cos ? ? cos ?j
?j ??2
36?
c(?, ?)c1 ?n1/2 О
5
?
?
2
X ?j?+1/2
X ?j??1/2
?
О ?|Pn?,? (cos ?)|
??j + ? |Pn?+1,? (cos ?)|
??j ? 6
?j ? ?
?j ? ?
2
?j ??2
?j ??2
36?
4
1/2 3?
c(?, ?)c1 ?n
+ 1 |Pn?,? (cos ?)|+
6
5
2 5(2? + 1)
??+3/2 ?+1,?
3?
3?
?+1/2
?,?
+ ?
|Pn (cos ?)| + ? |Pn
(cos ?)|
ln
n+
.
5
2
2
6
(34)
?? (15) ??? 0 < 1/n < ? 6 ?/2
|Pn?,? (cos ?)| 6
6
c(?, ?)
(1 ? cos ?)??/2?1/4 6
n1/2
?
2c(?, ?)
?c(?, ?) ???1/2
?
.
(sin ?)???1/2 6 ?
n1/2
2n1/2
??????????,
|Pn?+1,? (cos ?)| 6
? 2 c(?, ?) ???3/2
?
?
.
2 2n1/2
??????? ?? (34) ???????? ??????
54? 2
4
U2 6
+ 1 c(?, ?)c1 ?n1/2 |Pn?,? (cos ?)|+
5
5(2? + 1)
9? 2 ?
3?
3?
2
+
(2 2 + ?)c (?, ?)c1 ? ln
n+
.
5
5
2
(35)
?????? ???????? U3 ??? ? > 1/n. ?? (24), ????????? (12) ? (21), ?????
U3 6 ?
n
X
2N + 1 X
?1
(sin ?j )2?+1
{h?,?
|Pk?,? (cos ?)| и |Pk?,? (cos ?j )|??j =
k }
N?
?j ??3
k=0
=
3? X
?1
{h?,?
(sin ?j )2?+1 ??j +
0 }
?
?j ??3
+
3? X
(sin ?j )2?+1
?
?j ??3
n
X
k=1
(1)
(2)
?1
{h?,?
|Pk?,? (cos ?)| и |Pk?,? (cos ?j )|??j = U3 + U3 .
k }
(36)
67
(1)
?????? ????? U3 . ????????, ??? ??? ?1/2 < ?, ? < 1/2 (??. (2))
?+?+1
?(? + ? + 1)
и
=
?+?+1
2
?(? + 1)?(? + 1)
?(? + ? + 2)
?(3)
= ?+?+1
6
2 6 2.56,
2
?(? + 1)?(? + 1)
[?(1, 462)]
?1
{h?,?
=
0 }
????????
(1)
U3
6
7.68? X
7.68? X
15.36?
(sin ?j )2?+1 ??j 6
??j 6
.
?
?
?
?j ??3
(37)
?j ??3
(2)
?1
??? ?????? ???????? U3 ???????, ??? ? ???? ??????????? 1 ???????? {h?,?
k }
?,? ?1
????? ??????? O(k), ??? ??? {hk }
6 c2 k. ??? ??? ??? ?j ? ?3 ????? (1 +
cos ?j )???/2+1/4 6 2, ?? (36) ? ?????? (17) ????????
(2)
U3
6
n
X
6?c2 c2 (?, ?) X
(1 ? cos ?j )?/2+1/4 ??j 6
(1 ? cos ?)??/2?1/4
?
?j ??3
k=1
n
6?c2 c (?, ?) X
6
(1 ? cos ?)??/2?1/4 О
?
k=1
"
?/2+1/4 ?/2+1/4 #
1
1
2
1
+ cos ? ?
? cos ? +
6
О
1 ? cos ? ?
n
n
n
n
2
n
12?c2 c2 (?, ?) X
6
(1 ? cos ?)??/2?1/4 О
?n
k=1
"
?/2+1/4 ?/2+1/4 #
1
1
+ 2 sin ? sin
6
О
1 ? cos ? ?
n
n
?
?
!?/2+1/4
n
1
2
X
1 ? cos ? ? n
12?c2 c (?, ?)
?
6
+ 2(sin ?)??/2?1/4 n??/2?1/4 ? 6
?n
1 ? cos ?
k=1
?
?
n
12(1 + 2?)?c2 c2 (?, ?) X
12(1 + 2?)
6
16
?c2 c2 (?, ?).
?n
?
k=1
(38)
????????? (37), (38), ???????
U3 6
i
?
?h
12(1 + 2?)c2 c2 (?, ?) + 15.36 .
?
(39)
???? ?? ? 6 1/n, ?? ???????? U3 ??????? ?? ?????????? ?5 = 0, ? + n1 . ? ????
(1)
(2)
?????? U3 ??????????? ??? ??, ??? ??? ? > n1 , ? U3 ? ?????? |Pk?,? (cos ?)| 6 c(?, ?)n?
??????????? ???:
!
n
6?c2 c2 (?, ?) X ?+1/2 X
(2)
U3 6
k
(1 ? cos ?j )?/2+1/4 ??j 6
?
k=1
68
?j ??5
?/2+1/4 X
1
6?c2 c2 (?, ?) ?+3/2
n
1 ? cos ? +
??j 6
6
?
n
?j ??5
?+1/2
6?c2 c2 (?, ?) ?+3/2
1
2
24? 2
6
n
?+
6
c2 c (?, ?).
?
n
n
?
(40)
? ????? ????????
U3 6
i
?
?h
12(1 + 2?)c2 c2 (?, ?) + 15.36 .
?
(41)
?????? ???????? U4 . ??????? ?? ?? ???????????, ??? ??? ?????? ???????? U2 ,
???????
36?
U4 6
c(?, ?)c1 ?n1/2 О
5
?
?
2
X ?j?+3/2
X ?j?+1/2
?
??j + ? |Pn?+1,? (cos ?)|
??j ? 6
О ?|Pn?,? (cos ?)|
?(? ? ?j )
?(? ? ?j )
2
?j ??4
?j ??4
6
X
36?
??+3/2
1
c(?, ?)c1 ?n1/2 ??+1/2 |Pn?,? (cos ?)| + ? |Pn?+1,? (cos ?)|
??j .
5
? ? ?j
2
? ??
j
4
(42)
????????? ??????? g(?) = 1/(? ? ?) ????????? ??????? ?? ?????????? (0, ?), ????????? ?????? (11) ? (20), ????? ????????
X
?j ??4
1
??j 6
? ? ?j
1
?? n
Z
0
d?
3?
3?
?
3?
+
6 ln ?n +
6 ln n +
.
???
2
2
2
2
(43)
? ?????? ?????? (43) ?? (42) ???????
??+3/2
?
3?
36?
c(?, ?)c1 ?n1/2 ??+1/2 |Pn?,? (cos ?)| + ? |Pn?+1,? (cos ?)| ln n +
.
5
2
2
2
(44)
???1/2
?+1,?
?
??? ??? ? 6 1/n, ?? (15) ????? |Pn?,? (cos ?)| 6 ?c(?,?)
?
?
|P
(cos
?)|
6
n
2n1/2
U4 6
? 2 c(?,?) ???3/2
?
?
.
2 2n1/2
??????? ?? (44) ????????
U4 6
9? 2 ?
?
3?
(2 2 + ?)c2 (?, ?)c1 ? ln n +
.
5
2
2
(45)
??????? ?????? (30), (35), (41), (45) ? ??????????? ?? ? (24), ??? x ? [0, 1], ?1/2 <
?, ? < 1/2, n 6 N ? 1 ???????? ? ??????
L?,?
n,N (x)
3? 2
6 H1 (?, ?) ln
n + 3? +
10
h
i
?,?
+ H2 (?, ?)|Pn?,? (x)| + H3 (?, ?)|Pn+1
(x)| n1/2 + H4 (?, ?),
(46)
69
???
9? 2 ?
(2 2 + ?)c2 (?, ?)c1 ?,
5
4
6
8 + 9? 2
+1
c(?, ?)c1 ?,
H2 (?, ?) =
5
5(2? + 1)
H1 (?, ?) =
48
c(?, ?)c1 ?,
5
i
?
?h
H4 (?, ?) =
12(1 + 2?)c2 c2 (?, ?) + 15.36 .
?
H3 (?, ?) =
2. x ? [?1, 0]. ???????? ?????? ? ?????? ?1 6 x 6 0. ???????, ??? ??? ?????
?????? ? ??? ?????????????? ?????? 0 6 x 6 1. ????????? ???????? (14) ? (21), ???
????????????? x ? [0, 1] ?? (22) ?????
L?,?
n,N (?x)
n
X
?,? ?1 ?,?
?,?
=
хj {hk } Pk (?x)Pk (xj ) =
j=1
k=0
N
n
X
X
?,? ?1 ?,?
?,?
=
хj {hk } Pk (x)Pk (?xj ) .
N
X
j=1
(47)
k=0
?????? ?????? ?????????? x = cos ?, xj = cos ?j , ?? (47) ???????
L?,?
n,N (? cos ?)
n
X
?,? ?1 ?,?
?,?
=
хj {hk } Pk (cos ?)Pk (cos(? ? ?j )) =
j=1
k=0
N
n
X
X
?,?
?,?
?1 ?,?
=
хj {h?,?
}
P
(cos
?)P
(cos
?
)
= L?n,N (cos ?).
j
k
k
k
N
X
j=1
k=0
??? ??? ?j = ? ? ?j ????? ?????????? ???????? ? ?j , ??????? ?? ?? ???????????,
??? ??? ?????? ???????? L?,?
n,N (cos ?) ? ????????? ????????? (14), ???????, ???
3? 2
L??,?
(cos
?)
6
H
(?,
?)
ln
n
+
3?
+
1
n,N
10
h
i
?,?
+ H2 (?, ?)|Pn?,? (cos ?)| + H3 (?, ?)|Pn+1
(cos ?)| n1/2 + H4 (?, ?) =
3? 2
n + 3? +
= H1 (?, ?) ln
10
h
i
?,?
+ H2 (?, ?)|(?1)n Pn?,? (? cos ?)| + H3 (?, ?)|(?1)n+1 Pn+1
(? cos ?)| n1/2 + H4 (?, ?) =
3? 2
= H1 (?, ?) ln
n + 3? +
10
70
h
i
?,?
+ H2 (?, ?)|Pn?,? (? cos ?)| + H3 (?, ?)|Pn+1
(? cos ?)| n1/2 + H4 (?, ?).
(48)
??????????? ? ?????????? x, ?? (47) ? (48) ????????
3? 2
?,?
n + 3? +
Ln,N (x) 6 H1 (?, ?) ln
10
h
i
?,?
+ H2 (?, ?)|Pn?,? (x)| + H3 (?, ?)|Pn+1
(x)| n1/2 + H4 (?, ?)
(49)
??? x ? [?1, 0].
? ???????? ?????, ?? ?????? (46) ? (49) ??? ?1/2 < ?, ? < 1/2, n 6 N ? 1 ???
x ? [?1, 1] ???????
3? 2
L?,?
(x)
6
H
(?,
?)
ln
n
+
3?
+
1
n,N
10
h
i
?,?
+ H2 (?, ?)|Pn?,? (x)| + H3 (?, ?)|Pn+1
(x)| n1/2 + H4 (?, ?)
(50)
???
h
i
?,?
1/2
L?,?
|Pn?,? (x)| + |Pn+1
(x)| + 1 .
n,N (x) = O(1) ln n + n
??????? 1 ????????.
?? (7), (8) ? ??????????? ??????? 1 ????????
??????? 2. ????? ?1/2 < ?, ? < 1/2, m + n 6 N ? 1, q = max{?, ?}, ????? ???
m, n ? ?
?,?
kSm,n,N
k = O (mn)q+1/2 .
(51)
???????, ????????? (7)?(9) ? ????????? ??????? 2, ???????? ????????? ???????????.
?,?
?,?
??????? 3. ???? f (x, y) ? C[?1,1]2 , Sm,n,N
(f ) = Sm,n,N
(f ; x, y) ? ?????????? ??????? ????? ??????????? ?????????????? ????, lim (mn)q+1/2 Emn (f ) = 0, ?? ???
m,n??
?1/2 < ?, ? < 1/2, m > n, m + n 6 N ? 1
f (x, y) =
?,?
lim Sm,n,N
(f ; x, y)
m,n??
(52)
??? (x, y) ? [?1, 1]2 .
?,?
?,?
?????????. ???? f (x, y) ? C[?1,1]2 , Sm,m,N
(f ) = Sm,m,N
(f ; x, y) ? ??????????
2
??????? ????? ??????????? ??????????? ????, lim ln mEm (f ) = 0, ?? ??? ?1/2 <
?, ? < 1/2, 2m 6 N ? 1 ?????? ???????? [?1, 1]2
m??
?,?
f (x, y) = lim Sm,m,N
(f ; x, y).
m??
(53)
Summary
F. ?. ??rkmasov. On the approximation of continuous functions of two variables.
?,?
It is shown that if Pm
(x) (?, ? > ?1, m = 0, 1, 2, . . . ) are the classical Jacobi polynomir
?,?
?,?
r
als, then the system of polynomials of two variables {??,?
mn (x, y)}m,n=0 = {Pm (x)Pn (y)}m,n=0
N
(r = m + n 6 N ? 1) is an orthogonal system on the set ?NОN = {(xi , yj )}i,j=0 , where xi , yj
71
are the zeros of the Jacobi polynomial PN?,? (x). Given an arbitrary continuous function f (x, y)
on the square [?1, 1]2 , we construct the discrete partial Fourier?Jacobi sums of the rectangular
?,?
type Sm,n,N
(f ; x, y) by the orthonormal system introduced above. We prove that the order of
?,?
?,?
the Lebesgue constants kSm,n,N
k of the discrete sums Sm,n,N
(f ; x, y) for ?1/2 < ?, ? < 1/2,
?
?
q+1/2
m + n 6 N ? 1 is O (mn)
, where q = max{?, ?}.As a consequence of this result, several
?,?
approximate properties of the discrete sums Sm,n,N
(f ; x, y) are considered.
??????????
1. ????????? ?. ?. ???????????????? ???????? ??????? ?????-??????? ??? ??????????
???? ??????????? // ???. ???. ??????. 2004. ?. 45. ? 2. ?. 334?355.
2. ?????? ?. ?. ????????????? ?????????? ?? ???? ??????????. ?.: ?????, 1988. 384 ?.
3. ?????????? ?. ?. ??????? ?????????????????? ????. ???????????: ?????, 1986. 272 ?.
4. ???????? ?. ?. ? ?????????? ?????? ??? ??????? ????? ????? // ?????? ??????????.
???. 6. ?.: ???-?? ???????. ??-??, 1970. ?. 8?13.
5. ????????? ?. ?. ?? ??????????????? ????????? ???????? ?????? ??? ?????????????????? ??????????? ??????? ???? ??????? ????? ????? // ???. ???. ??????. 1977. ?. 18. ? 3.
?. 629?636.
6. ?????????? ?. ?. ????????? ?????? ?????????????? // ???. ???????. 1982. ?. 32. ? 6.
?. 817?822.
7. ?????????? ?. ?. ??????? ????? ???????? ?????? ???? ????? ?? ????????? // ?????.
???????. ??-??. ???. 1. 1982. ? 7. ?. 110?111.
8. ???????? ?. ?., ???????? ?. ?. ??????? ?????? ???? ??????????? // ?????. ???????.
??-??. ???.: ???. 1968. ???. 1. ?. 11?23.
9. ?????? ?. ?. ?????? ??????? ?????? ? ??????? ???? ??????????? // ???. ???. ??????.
1968. ?. 9. ? 6. C. 1263?1283.
10. ???? ?. ????????????? ??????????. ?.: ?????????, 1962. 500 ?.
11. ?????? ?. ?. ???????????? ????????????? ??????????. ?.: ?????, 1976. 328 ?.
12. ??????? ?. ???? ????? ? ????????????? ????????. ?.: ??, 1948. 260 c.
13. ???????? ?. ?. ?????????????? ?????? ???????. ?.: ???????????, 1949. 688 ?.
14. ????? ?. ?. ? ???????? ?????? ?????????? ? ???? ?? ????????? ????? ??? ???????
? = ? = 1/2; ? = ?1/2, ? = 1/2; ? = 1/2, ? = ?1/2 // ?????? ???. ????, 1958. ?. 13. ? 6.
?. 207?211.
15. ????? ?. ?. ?????? ??????????? ??????? ??????????????? ???????????. ?.: ?????????, 1960. 624 c.
16. Gronwall T. U?ber die Laplacesche Reiche // Math. Ann. 1913. Vol. 74. P. 213?270.
17. Rau H. U?ber die Lebesgueschen Konstanten der Reihenentwicklugen nach Jacobischen Polynomen // J. fu?r Math. 1929. Vol. 161. P. 237?254.
?????? ????????? ? ???????? 12 ??????? 2006 ?.
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