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ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
ȿɳɟ ɨ ɧɚɢɥɭɱɲɢɯ ɩɪɢɛɥɢɠɟɧɢɹɯ ɚɧɚɥɨɝɚɦɢ «ɫɜɨɢɯ» ɢ «ɧɟ ɫɜɨɢɯ»
ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ
ɤ.ɮ-ɦ.ɧ. ɞɨɰ. ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ.
ɍɧɢɜɟɪɫɢɬɟɬ ɦɚɲɢɧɨɫɬɪɨɟɧɢɹ
Ⱥɧɧɨɬɚɰɢɹ. ȼ ɫɬɚɬɶɟ ɢɡɭɱɚɸɬɫɹ ɧɚɢɥɭɱɲɢɟ ɩɪɢɛɥɢɠɟɧɢɹ ɤɥɚɫɫɨɜ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɦɧɨɝɢɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɫ ɡɚɞɚɧɧɨɣ ɦɚɠɨɪɚɧɬɨɣ ɫɦɟɲɚɧɧɵɯ ɦɨɞɭɥɟɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɚɧɚɥɨɝɚɦɢ «ɫɜɨɢɯ» ɢ «ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ. Ɇɚɠɨɪɚɧɬɚ ɫɨɞɟɪɠɢɬ ɫɬɟɩɟɧɧɵɟ ɦɧɨɠɢɬɟɥɢ ɜ ɪɚɡɧɵɯ ɫɬɟɩɟɧɹɯ ɢ
ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɟ ɦɧɨɠɢɬɟɥɢ ɜ ɪɚɡɧɵɯ ɫɬɟɩɟɧɹɯ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɩɟɪɢɨɞɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ, ɧɚɢɥɭɱɲɢɟ
ɩɪɢɛɥɢɠɟɧɢɹ, ɫɦɟɲɚɧɧɵɣ ɦɨɞɭɥɶ ɧɟɩɪɟɪɵɜɧɨɫɬɢ, ɚɧɚɥɨɝɢ «ɫɜɨɢɯ» ɢ «ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ.
§1. Ɉɩɪɟɞɟɥɟɧɢɹ. Ɋɚɧɟɟ ɩɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ.
Ȼɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɮɭɧɤɰɢɢ ɨɬ d ɜɟɳɟɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ f ( x) f ( x1 ,!, xd ) ,
ɢɦɟɸɳɢɟ ɩɟɪɢɨɞ 2S ɩɨ ɤɚɠɞɨɣ ɩɟɪɟɦɟɧɧɨɣ.
ɋɦɟɲɚɧɧɵɣ ɦɨɞɭɥɶ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɩɨɪɹɞɤɚ l ɨɬ ɮɭɧɤɰɢɢ f ( x) ɨɩɪɟɞɟɥɢɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
: ( l ) ( f , t ) q sup 'lh f ( x) q ,
h j dt j
j 1,! , d
ɝɞɟ t
(t1 ,!, t d ), h (h1 ,!, hd ) , ɪɚɡɧɨɫɬɶ 'lh f (x) ɨɡɧɚɱɚɟɬ ɜɡɹɬɢɟ ɪɚɡɧɨɫɬɢ ɩɨɪɹɞɤɚ l ɫ
ɲɚɝɨɦ h j ɩɨ ɩɟɪɟɦɟɧɧɨɣ x j , j 1,!, d , ɧɨɪɦɚ ɪɚɜɧɚ:
·
§ 1
q
d
¸
¨
>
@
S
S
S
f
x
dx
(
)
,
;
, 1 q f .
d
³
d
q
¸
¨ (2S )
S
¹
©
ȼɜɟɞɟɦ ɮɭɧɤɰɢɸ :(t ) :(t1 ,!, t d ) ɞɥɹ t j t 0, j 1,!, d ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɭɥɨɣ:
1 b
­d r
ɩɪɢ t j ! 0, j 1,!, d ,
t j (log ) °°–
tj
j 1
:(t ) ®
(1)
d
°0, – t j 0.
°¯
j 1
Ɂɞɟɫɶ 0 r j l , b j - ɩɪɨɢɡɜɨɥɶɧɵɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɢɫɥɚ, ɥɨɝɚɪɢɮɦɵ ɛɟɪɭɬɫɹ (ɤɚɤ ɢ
ɜɫɸɞɭ ɧɢɠɟ) ɩɨ ɨɫɧɨɜɚɧɢɸ 2, ɚ ɬɚɤɠɟ ɩɪɢ W ! 0 ɩɨɥɚɝɚɟɦ (logW ) max ^ logW ; 1 `.
1
q
f ( x)
d
j
j
^ f ( x ,!, x
Ɋɚɫɫɦɨɬɪɢɦ ɤɥɚɫɫɵ ɮɭɧɤɰɢɣ
H q:
) : f ( x)  L0q (S d ), : ( l ) ( f ; t ) q d : (t ) `,
ɝɞɟ ɡɚɩɢɫɶ f ( x)  Lq (S d ) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɧɨɪɦɚ
1
d
0
³ f ( x) d x
S
S
0,
f ( x)
q
ɤɨɧɟɱɧɚ ɢ
j 1,!, d , : (t ) ɡɚɞɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ (1).
Ɇɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɭɱɚɣ 1 q f .
j
ɑɟɪɟɡ A ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɤɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ Ⱥ. Ɇɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢ-
)
(
ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
105
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
ɜɚɬɶ ɩɨɪɹɞɤɨɜɵɟ ɧɟɪɚɜɟɧɫɬɜɚ ɢ ɩɨɪɹɞɤɨɜɵɟ ɪɚɜɟɧɫɬɜɚ. Ɂɚɩɢɫɶ A ( N ) B ( N ) ɨɡɧɚɱɚɟɬ,
ɱɬɨ A( N ) d C ˜ B ( N ) , ɝɞɟ ɋ ɧɟ ɡɚɜɢɫɢɬ ɨɬ N. Ɂɚɩɢɫɶ A (N ) B (N ) ɨɡɧɚɱɚɟɬ, ɱɬɨ
A ( N ) B ( N ) ɢ B ( N ) A ( N ) .
Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɚ, ɩɨɪɨɠɞɟɧɧɵɟ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɭɪɨɜɧɹ ɮɭɧɤɰɢɢ : (t ) ɢɡ (1).
Ɉɛɨɡɧɚɱɢɦ
1½
­
( s1 ,!, sd ) : s j  ` , j 1,!, d , : (2 s ) t ¾ ,
N¿
¯
sd
s1
(2 ,!, 2 ).
ɠ(N)= ® s
ɝɞɟ: 2
s
­
Ɍɨ ɟɫɬɶ: ɠ(N)= ® s ( s1 ,!, sd ) : s j  ` , j 1,!, d ,
¯
d
ɋɠ(N) N \ ɠ(N).
ɉɨɥɨɠɢɦ Q (N )
‰ U (s ) ,
s ɠ(N)
U ( s)
ɝɞɟ
^k
(k1 ,!, k d ) : k j  N , 2
s j 1
d kj 2 ,
sj
½
rj s j b j
d
2
s
N
¾.
–
j
j 1
¿
d
`
j 1,!, d .
Ɍɚɤɠɟ ɪɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɚ
)(
­°
1
1
1 ½°
Ƚ ( N ) ® k (k1 ,!, kd ) : k j  ` , j 1,!, d , : ( ,!,
) t ¾,
k1
kd
N ¿°
¯°
d
­
½
rj
b
ɬɨ ɟɫɬɶ: Ƚ ( N ) ® k ( k1 ,!, kd ) : k j  ` , j 1,!, d , – k j (log k j ) j d N ¾ .
j 1
¯
¿
ed
ed
Ɇɨɠɧɨ ɥɟɝɤɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ Ƚ ( N )  Q (2 N )  Ƚ ( 2 N ) .
Ɉɬɤɭɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ Ƚ (N )
Q(N ) .
Ɍɚɤɠɟ ɧɚɦ ɩɨɧɚɞɨɛɹɬɫɹ ɦɧɨɠɟɫɬɜɚ
­
¯
T ( N ) ® s ( s1 ,!, sd ) : s j  ` , j 1,!, d ,
ȼ ɪɚɛɨɬɟ [1] ɞɨɤɚɡɚɧɨ, ɱɬɨ
T (N )
(log N ) d 1 .
1
1½
s
(2
)
d
:
¾.
N¿
2l N
)
(
Ɉɬɦɟɬɢɦ, ɱɬɨ ɮɭɧɤɰɢɹ : (t ) ɢɡ (1) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ (S) ɢ ( S l ) (ɫɦ. [1]). ɉɨɷɬɨ:
ɦɭ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɡɞɟɫɶ ɤɥɚɫɫɨɜ H q ɜɟɪɧɚ ɬɟɨɪɟɦɚ ɨ ɩɪɟɞɫɬɚɜɥɟɧɢɢ (ɫɦ. [2]).
Ɍɟɨɪɟɦɚ Ⱥ. Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ f (x) ɢɡ Lq ( S d ) ɩɪɢɧɚɞɥɟɠɚɥɚ ɤɥɚɫɫɭ H q ɧɟɨɛɯɨɞɢɦɨ
ɢ
ɞɨɫɬɚɬɨɱɧɨ
ɩɪɢ
1 q f , ɱɬɨɛɵ
0
G s ( f , x)
: (2 s )
s ( s1 ,!, sd ) ɫ ɧɚɬɭɪɚɥɶɧɵɦɢ ɤɨɦɩɨɧɟɧɬɚɦɢ, ɢ ɩɪɢ 1 d q d f
ɝɞɟ:
G s ( f , x)
¦
U
k ( s )
q
:
ɞɥɹ
ɜɫɟɯ
As ( f , x)
q
ɜɟɤɬɨɪɨɜ
: (2 s ) ,
f€(k ) e i ( k , x ) ,
f€(k ) - ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f (x) ,
As ( f , x) ɨɛɨɡɧɚɱɚɟɬ ɫɜɟɪɬɤɭ f ( x) ɫ ɹɞɪɨɦ As ( x) 2d – (V2 s j 1 ( x j ) V2 s j 2 ( x j )) , ɩɨɪɨɠɞɟɧɧɵɦ ɹɞɪɚɦɢ ȼɚɥɥɟ –ɉɭɫɫɟɧɚ Vm (W ) ɩɨɪɹɞɤɚ 2m 1 , ɤɪɨɦɟ ɬɨɝɨ, ɫɱɢɬɚɟɦ
j 1
106 ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
V2 1 (W ) { 1 .
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
Ɇɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɧɚɢɥɭɱɲɢɟ ɩɪɢɛɥɢɠɟɧɢɹ EQ ( N ) ( f ) q ɮɭɧɤɰɢɣ f (x) ɜ ɧɨɪɦɟ
Lq ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɦɢ ɩɨɥɢɧɨɦɚɦɢ ɫɨ ɫɩɟɤɬɪɨɦ ɢɡ Q(N ) .
EQ ( N ) ( H q: ) q sup EQ ( N ) ( f ) q
f H q
Ɍɚɤɠɟ ɨɛɨɡɧɚɱɢɦ
ɂɡ ɬɟɨɪɟɦɵ 1 ɪɚɛɨɬɵ [1] ɜɵɬɟɤɚɟɬ
:
.
Ɍɟɨɪɟɦɚ Ȼ. ɉɪɢ 1 q f EQ ( N ) ( H q: ) q
ɝɞɟ q0
)
(
min ^q ; 2`.
d 1
1
(log N ) q
N
,
0
r
ɉɪɢ ɢɡɭɱɟɧɢɢ ɩɪɢɛɥɢɠɟɧɢɣ ɮɭɧɤɰɢɣ ɢɡ ɤɥɚɫɫɨɜ ɋ.Ɇ. ɇɢɤɨɥɶɫɤɨɝɨ H q , ɤɨɬɨɪɵɟ ɫɨɜ-
:
ɩɚɞɚɸɬ ɫ H q ɩɪɢ : (t )
–t
d
rj
j
j 1
, ɜɵɹɫɧɢɥɨɫɶ (ɫɦ. [3, 4]), ɱɬɨ ɜɦɟɫɬɨ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ
r1 ! rQ rQ 1 d ! d rd ) , ɩɨɪɨɠɞɟɧɧɵɯ ɩɨ-
ɩɨɥɢɧɨɦɨɜ ɫɨ ɫɩɟɤɬɪɨɦ ɢɡ ɦɧɨɠɟɫɬɜ Qn ( r
r
rc
n
ɦɧɨɠɟɫɬɜɚ Q
rc
n
ɲɢɪɟ, ɱɟɦ Q , ɧɨ Q
r
n
–t
d
rj
rc
j
, ɥɭɱɲɟ ɛɪɚɬɶ ɦɧɨɠɟɫɬɜɚ Qn . Ⱦɟɥɨ ɜ ɬɨɦ, ɱɬɨ
)
(
ɜɟɪɯɧɨɫɬɹɦɢ ɭɪɨɜɧɹ ɮɭɧɤɰɢɢ : (t )
Qnr , ɚ ɧɚɢɥɭɱɲɢɟ ɩɪɢɛɥɢɠɟɧɢɹ EQ ( H qr ) q ɩɨ
j 1
rc
n
ɩɨɪɹɞɤɭ ɥɭɱɲɟ, ɱɟɦ EQ r ( H q ) q . ɗɬɨɬ ɷɮɮɟɤɬ ɩɨɡɜɨɥɹɟɬ ɭɥɭɱɲɢɬɶ ɨɰɟɧɤɢ ɩɨɩɟɪɟɱɧɢɤɨɜ.
r
rc
n
Ɇɧɨɠɟɫɬɜɚ Qn ɢ Qn ɧɚɡɵɜɚɸɬɫɹ «ɫɜɨɢɦɢ» ɢ «ɧɟ ɫɜɨɢɦɢ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦɢ ɤɪɟɫɬɚɦɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
ɗɮɮɟɤɬ «ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ ɞɥɹ : (t ) ɢɡ (1) ɩɪɢ ɭɫɥɨɜɢɢ
r
! rd ɪɚɫɫɦɨɬɪɟɧ ɚɜɬɨɪɨɦ ɪɚɧɟɟ (ɫɦ. [5]). ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɦɵ ɢɡɭɱɢɦ ɷɬɨɬ ɷɮɮɟɤɬ
ɜ ɫɥɭɱɚɟ ɪɚɡɥɢɱɧɵɯ r j .
r1
r2
Ⱥɜɬɨɪɨɦ ɞɨɤɚɡɚɧɚ (ɫɦ. [5])
Ɍɟɨɪɟɦɚ ȼ.
ɉɭɫɬɶ : (t ) ɡɚɞɚɧɨ ɪɚɜɟɧɫɬɜɨɦ (1), ɩɪɢɱɟɦ r1
r2 ! rd
b1 d b2 d ! d bd . Ɍɨɝɞɚ ɤɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ Ƚ (N ) ɩɨ ɩɨɪɹɞɤɭ ɪɚɜɧɨ
r,
N r ˜ I 1 ( N , r , b1 ,!, bd ) , ɝɞɟ: I1( d ) ( N , r , b1 ,!, bd ) ɪɚɜɧɨ (log N ) 1 (log log N )Q 1 ɩɪɢ
1
b1
! bQ
r bQ 1 d ! d bd ;
(log N )
b1
r
!
bQ
(Q 1)
r
(log log N ) P ,
ɩɪɢ
b1 d ! d bQ r bQ 1 ! bQ P bQ P 1 d ! d bd ; (log N ) r ɩɪɢ r d b 1 d ! d bd , b 2 ! r .
ȿɫɥɢ ɜ ɫɥɭɱɚɟ r1 ! rd r ɧɟɤɨɬɨɪɵɟ b j ɛɨɥɶɲɟ r , ɬɨ ɜɨɡɧɢɤɚɟɬ ɷɮɮɟɤɬ «ɧɟ ɫɜɨ
b1
ɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ.
ȼ ɫɥɭɱɚɟ r r1 ! rd ɪɨɥɶ «ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ ɢɝɪɚɸɬ ɦɧɨɠɟɫɬɜɚ
d
­
½
rs b c
‰
U
(s
)
, ɝɞɟ: ɠ`(N)= ® s ( s1 ,!, sd ) : s j  ` , j 1, !, d , – 2 j s j j d N ¾ ,
s
j 1
¯
¿
c
c
ɩɪɢɱɟɦ b j b j , j 1,!, Q , r b j b j , j Q 1,!, d .
Qc(N )
ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
107
c
c
ȼ ɫɥɭɱɚɟ r b1 ! bQ bQ 1 d!d bd ɛɟɪɟɦ bj b1 , j 1,!,Q, b1 bj bj , j
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
min ^ q ; 2 `. Ɍɨɝɞɚ
Ⱥɜɬɨɪɨɦ ɞɨɤɚɡɚɧɚ (ɫɦ. [5])
Ɍɟɨɪɟɦɚ Ƚ. ɉɭɫɬɶ 1 q f , q0
1
I2 ( N , d , q0 , b, bc) , ɝɞɟ I 2 ( N , d , q0 , b, bc) ɩɨ ɩɨɪɹɞɤɭ ɪɚɜɧɨ:
N
c ) ! ( b b c )
c
c 1
, ɟɫɥɢ bQ 1 bQ 1 d ! d b d b d ;
q0
d 1
( b Q 1 b Q 1
q0
Q P 1
q0
2) (logN)
d
d
(bQ 1 bQ 1c ) ! (bQ P bQ Pc )
Q P 1
q0
3) (logN )
)(
EQc ( N ) ( H q: ) q
1) (log N )
(loglogN)
d P Q
q0
[
( bQ 1 bQ 1c ) ! (bQ P bQ P c )
c
c
c
c 1
, ɟɫɥɢ bQ1 bQ1 d!dbQP bQP bQP1 bQP1 ! bd bd ;
q0
q0
(loglogN) ,
c
c
c
c
bQ 1 bQ 1 d!d bQ P bQ P bQ P1 bQ P1 ! bQ P[ bQ P[
Q 1
q0
4) (logN)
c
c
(loglogN) , ɟɫɥɢ bQ 1 bQ 1 ! bQ P bQ P
q0
, ɟɫɥɢ
Q P 1
6) (logN )
q0
1
c
c
bQ 1 bQ 1 d ! d b d b d ;
q0
(bQ 1 bQ 1c ) ! (bQ P bQ P c )
Ɂɚɦɟɱɚɧɢɟ.
EQc( N ) ( H q: ) q Ʉɚɤ
c
ɟɫɥɢ
1
c
c
bQP[1 bQP[1 d!dbd bd ;
q0
1
c
c
bQ P1 bQ P1 d!d bd bd ;
q0
P
q0
Q 1
5) (log N )
Q 1,!, d.
c 1
c
c
bQP1 bQP1 d!dbd bd .
q0
q 1, f ɜɟɪɧɚ ɨɰɟɧɤɚ ɫɜɟɪɯɭ
, ɟɫɥɢ bQ1 bQ1 d!dbQP bQP ɨɬɦɟɱɟɧɨ
1
I 2 ( N , d ,1, b, bc) .
N
ɜ
ɩɪɢ
[5],
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɬɟɨɪɟɦɵ Ƚ, ɨɩɪɟɞɟɥɟɧɧɵɟ ɪɚɡɥɢɱɢɹ ɜ ɩɨɤɚɡɚɬɟɥɹɯ ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɯ ɦɧɨɠɢɬɟɥɟɣ ɩɨɪɨɠɞɚɸɬ ɷɮɮɟɤɬ «ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ.
ɋɮɨɪɦɭɥɢɪɭɟɦ ɥɟɦɦɭ, ɞɨɤɚɡɚɧɧɭɸ ɜ [5]. ɗɬɚ ɥɟɦɦɚ ɢɫɩɨɥɶɡɨɜɚɥɚɫɶ ɩɪɢ ɞɨɤɚɡɚɬɟɥɶɫɬɜɟ
ɬɟɨɪɟɦɵ Ƚ, ɢ ɨɧɚ ɧɚɦ ɬɚɤɠɟ ɩɨɧɚɞɨɛɢɬɫɹ ɩɪɢ ɞɨɤɚɡɚɬɟɥɶɫɬɜɟ ɧɨɜɵɯ ɬɟɨɪɟɦ.
Ʌɟɦɦɚ Ⱥ. Ⱦɥɹ ɫɭɦɦɵ V
ɟɫɥɢ
ɟɫɥɢ
ɟɫɥɢ
s ( N ) j 1
J d ! d J 1 , ɬɨ V ԅ (log N )
(ɫɱɢɬɚɟɦ
ɟɫɥɢ
J 1 d ! d J d ):
¦
–s
T
d
J d! d JQ JQ
1
d
J d!dJQ JQ
1
1
J
1
! JQ
1
1
! Jd
ɜɟɪɧɵ ɫɥɟɞɭɸɳɢɟ ɩɨɪɹɞɤɨɜɵɟ ɪɚɜɟɧɫɬɜɚ
d 1J 1 !J d
;
1, ɬɨ V ԅ (log N )
Q 1J 1 !J Q
1 J Q 1
Q 1J1 !JQ
(log log N )Q 1
;
d ! d J d , ɬɨ V ԅ
log N
J 1
;
J d ! d J Q 1 J Q d ! d J , ɬɨ V ԅ (log N )
1
(log log N ) d Q ;
! JQP 1JQP1 d!dJ d ,Q P d d, ɬɨ V ԅ (log N)
ɟɫɥɢ 1 J 1 d ! d J d , ɬɨ V ԅ (log N )
ɟɫɥɢ
J j
j
1
d
108 ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
Q 1J 1 !J Q
.
(loglog N)P ;
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
§2. ɇɨɜɵɟ ɪɟɡɭɥɶɬɚɬɵ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɦɚɠɨɪɚɧɬɭ ɫɦɟɲɚɧɧɵɯ ɦɨɞɭɥɟɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɩɨɪɹɞɤɚ l ɢɡ (1) ɫ
rj . Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ r r1 ! rK rK1 d!drd, 1dKdd, 0r drd l,
ɪɚɡɥɢɱɧɵɦɢ
b1 d ! d b Q d r bQ 1 d ! d bK , 1 d Q d K , b j - ɩɪɨɢɡɜɨɥɶɧɵɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɢɫɥɚ ɩɪɢ
j K 1,!, d .
:
q
EQ ( N ) ( H ) q
)
(
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɬɟɨɪɟɦɚ Ȼ ɞɚɟɬ ɫɥɟɞɭɸɳɭɸ ɨɰɟɧɤɭ:
d 1
1
(log N ) q
N
0
Q(N ) , ɨɩɪɟɞɟɥɟɧɧɵɟ ɜɵɲɟ, - ɚɧɚɥɨɝɢ «ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ. Ⱥɧɚɥɨɝɚɦɢ
«ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ ɛɭɞɭɬ ɦɧɨɠɟɫɬɜɚ
Qcc ( N ) ‰ U ( s ) ,
sɠ``(N)
­
ɩɪɢɱɟɦ rj
s
–2
( s1 ,!, sd ) : s j  ` , j 1,!, d ,
ɝɞɟ: ɠ``(N)= ® s
¯
d
r jss j
j 1,!,K r rj s rj , j K 1,!, d ,
b j s b j , j 1,!,Q , r b j s b j , j Q 1,!, K ;
j 1
r,
½
bs
sjj d N ¾,
¿
ɜ ɫɥɭɱɚɟ r b1 ! bQ bQ 1 d ! d bK ɛɟɪɟɦ bjc b1, j 1,!,Q, b1 bjc bj , j
b jcc ɥɸɛɵɟ ɩɪɢ j K 1,!, d .
Ɍɟɨɪɟɦɚ 1.
Q 1,!,K,
ɉɭɫɬɶ : (t ) ɡɚɞɚɟɬɫɹ (1), ɩɪɢɱɟɦ r r1 ! rK rK1 d!d rd , b1 d!d bK .
Ɍɨɝɞɚ ɤɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ Q (N ) ɩɨ ɩɨɪɹɞɤɭ ɪɚɜɧɨ: N r <I 1 ( N , r , b1 ,!, bK ) ,
1
ɝɞɟ ɮɭɧɤɰɢɹ
I 1 ( N , r , b1 ,!, bK ) ɫɦ. ɜ ɮɨɪɦɭɥɢɪɨɜɤɟ ɬɟɨɪɟɦɵ ȼ ɫ ɡɚɦɟɧɨɣ d ɧɚ K .
Ȼɭɞɟɦ ɨɰɟɧɢɜɚɬɶ Ƚ (N ) ,
­
ɝɞɟ Ƚ ( N ) ®k
¯
½
bj
rj
k
k
N
(log
)
d
¾.
–
j
j j 1
¿
(k1 ,!, kd ) : k j  ` , j 1,!, d ,
d
ɉɨɧɹɬɧɨ, ɱɬɨ ɷɬɨ ɞɚɫɬ ɞɨɤɚɡɵɜɚɟɦɨɟ ɩɨɪɹɞɤɨɜɨɟ ɪɚɜɟɧɫɬɜɨ. ȼɜɟɞɟɦ ɫɥɟɞɭɸɳɢɟ ɦɧɨɠɟ-
Ƚ
(K )
­
( N ) ®k
¯
(k1 ,!, kK ) : k j  ` , j 1,!, K ,
¦
)
(
)
(
ɉɨɥɭɱɢɦ ɨɰɟɧɤɭ ɫɜɟɪɯɭ:
Ƚ (N )
½
bj
rj
k
k
N
(log
)
d
¾.
–
j
j j 1
¿
K
( k K 1 , ! , k d ):
k ( k 1 , ! , k d ) Ƚ ( N )
§
¨
N
Ƚ (K ) ¨ d
r
b
¨
¨ – k j (log k j ) © j K 1
j
j
·
¸
¸
¸
¸
¹
(ɩɨ ɬɟɨɪɟɦɟ ȼ ɫ ɡɚɦɟɧɨɣ d ɧɚ K )
)
(
ɫɬɜɚ:
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ:
ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
109
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
§
¨
¨
¨
¨
©
·
§
·
¸
¨
¸
N
N
¸
¨
˜ I1 d
, r , b1 ,!, bK ¸ ¦
d
r
b ¸
rj
bj
¨
¸
( k K 1 ,! , k d ):
k j j (log k j ) j ¸
–
k Ƚ ( N )
¨ – k j (log k j ) j K 1
¹
© j K 1
¹̧
§
·
¨
¸
N
, r , b1 ,!, bK ¸
I1 ¨ d
rj
bj
¨
¸
¨ – k j (log k j )
¸
j K 1
(K )
©
¹ Ƚ (K ) ( N ) ,
Ƚ ( N ) ˜ ¦
r
b
j
j
d
1
r
( k Q 1 ,! , k d ):
kȽ ( N )
ɬ.ɤ.
rj
r
k
–
K
j
1
r
j
(log k j ) r ˜ I 1 ( N , r , b1 ,!, bK )
j K 1, ! , d .
! 1 ɩɪɢ
ɉɨɥɭɱɚɟɦ ɨɰɟɧɤɭ ɫɧɢɡɭ.
Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɨ
Ƚ (N )
^k
(k1 ,!, kK ,1,!,1) : k j  ` , j 1,!, K , ( k1 ,!, kK )  Ƚ (K ) ( N ) ` .
ɂɬɚɤ, Ƚ (N )
)
(
Ɉɱɟɜɢɞɧɨ, Ƚ ( N )  Ƚ ( N ) . ɉɨɷɬɨɦɭ Ƚ ( N ) t Ƚ ( N )
Ƚ (K ) ( N ) .
Ƚ (K ) ( N ) .
ɍɱɢɬɵɜɚɹ ɬɟɨɪɟɦɭ ȼ, ɩɨɥɭɱɚɟɦ ɬɟɨɪɟɦɭ 1.
:
Ⱦɚɥɟɟ ɨɰɟɧɢɦ E Q cc ( N ) ( H q ) q .
Ɍɟɨɪɟɦɚ 2. ȿɫɥɢ ɮɭɧɤɰɢɹ : (t ) ɡɚɞɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ (1), ɬɨ ɜɟɪɧɨ ɫɥɟɞɭɸɳɟɟ ɩɨɪɹɞɤɨ-
1
˜ I 2 ( N , K , q 0 , b, bcc) , ɮɭɧɤɰɢɸ I 2 ( N , K , q 0 , b, bcc) ɫɦ. ɜ
N
ɬɟɨɪɟɦɟ Ƚ ɫ ɡɚɦɟɧɨɣ d ɧɚ K ɢ bc ɧɚ bcc .
:
)
(
ɜɨɟ ɪɚɜɟɧɫɬɜɨ: E Q cc ( N ) ( H q ) q
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɋɧɚɱɚɥɚ ɩɨɥɭɱɢɦ ɨɰɟɧɤɢ ɫɜɟɪɯɭ. ɑɟɪɟɡ S Q cc ( N ) ( f ) ɨɛɨɡɧɚɱɢɦ ɫɭɦɦɭ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
f ( x) , ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɦɧɨɠɟɫɬɜɭ Qcc( N ) .
ɂɡ ɦɧɨɝɨɦɟɪɧɨɣ ɬɟɨɪɟɦɵ Ʌɢɬɬɥɜɭɞɚ-ɉɷɥɢ (ɫɦ.[4], Ɍɟɨɪɟɦɚ Ⱥ) ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ f ( x)
§
¦
© sC
``(N)
ɢɡ H q ɛɭɞɟɬ E Q cc ( N ) ( f ) q ɠ
¨
:
G s ( f , x)
q0
q
·
.
¹̧
1
q0
q ·q
§
ɂɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ Ⱥ ɢ ɥɟɦɦɭ Ȼ ɢɡ [5], ɩɨɥɭɱɚɟɦ: E Qcc( N ) ( f )q ¨ ¦ :2s .
© sT cc( N )
¹̧
l
Ɂɞɟɫɶ T cc(N ) ɠ`` ( 2 N ) \ ɠ``(N).
1
0
2
ɂɦɟɟɦ: : ( 2 ) – b
j 1
sj
s
d
r j s j
j
: 1 (2 ) ˜
s
–
d
j K 1
110 ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
2
s j (r j r js )
b j b j s
sj
,
0
–
ɝɞɟ : 1 ( 2 )
s
j 1
2
b js
.
sj
( s 1 ,!, s d )  T cc( N ) ɛɭɞɟɬ : 1 (2 s )
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
·
§ d ( r r cc ) s
( b b cc ) ·
V 1.
˜¨ – 2 j j j ˜ sj j j ¸ ¸
¸
j 1
© j K 1
¹ ¹
N
Ⱦɥɹ ɮɢɤɫɢɪɨɜɚɧɧɵɯ sK 1 ,!, sd ɨɛɨɡɧɚɱɢɦ d
ɱɟɪɟɡ N1 , ɚ ɬɚɤɠɟ ɩɨɥɨɠɢɦ
b cc
r jcc s j ˜s j j
–2
–sj
K
( b j b jcc ) q0
q0
j K 1
­
TK cc ( N1 ) ®( s1 ,!, sK ) : s j  `,
¯
½
r ccs
b cc
N1 – 2 j j ˜ s j j d 2l N1 ¾.
j 1
¿
K
j 1,!, K ,
Ɍɨɝɞɚ, ɩɪɢɦɟɧɢɜ ɥɟɦɦɭ Ⱥ, ɩɨɥɭɱɢɦ:
)(
V1
1
q0
§
q0
§ d
·
cc
cc
1¨
( rj rj ) s j
( b b )
˜ sj j j ¸ ˜
¨– 2
¦
¦
¨
N ¨ ( sK 1 ,!, sd ): © j K 1
¹ ( s 1 ,!, sK )TKcc ( N1 )
© ( s 1 ,!, sd )T cc( N )
–s
K
( b j b jcc ) q0
j
j 1
§
·
q0
§ d
1¨
q0 ¸
( r j r jcc ) s j
( b j b jcc ) ·
˜ sj
¸ ˜ I2 ( N1 , K , q0 , b, bcc) ¸
¦ ¨– 2
N ¨¨ ( sK 1 ,!, sd ): © j K 1
¹
¸
© ( s 1 ,!, sd )T cc( N )
¹
·
¸
¸
¸
¹
1
q0
1
q0
)(
E Qcc( N ) ( f ) q 1§
¨ ¦
N ¨ sT cc( N )
©
1
.
N
)
(
Ⱦɥɹ s
d
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
r js s j
§
q0 ·
§ d
1
¨
( r j r jcc ) s j
( b j b jcc ) · ¸
2
I2 ( N1 , K , q0 , b, bcc) ˜ ¨
˜ sj
¸ ¸ ¦ ¨ j–
N
( s K 1 ,!, sd ):
1
K
¹ ¸
¨ ( s ,!, s )T cc( N ) ©
© 1 d
¹
1
I2 ( N1 , K , q0 , b, bcc) .
(2)
N
Ɂɞɟɫɶ ɦɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɜ ɧɚɲɢɯ ɭɫɥɨɜɢɹɯ I2 ( N1 , K , q0 , b, bcc) ɜɨɡɪɚɫɬɚɟɬ
1
q0
ɫ ɪɨɫɬɨɦ N ɢ ɬɨ, ɱɬɨ rj rjcc ! 0 . Ɉɰɟɧɤɢ ɫɜɟɪɯɭ ɞɨɤɚɡɚɧɵ.
ɉɨɥɭɱɢɦ ɨɰɟɧɤɭ ɫɧɢɡɭ.
ɉɭɫɬɶ ɫɧɚɱɚɥɚ 1 q 2 .
^s
s
(s1,!, sK ,1,!,1): sj ` , j 1,!,K ` ,
Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ f q ( x )
Ɂɞɟɫɶ A
B ˜ ¦ : (2 ) ˜ 2
s
s A
1
(
1
1)
q
s
G ( x) – sin x j .
(K )
s
1
d
j K 1
s1 ! sd , s (K ) (s1 ,!, sK ) ,
ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
111
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
G ( x)
(K )
s
– ¦
K
j 1
2
k
j
k 1 x 1 ! kK xK .
s j 1
2
e i ( k , x ) , (k , x)
s j 1
Ʉɚɤ ɢɡɜɟɫɬɧɨ,
G ( x)
(K )
s
s (K )
2
1
(1 1
)
q
.
)
(
ɉɨɷɬɨɦɭ ɜ ɫɢɥɭ ɬɟɨɪɟɦɵ Ⱥ f q ( x )  H q ɩɪɢ ɧɟɤɨɬɨɪɨɦ B ! 0 .
q
:
§
·
(: (2 s )) q .
¦
© sT cc ( N )
¹̧
Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ [1], E Q cc ( N ) ( f q ) q !! ¨
1
q
§
1¨
E Qcc( N ) ( f q ) q !! ¨
N ¨s
©
¦
–s
K
( s 1 ,!, sK , 1,!, 1):
sT cc ( N )
( b j b jcc ) q0
j
j 1
ɉɨɥɭɱɢɥɢ ɨɰɟɧɤɭ ɫɧɢɡɭ ɩɪɢ 1 q 2 .
·
¸
¸
¸
¹
1
q0
1
I2 ( N1 , K , q0 , b, bcc) .
N
)
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ɂ ɞɚɥɟɟ, ɚɧɚɥɨɝɢɱɧɨ ɫɨɨɬɧɨɲɟɧɢɸ (2) ɩɨɥɭɱɚɟɦ:
Ɍɟɩɟɪɶ ɩɨɥɭɱɢɦ ɨɰɟɧɤɢ ɫɧɢɡɭ ɩɪɢ 2 d q f. Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ f (x)
s
s
) ˜ ei(k , x) ,
s
k  U ( s ), (k , x) k x ! k x , ɚ ɫɭɦɦɚ ɛɟɪɟɬɫɹ ɩɨ ɜɫɟɦ ɜɟɤɬɨɪɚɦ
( s1 ,!, s d ) ɫ ɧɚɬɭɪɚɥɶɧɵɦɢ ɤɨɦɩɨɧɟɧɬɚɦɢ. Ɍɨɝɞɚ ɩɨ ɬɟɨɪɟɦɟ Ʌɢɬɬɥɜɭɞɚ-ɉɷɥɢ ɢɦɟɟɦ:
s
s
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§
s 2 ·
E Qcc( N ) ( f ) q !! ¨ ¦ (: (2 )) ¸ t ¨¨
© sT cc( N )
¹ ¨s
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2
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( s 1 ,!, s K ,1,!,1):
sT cc ( N )
·2
K
cc
( b b )2
s j j j ¸¸
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j 1
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Ɍɟɨɪɟɦɚ 2 ɞɨɤɚɡɚɧɚ.
ɂɬɚɤ, ɦɵ ɜɢɞɢɦ, ɱɬɨ
1
2
2
1
I2 ( N1 , K , 2, b, bcc) .
N
E Qcc ( N ) ( H q: ) q ɩɪɢ 1 q f ɩɨ ɩɨɪɹɞɤɭ ɦɟɧɶɲɟ, ɱɟɦ
E Q ( N ) ( H q: ) q ɩɪɢ : (t ) ɢɡ (1), ɯɨɬɹ Qcc(N )
Ɂɚɦɟɱɚɧɢɟ. ɉɪɢ q
¦
·
(: (2 )) ¸¸ !!
¸
¹
s
( s 1 ,!, s K ,1,!,1):
sT cc ( N )
1
)(
§
1¨
!! ¨
N ¨s
©
s
s1
d d
s1
1 1
)
(
ɝɞɟ
¦:(2
Q (N ) .
1; f ɜɟɪɧɵ ɨɰɟɧɤɢ ɫɜɟɪɯɭ: EQcc( N ) ( H q: ) q 1
I2 ( N , K ,1, b, bcc) .
N
Ʌɢɬɟɪɚɬɭɪɚ
1. ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ. ɉɪɢɛɥɢɠɟɧɢɟ ɦɧɨɝɨɦɟɪɧɵɯ ɮɭɧɤɰɢɣ ɫ ɡɚɞɚɧɧɨɣ ɦɚɠɨɪɚɧɬɨɣ ɫɦɟɲɚɧɧɵɯ ɦɨɞɭɥɟɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ // Ɇɚɬɟɦ. ɡɚɦɟɬɤɢ. Ɍ. 65. ȼɵɩ. 1. 1999. ɫ. 107-117.
2. ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ. ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɢ ɩɪɢɛɥɢɠɟɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ ɫ ɡɚɞɚɧɧɵɦ ɫɦɟɲɚɧɧɵɦ ɦɨɞɭɥɟɦ ɧɟɩɪɟɪɵɜɧɨɫɬɢ // Analysis Mathematica,
v.20(1994), c.35-48.
3. Ɍɟɥɹɤɨɜɫɤɢɣ ɋ.Ⱥ. ɇɟɤɨɬɨɪɵɟ ɨɰɟɧɤɢ ɞɥɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɪɹɞɨɜ ɫ ɤɜɚɡɢɜɵɩɭɤɥɵɦɢ
ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ // Ɇɚɬɟɦ. ɫɛɨɪɧɢɤ. 1964. Ɍ.63(105). ɫ.426-444.
4. Ɍɟɦɥɹɤɨɜ ȼ.ɇ. ɉɪɢɛɥɢɠɟɧɢɟ ɮɭɧɤɰɢɣ ɫ ɨɝɪɚɧɢɱɟɧɧɨɣ ɫɦɟɲɚɧɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ // Ɍɪɭɞɵ
ɆɂȺɇ ɋɋɋɊ. 1986. Ɍ. 178. ɫ. 1-112.
112 ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
5.
6.
7.
8.
9.
ɋɟɪɢɹ 3. ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ.
ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ. Ɉ ɧɚɢɥɭɱɲɢɯ ɩɪɢɛɥɢɠɟɧɢɹɯ ɚɧɚɥɨɝɚɦɢ «ɫɜɨɢɯ» ɢ «ɧɟ ɫɜɨɢɯ» ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɤɪɟɫɬɨɜ // Ɇɚɬɟɦ. ɡɚɦɟɬɤɢ. 2013. Ɍ.93. ȼɵɩ. 3. ɫ.460-470.
ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ. Ɉɪɬɨɩɨɩɟɪɟɱɧɢɤɢ ɤɥɚɫɫɨɜ ɦɧɨɝɨɦɟɪɧɵɯ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ, ɦɚɠɨɪɚɧɬɚ ɫɦɟɲɚɧɧɵɯ ɦɨɞɭɥɟɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɤɨɬɨɪɵɯ ɫɨɞɟɪɠɢɬ ɤɚɤ ɫɬɟɩɟɧɧɵɟ, ɬɚɤ ɢ ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɟ ɦɧɨɠɢɬɟɥɢ // Analysis Mathematica, v.34(2008), c.187-224.
Ȼɭɝɪɨɜ ə.ɋ. ɉɪɢɛɥɢɠɟɧɢɟ ɤɥɚɫɫɚ ɮɭɧɤɰɢɣ ɫ ɞɨɦɢɧɢɪɭɸɳɟɣ ɫɦɟɲɚɧɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ //
Ɇɚɬɟɦ. ɫɛɨɪɧɢɤ. 1964. Ɍ. 64(106). ɫ.410-418.
Ȼɭɝɪɨɜ ə.ɋ. Ʉɨɧɫɬɪɭɤɬɢɜɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɤɥɚɫɫɨɜ ɮɭɧɤɰɢɣ ɫ ɞɨɦɢɧɢɪɭɸɳɟɣ ɫɦɟɲɚɧɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ// Ɍɪɭɞɵ ɆɂȺɇ ɋɋɋɊ. 1974. Ɍ.131. ɫ.25-32.
r
ɇɢɤɨɥɶɫɤɚɹ ɇ.ɋ. ɉɪɢɛɥɢɠɟɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɤɥɚɫɫɚ SH p ɫɭɦɦɚɦɢ Ɏɭɪɶɟ //
ɋɢɛ. ɦɚɬɟɦ. ɠɭɪɧ. 1975. Ɍ. 16. ʋ4. ɫ.761-780.
10. ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ. Ɉɪɬɨɩɨɩɟɪɟɱɧɢɤɢ ɧɟɤɨɬɨɪɵɯ ɤɥɚɫɫɨɜ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɞɜɭɯ
ɩɟɪɟɦɟɧɧɵɯ ɫ ɡɚɞɚɧɧɨɣ ɦɚɠɨɪɚɧɬɨɣ ɫɦɟɲɚɧɧɵɯ ɦɨɞɭɥɟɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ // ɂɡɜ. ɊȺɇ. ɋɟɪɢɹ ɦɚɬɟɦ. Ɍ.64(2000). ɫ.123-144.
11. ɉɭɫɬɨɜɨɣɬɨɜ ɇ.ɇ. Ɉ ɩɪɢɛɥɢɠɟɧɢɢ ɢ ɯɚɪɚɤɬɟɪɢɡɚɰɢɢ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ, ɢɦɟɸɳɢɯ ɦɚɠɨɪɚɧɬɭ ɫɦɟɲɚɧɧɵɯ ɦɨɞɭɥɟɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɫɩɟɰɢɚɥɶɧɨɝɨ ɜɢɞɚ
// Analysis Mathematica, v.29(2003), c.201-218.
ɂɡɜɟɫɬɢɹ ɆȽɌɍ «ɆȺɆɂ» ʋ 1(15), 2013, ɬ. 3
113
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