Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой
Saved in:
| Published in: | Український математичний журнал |
|---|---|
| Date: | 2011 |
| Main Author: | |
| Format: | Article |
| Language: | Russian |
| Published: |
Інститут математики НАН України
2011
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/166404 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой / С.А. Пичугов // Український математичний журнал. — 2011. — Т. 63, № 12. — С. 1657–1671. — Бібліогр.: 12 назв. — рос. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860085542125830144 |
|---|---|
| author | Пичугов, С.А. |
| author_facet | Пичугов, С.А. |
| citation_txt | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой / С.А. Пичугов // Український математичний журнал. — 2011. — Т. 63, № 12. — С. 1657–1671. — Бібліогр.: 12 назв. — рос. |
| collection | DSpace DC |
| container_title | Український математичний журнал |
| first_indexed | 2025-12-07T17:19:28Z |
| format | Article |
| fulltext |
© ! . A. "#$%&'( , 2011
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12 1657
!"# 517. 5
!. ". #$%&'() ("$%&'(&%)'. ) *+. ,$ -) -. -.. )'*$ / &.)
*+,"-+*!.-" /01 .,23 4*45+.,26+!728
#402*454- - #,4!.,"* !.-"8
! 2*. +3,"09*4: 5+.,274:
In the spaces L! (T ) of periodic functions with metric ! ( f , 0)" = " (| f (x) |)
T# dx , where ! is a
function of the modulus-of-continuity type, we investigate analogs of the classic Bernstein inequalities for the
norms of derivatives and increments of trigonometric polynomials.
! &'(/)('*0 L! (T ) &%'1(.23$20 4,$5+16 7 8%)'25(9 ! ( f , 0)" = " (| f (x) |)
T# dx , .% ! Ñ 4,$5 -
+1: )2&, 8(.,;: $%&%'%'<$(/)1, .(/; 1.-e$( *$*;(=2 5;*/23$20 $%'1<$(/)%6 >%'$?)%6$* .;: $('8 & ( 01.-
$20 )* &'2'(/)1< )'2=($ ( 8%)'23$20 &(;1$(81<.
1. -);<;=$;. "*$$*: /)*)@: :<;:%)/: &'(.(;-%$2%8 '*A() [1, 2]. B/% ( / $(<$C%
(A(7$*3%$2: 2 &($:)2: /8. < [2].
";: .%6/)<2)%;@$(7$*3$C0 4,$5+26 f (x), x !R1, 28%9D20 &%'2(. 1, L0 !
! L0(T) Ñ 8$(- %/)<( 278%'28C0 2 &(3)2 </9., 5($%3$C0 4,$5+26 $* )('% &%-
'2(.(< T = [0,1] ; ! Ñ 8$(-%/)<( 4,$5+26 ! : R+
1 " R+
1 , :<;:9D2 0/: 8( ., ;%8
$%&'%'C<$( /)2;
L! = L! (T) = f " L0(T) : f ! = ! (| f (x) |)
T
# dx < $
%
&
'
('
)
*
'
+'
Ñ 8%)'23%/52% &'(/)'*$/)<* (< /;,3*% ! " # ).
B E)20 &'(/)'*$/)<*0 '*//8()'28 &(.&'(/)'*$/)<* !T
2n+1 )'2=($(8%)' 23%-
/520 &(;2$(8(< Tn(x) = ckei2! kx
k=" n
n# , c! k = ck , 2 ;2$%6$C% (&%'*)('C A :
!T
2n+1 ! !T 2n+1. FC A,.%8 27,3*)@ $('8C E)20 &( ;2$(82*;@$C0 (&%'*)('(<, ). e.
<%;232$C
A ! , n := sup
Tn" !T 2n+1,Tn#0
ATn !
Tn !
. (1)
G'2 E)(8 $*/ < &%'<,9 (3%'%.@ 2$)%'%/,9) *$*;(=2 5;*//23%/520 $%'*<%$/)< )2&*
>%'$?)%6$* .;: &'(27<(.$C0 2 &'2'*D%$26 &(;2$(8(<; E)28 *+,-.*/.0) <CA('
5;*//(< (&%'*)('(< A , 5()('C% 8C 27,3*%8.
H//;%.(<*$29 )*520 $%'*<%$/)< < $('82'(<*$$C0 &'(/)'*$/)<*0 & ( /<:D%$(
8$(=( '*A() (/8., $*&'28%', 8($(='*422 [3, 4]). O)8%)28 )(; @5(, 3)( < 8%)'23%-
/520 &'(/)'*$/)<*0 Lp, p ! (0,1) , )(3$C% &( &(':.5, $ %'*<%$/)<* >%'$?)%6$*
.;: &'(27<(.$C0 Tn!(x)
1658 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
!Tn (x) p " Cnp Tn p (2)
2 &'2'*D%$26 !hTn(x) = Tn x +
h
2
"
#$
%
&' ( Tn x ( h
2
"
#$
%
&'
! hTn p " C(nh)p Tn p , 0 < nh !
1
2
, (3)
.(5*7*$C < [5, 6], * < '*A()% [7], < 3*/)$(/)2, $*6.%$* )(3$*: 5($/)*$)* < (2).
B )1-2*3405 '*A()% &(;,3 0)6 *$*;(=2 $%'*<%$/)< (2), (3) < &'(/)'*$/)<*0
L! . G'2;(-%$29 E)20 '%7,;@)*)(< 5 2//;%.(< *$29 (A'*)$C0 )%('%8 "-% 5/($* <
&'( /)'*$/)<*0 L! A,.%) &(/<:D%$* ().%;@$*: /)*)@:.
2. 2=>;?@(ABC$(==DB E(?F&AD. M)8%)28 (.$( <*-$(% &'%.&( ;(-%$2% ()-
$(/2)%;@$( (&%'*)('(< A . B/9., < .*;@$%6?%8 (2 E)( $% A,.%) (=(<*'2<*)@/:
().%;@$() 27,3*9)/: (&%'*)('C A, 5()('C% (&'%.%;:9)/: 8$(-2)%;:82
! k " C; k # n{ } , ! k = ! " k , &( 4('8,;%
A ckei2! kx
k " n
#
$
%
&
'
(
) = * kckei2! kx
k " n
# .
M3%<2.$(, 3)( 5*-.C6 )*5(6 (&%' *)(' A &%'%/)*$(<(3%$ /( /.<2=(8; E)(
(7$*3*%), 3)( !t A = A!t .;: </%0 (&%'*)('(< ! t /.<2=* $* &*' *8%)' t .
B<%.08 %D0 *$*;(=2 5;*//23%/520 &(;2$(8(< (:. %') B*;;% G,//%$* (/8., $* -
&'28%', [3]).
MA(7$*328 3%'%7 P! 5;*// 4,$5+26 ! : R" R )*520, 3)(:
1) ! (s) = 1 .;: s ! [" 1,1] ; ! (s) = 0 .;: s ! 2 ;
2) ! (" s) = ! (s) ;
3) ! " C(R) .
#*-.*: 4,$5+2: ! E)(=( 5;*//* &('(-.*%) )'2=($(8%)'23%/526 &(; 2$(8
Vn(x) ! Vn(x; " ) := "
k
n
#
$%
&
'(
ei2) kx
|k|<2n
* (4)
/)%&%$2 $% <C?% 2n ! 1.
";: (&%'*)('* A , &%'<($*3*;@$( 7*.*$$(=( $* &(;2$(8*0 /)%&%$2 n , A,.%8
2/&(;@7(<*)@ %=( &'(.(;-%$2% $* &(;2$(8C /)%&%$2 2n &( &'*<2;,
! n+k := ! n" k ; ! " (n+k) := ! n+k .;: k = 1,É , n . (5)
";: <$(<@ &(;,3%$$(=( (&%'*)('* / 8$(-2)%;:82 ! k ; k " 2n{ } /(0'*$28
&'%-$%% (A(7$*3%$2% A.
N*?2 (+%$52 $('8 (&%'*)('(< A A*72',9)/: $* /;%.,9D%6 2$)%'&(; : +2($-
$(6 4('8,;%.
.;(?;FD 1. +,- ,./010 20,3)0$% Tn ! !T 2n+1 3 4567 x, t ! R 52#%468,340
500&)096)36
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1659
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
ATn(x + t) =
1
3n
Tn(t + xj )AVn(x ! xj )
j =1
3n
" , (6)
186 x j =
j
3n
Ñ 535&6$% #%4)00&5&0-:37 &0;6" )% 26#3086 T = [0,1] , Vn 02-
#686,6)< 4 (4), ! "P# , % =)%;6)3- AVn 02#686,-.&5- 5 20$0:>. (5).
Доказательство. ";: &(;2$(8* Tn /&'*<%.;2<( 2$)%='*;@$(% &'%. /)* <;%-
$2%
Tn(x) = Tn(u) !Vn(x " u)du
T
#
(E)( /;%.,%) 27 )(=(, 3)( ! (s) = 1 &'2 s ! 1 ). G(. 6$)%='*;@$*: 4,$5+2: %/)@
&(;2$(8 /)%&%$2 $% <C?% 3n ! 1 . ";: <C32/;%$2: 2$)%='*;* 2/&(;@7,%8 5<*.'* -
),'$,9 4('8,;, &':8(,=(;@$25(< / 3n ,7;*82, )(3$,9 $* &(;2$ ( 8*0 /)%&%$2
3n ! 1 :
Tn(x) =
1
3n
Tn(xj )Vn(x ! xj )
j =1
3n
" .
"*-7*.87, E)( /(()$(?%$2% /&'*<%.;2<( .;: ; 9A(=( &(;2$(8*, &'28%$28 %=(
.;: !tTn :
!tTn(x) = 1
3n
Tn(t + xj )Vn(x " xj )
j=1
3n
# . (7)
G(.%6/)<,%8 (&%'*)('(8 A $* (A% 3*/)2 (7) 2 &(;,328 (6).
3. 4C;=G$ =(?FH E$GI$?()D==('( (@;?D>(?D. ";: 7*.*$$(6 4,$5+22 ) 2&*
8(. , ;: $%&'%'C<$(/)2 ! (&'%.%;28 %% 4,$5+29 '*/):-%$2: M ! (s) , s ! (0, " )
[8]:
M! (s) = sup
0<t<"
!(st)
!(t)
.
M3%<2.$(, 3)(
! (st) " ! (s)M! (t) . (8)
.;(?;FD 2. ?#3 ,./0@ ! "# 4 2#05&#%)5&46 L! 8,- )0#$< 026#%&0#%
A 3$6.& $65&0 84(5&0#0))36 )6#%46)5&4%
1
2M! ( 2)
M! max
k"n
#k( ) " A !, n " inf
$%P&
3n M!
1
3n
AVn(x)'
()
*
+, dx
T
- . (9)
Доказательство. ";: (+%$52 /<%'0, 2/&(;@7,%8 )%('%8, 1.
H7 &(;,*. . 2)2<$(/)2 U, (6) 2 (8) /;%.,%)
! ATn(x + t )( ) " ! Tn(t + xj )( ) M!
1
3n
AVn(x # xj )
$
%&
'
()j=1
3n
* .
1660 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
H/&(;@7,: 2$<*'2*$)$(/)@ &( /.<2=, ! -8%)'252, ()/9.* &(;,3*%8 &' *<,9 3*/)@
(9):
ATn ! = ! ATn (x + t)( ) dtdx "
t#T
$
x#T
$
! " Tn(t + xj )( ) dt
t#T
$
j =1
3n
% M"
1
3n
AVn(x & xj )
'
()
*
+,
dx =
x#T
$
= Tn ! 3n M!
1
3n
AVn(x)"
#$
%
&'
dx
x( T
) .
";: .(5*7*)%;@/)<* $2-$%6 (+%$52 < (9) '*//8()'28 &(;2$(8
pk(x) = cos(2!kx) , k = 0,1,É , n .
G,/)@ k ! 0 , ! k = ! k ei" k . Q(=.*
Apk(x) =
1
2
A(e2!ikx + e"2!ikx ) =
1
2
#k ei(2!kx+$k ) + #k e"i (2!kx+$k )( ) =
= ! k cos(2" kx + #k ) ,
Apk ! = ! " k sin 2#x( ) dx $ 2 ! " k sin 2#x( ) dx $
1/8
3/8
%
0
1
%
!
1
2
" #k
2
2
$
%&
'
()
.
O/;2 -% k = 0 , )( Apk ! = ! "0( ) # 1
2
! "0
2
2
$
%&
'
()
.
Q%&%'@ < (+%$5% /$27, 2/&(;@7,%8 /%8%6/)<( &(;2$(8(< !pk(x);{
k = 0,1,…, n; ! > 0} :
A ! , n " sup
{ #pk (x)}
A#pk !
#pk !
" sup
#>0
max
0$k$n
1
2
! # %k
2
2
&
'(
)
*+
! (#)
=
=
1
2
sup
! >0
" ! max
k#n
$k
2
2
%
&'
(
)*
" (! )
=
1
2
M"
2
2
max
k#n
$k
%
&'
(
)*
+
!
1
2M " ( 2)
M " max
k#n
$k( ) .
N* &(/;%.$%8 E)*&% 2/&(;@7(<*$( /<(6/)<( M ! (xy) " M! (x)M! (y).
Q%( '%8* 2 .( 5*7*$*.
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1661
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
"*-7*.87, (&%'*)(' A (.$(7$*3$( (&'%.%;:%)/: 8$(-2)%;:82 !k; k " n{ } ,
(+%$52 %=( $('8 -%;*)%;@$( &(;,32)@ < )%'82$*0 ! k{ } . B E)(8 /8C/;% &'* <,9
(+%$5, (9) %D% $%;@7: /32)*)@ ,,0('(?%6 Ó. FC &'(.<2$%8/: .*; @?% < (+%$5*0
/<%'0, $('8 (&%'*)('(<, $*5;*.C<*: $%5()('C% .(&(;$ 2)%;@$C% (='*$23%$2: 5*5
$* (&%'*)('C, )*5 2 $* ! -8%)'252.
4. 4C;=GD =(?F @(IA;<()D>;AJ=(I>;K (@;?D>(?(). N*&(8$28 [8], 3)(
&(<%.%$2% 4,$5+22 '*/):-%$2: MU .;: ! " # < &'*<(6 (5'%/)$(/)2 $,;: 0*'*5 -
)%'27,%)/: )*5 $*7C<*%8C8 $2-$28 &(5*7*)%;%8 '*/):-%$2: ! " , 9:0 9D28
/<( 6/)<*:
a) ! " # [0,1] ;
A) M ! (s) " s#! !s"(0,1] ;
<) ;.3 .<+6= ! > 0 9 s ! (0,1] - )07*2*>*5 7*)-21)2*5 C!
M! (s) " C#s
$ ! %# .
M)/9.* /;%.,%), < 3*/)$(/)2, 3)( < /;,3*% ! " = 0 M ! (s) " 1 .;: s ! [0,1] ,
* &'2 ! " > 0 M ! (+ 0) = 0 .
B E)(8 &,$5)% 2//;%.,%8 &(/;%.(<*)%;@$(/)2 (&%'*)('(< An; n = 1, 2,É{ } ,
(A'*7(<*$$C% &( /;%.,9D%8, &'*<2;,: 7*.*$* $%5()('*: 4,$ 5+2: µ(s) : R ! C ,
µ(! s) = µ(s) , 2 (&%'*)(' An , n = 1, 2,É , .%6/)<,9D 96 $* !T 2n+1, (&'%.%;:%)/:
8$(-2)%;:82 ! k := µ(k) , k ! n . ";: &(/;%.(<*)%;@$(/)2 )*520 (&%'*)('(<
$%) $%(A0(.28(/)2 < &'(+%.,'% 20 &'(.(;-%$2: (5), 2 (+%$5* /<%'0, (9) &'2$28*%)
<2.
An ! , n " inf
#$P%
3n M!
1
3n
&n
k
n
'
()
*
+,
ei2- kx
|k|<2n
.
'
()
*
+,
dx
T
/ , (10)
=.% ! n(s) := µ(ns) " (s) .
B .*;@$%6?%8 (='*$2328/: =;*.5282 4,$5+2:82 ! " P# ! C$ (R) 2 ; ( 5*;@$(
2$)%='2',%8C82 4,$5+2:82 µ . B E)(8 /;,3*% &'%(A'*7(<*$2% V, '@% 4,$5+22
! n ,
ö! n (x) = ! n (s)e" i2#sxds
R
$ ,
:<;:%)/: 4,$5+2%6, 2$)%='2',%8(6 $* (/2.
.;(?;FD 3. ?(5&> ! " > 0 , % A()"B3- µ(s) &%"04%, ;&0 8,- 8%))010 n
)%@8(&5- ! " P# ! C$ (R) , ! > 0 3 "0)5&%)&% K(n, ! ) &%"36, ;&0 8,- x ! R
4<20,)6)0 )6# %46)5&40
ö! n(x) "
K(n,#)
1+ x( )
1
$%
+#
. (11)
1662 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
C018% 8,- )0#$< 500&46&5&4(.:610 026#%&0#% An 52#%468,34% 0B6)"%
An ! , n " 3M!
1
3
#
$%
&
'(
M!
ö) n(x)( )
R
* dx . (12)
Доказательство. G( 4('8,;% /,882'(<*$2: G,*//($* (/8., $*&' 28%', [9])
! n
k
n
"
#$
%
&'
ei2( kx = n!̂ n(n(x ) j ))
j * Z
+
|k|, 2n
+ ,
&'2 E)(8 ' :. /&'*<* '*<$(8%'$( /0(.2)/: A;*=(.*': ,/;(<29 (11).
"*-7*.87, ! " > 0, 27 (11) 2 -/*5 /)<* <) .;: 4,$5+22 '*/):-%$2: &'2 &(.0( -
.:D%8 <CA('% ! /;%.,9) '*<$(8%'$*: /0(.28(/)@ ':.*
M !
1
3
ö" n(n(x # j ))$
%&
'
()
j * Z
+
2 /0(.28(/)@ 2$)%='*;*
M !
ö" n(x)( ) dx
R
# .
Q*5 5*5 ! &(;,*..2)2<$*, 27 (&'%.%;%$2: M! <2.$(, 3)( 4,$ 5+2: M !
)*5-% &(;,*..2)2<$*. G(E)(8,
M !
1
3n
" n
k
n
#
$%
&
'(
ei2) kx
|k|* 2n
+
#
$%
&
'(
* M!
1
3
ö" n(n(x , j ))#
$%
&
'(
j - Z
+ . (13)
Q%&%'@ 27 (10) 2 (13) /;%.,%), 3)(
An ! , n " 3n M!
1
3
ö#n(n(x $ j ))%
&'
(
)*
dx = 3 M!
1
3
ö#n(x)%
&'
(
)*
dx "
R
+
T
+
j, Z
-
! 3M "
1
3
#
$%
&
'(
M " )̂ n (x)( ) dx
R
* .
Q%('%8* 3 .(5*7*$*.
# 7<%/)$( (/8., $*&'28%', [9]), 3)( %/;2 4,$5+2: f 27 L1(R) )*5( <*, 3)(
4,$5 +22 f , !f , ..., f (l"1) *A/(;9)$( $%&'%'C<$C $* 5*-.(8 5($%3$(8 2$)%'<*;%
(l ! N) , * f (l ) ! L1(R) , )( .;: </%0 x !R <C&(;$:%)/: $%'*<%$/)<(
öf (x) !
"K
1+ x( )l .
Q*528 (A'*7(8, A;*=(.*': )(8,, 3)( ! Ñ A%/5($%3$( .244%'%$+2',%8*:
4,$5+2: / 5($%3$C8 $(/2)%;%8, .;: /6?*.)0)93 $%'*<%$/)<* (11) 8(-$( ,5*7*)@
.(/)*)(3$C% ,/;(<2: < )%'82$*0 =;*.5(/)2 4,$5+22 µ .
Следствие 1. ?(5&> ! " > 0 , % A()"B3- µ )% 0="6 [ ! 2n, 2n]
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1663
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
%/50,. &)0 )62#6#<4)% 4$65&6 50 5403$3 2#03=408)<$3 !µ , !!µ , ...,µ 1/" #$% &' +1
.
C018% 8,- D&010 =)%;6)3- n 4<20,)-6&5- )6#%46)5&40 (12).
H7 E)(=( 4*5)* ;%=5( /;%.,%) *$*;(= $%'*<%$/)< >%'$?)%6$*.
Следствие 2. ?(5&> ! " > 0 3 r ! N . C018% 3$6.& $65&0 )6#%46)5&4%
C1(r )M! (nr ) " sup
Tn# !T 2n+1,Tn$0
Tn
(r )
!
Tn !
" C2(r )M! (nr ) (14)
5 "0)5&%)&%$3 C1(r ) , C2(r ) , )6 =%435-:3$3 0& n .
Доказательство. V,$5+2: µ(s) = (i2! s)r &'2$*.;%-2) C! (R), &(E)( 8,
.;: (+%$52 /<%'0, 8(-$( 2/&(;@7(<*)@ (12).
"*;%%, )*5 5*5 µ(s) Ð (.$('(.$*: 4,$5+2: /)%&%$2 r , )(
! n(s) = µ(ns) " (s) = nrµ(s) " (s) ,
ö! n(x) = nr (i2" x)r!
#(x) ,
Tn
(r )
!
" Tn ! #3M!
1
3
$
%&
'
()
M! (i2* x)r!
+(x)$
%
'
( dx M! (nr )
R
, .
M+%$5* /$27, /;%.,%) 27 (9).
Следствие 3. ?(5&> ! " > 0 3 nh ! (0,1/2] . C018% 8,- k !N 3$6.& $65 -
&0 ) 6#%46)5&4%
C1(k)M! ((nh)k ) " sup
Tn#T 2n+1,Tn$0
%h
kTn !
Tn !
" C2(k)M! ((nh)k ) (15)
5 "0)5&%)&%$3 C1(k) , C2 (k) , )6 =%435-:3$3 0& n 3 h .
Доказательство. ";: </%0 k ! N '*//,-.%$2: (.2$*5(<C%, &(E)(8, .;:
&'(/ )()C (='*$2328/: /;,3*%8 k = 1 .
V,$5+2: µ(s) = 2i sin(! hs) &'2$*.;%-2) C! (R), 2 &( /;%./)<29 1
! hTn " # Tn " $3M"
1
3
%
&'
(
)*
M"
ö+n(x)( ) dx
R
, .
"*-7*.87,
ö! n (x) = (µ(n") #("))! (x) = $ nh ö#(x) , )(
M !
ö" n(x)( ) dx # M! (nh) M!
$ nh ö%(x)
nh
&
'(
)
*+
dx
R
,
R
, ,
2 .;: (+%$52 /<%'0, (/)*;(/@ &(5*7*)@, 3)( 4,$5+2:
! (y) := M"
# y$̂ (x)
y
%
&
'
(
)
* dx
R
+
1664 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
'*<$(8%'$( (='*$23%$* .;: y ! [0,1/2] . B-.0;-2/90 )(=(, 3)( ! (y) $%&'%'C<$*
&'2 y > 0 , .(/)*)(3$( .(5*7*)@ /,D%/)<(<*$2% 5($%3$(=( &'%.%;* !(y) &'2
y ! 0 .
Q*5 5*5 42$2)$*: 4,$5+2: (iy)2! (y) " C# , %% &'%(A'*7(<*$2% V,'@%, '*<$(%
D2 ö!(x) , ,AC<*%) $* A%/5($%3$(/)2 AC/)'%% ;9A(6 /)%&%$2:
D2 ö! (x) "
CN
1+ x( )N .
Q(=.* &( 4('8,;% Q%6;('* .;: $%5()('(6 )(352 ! " [ x, x + y] 28%%8
! y"̂ (x)
y
# D"̂ (x) =
1
2
y D2"̂ ($) %
CN&y
1+ $( )N %
CN&y
1+ x( )N ,
M !
" y ö#(x)
y
$ D ö#(x)
%
&'
(
)*
dx +
R
, M! (CN-)M! (y) M!
1
1+ x( )N
%
&
'
(
)
* dx
R
, . (16)
"*-7*.87, ! " > 0 , &'2 .(/)*)(3$( A(;@?20 N 2$)%='*; < &'*<(6 3*/)2 (16)
5($%3%$, * M! (y) " 0 &'2 y! 0. M)/9.* /;%.,%), 3)(
lim
y! 0
" (y) = M# D ö$(x)( ) dx < %
R
& ,
2 (+%$5* /<%'0, < (15) .(5*7*$*. M+%$5* /$27, /;%.,%) 27 (9).
J$*;(=23$( .(5*7C<*%)/: 2 /;%.,9D26 A(;%% (AD26 4*5).
.;(?;FD 4. ?(5&> ! " > 0 , k = 0,1, …, r = 0,1, É , nh!(0,1/2] . C018% 3$6-
.& $ 65&0 )6#%46)5&4%
C1(k, r )M! (nr +khk ) " sup
Tn# !T 2n+1,Tn$0
%h
kTn
r( )
!
Tn !
" C2 (k, r )M! (nr +khk ) , (17)
C3M! max
|k|"n
k r sin#kh
k
$ #k
%
&'
(
)*
%
&'
(
)*
" sup
Tn+ !T 2n+1,Tn,0
-h
h
$ D%
&'
(
)* Tn
(r )
!
Tn !
"
! C4M" max
|k|! n
k r sin #kh
k
$ #k
%
&'
(
)*
%
&'
(
)*
. (18)
W*8%)28 %D%, 3)( &'*<C% (+%$52 < (17), (18) /&'*<%.;2<C &'2 </%0 h ! (0,1/2] .
5. *;?D);=I>)D <AB @?($L)(<=HM $ @?$?DN;=$K ) IA&%D; ! " = 0. W*8%-
)28, 3)( (+%$52 /$27, < (14), (15), (17) (/)*9)/: /&'*<%.;2<C82 2 < /;,3*%
! " = 0. I .',=(6 /)('($C, (3%<2.$(, 3)( !h
kTn "
# 2k Tn " .
B-.0;-2/90 )(=(, 3)( &'2 ! " = 0 M! (y) " 1 .;: </%0 y > 0 , 27 (+%$52
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1665
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
/$27, < (15) )0?*->0;-2/0))* <C)%5*%) /;%.,9D26 4*5).
O>);?P<;=$; 1 . E ,./0$ 2#05&#%)5&46 L! 2#3 ! " = 0 )%@8(&5- "0) -
5&%)&< Ck > 0 &%"36, ;&0 8,- 4567 n = 1, 2,É 52#%468,34< 500&)096)3-
2k ! lim
h" 0
sup
Tn# !T 2n+1,Tn$0
%h
kTn &
Tn &
! Ck > 0 . (19)
Q*528 (A'*7(8, ,)<%'-.%$2% 1 (7$*3*%), 3)( $%'*<%$/)< >%'$?)%6$* .;: &'2-
'*D%$26 < 4('8%, *$*;(=23$(6 (3), < &'(/)'*$/)<*0 L! < /;,3*% ! " = 0 $%).
J <() /2),*+2: / $%'*<%$/)<*82 .;: &'(27<(.$C0 2$*:: ,/;(<2% ! " = 0 $% 2/-
5;9 3*%) $*;232: $%'*<%$/)< )2&* (2). M)8%)28 '*A(), [7], < 5( )('(6, < 3 */) $( /)2,
.( 5*7*$C )(3$C% $%'*<%$/)<*
sup
Tn! !T 2n+1,Tn" 0
# $Tn(t)( ) dt
T%
# Tn(t)( ) dt
T%
=
# 2&n sin(2&t)( ) dt
T%
# sin(2&t)( ) dt
T%
(20)
.;: </%0 4,$5+26 ! 27 5;*//* ! 4,$5+26, $%,AC<*9D20 $* (0,!) , *A/( ;9) -
$( $%&'%'C<$C0 $* 5*-.(8 ()'%75% [a, b] ! (0,") 2 )*520, 3)( 4,$5+2: x !" (x)
$% ,AC<*%) $* (0, ! ) .
B 3*/)$(/)2, 4,$5+2: ! (x) = ln(1+ x) , (&'%.%;:9D*: &'(/)'*$/)<( ln(1+ L) ,
&'2$*.;%-2) 5;*//, ! ! " , 2 .;: $%e ! " = 0 .
FC $% /8(=;2 $*6)2 )(3$C% &( &(':.5, $%'*<%$/)<* >%'$?)%6$* .;: &'(27<(. -
$C0 <( </%0 &'(/)'*$/)<*0 L! / ,/;(<2%8 ! " = 0 . M.$*5(, 8C $2-% ,5*-%8
5;*// &'(/)'*$/)< L! , < 5()('C0 ,.*;(/@ .(5*7*)@ $%'*<%$/)<* .;: &'(27<(.$C0
.*-% / )(3$C82 5($/)*$)*82. X)(8, 5;*//,, < 3*/)$(/)2, &'2$*.;%-2) $ *':., /
&'(/)'*$/)<(8 ln(1+ L) %D% 9 &'(/)'*$/)<( L0 / 8%)'25(6
f 0 := ! f (x)( ) dx
T
" , ! (x) =
x
1+ x
, x > 0 ,
&('(-.*9D%6 /0(.28(/)@ &( 8%'%. M)8%)28, 3)( ! " = 0 .
N( /$*3*;* &'2<%.08 (.$, (AD,9 (+%$5, $('8 (&%'*)('(< < &'(27<(; @$C0
&'(/)'*$/)<*0 L! .
MA(7$*328
I n := sup
Tn! !T 2n+1,Tn" 0
Tn C
Tn L1
. H7<%/)$( [10], 3)( n +1 ! I n ! 2n +1.
G,/)@, 5*5 2 '*$%%, .;: 425/2'(<*$$(=( n A: !T 2n+1 ! !T 2n+1 Ñ (&%'*)(' / 8$ ( -
-2)%;:82 ! k; k " n{ } 2
A 1!1 := sup
Tn" !T 2n+1,Tn#0
ATn 1
Tn 1
.
MA(7$*328 %D0 3%'%7 ! 5;*// </%0 <C&,5;C0 <<%'0 8(.,;%6 $%&'%'C<$(/ )2
! : R+ " R+ .
.;(?;FD 5. ?(5&> ! " # . C018% 4<20,)-.&5- )6#%46)5&4%
1666 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
1
2 /M! ( 2)
M! max
|k|" n
#k( ) " A ! , n " I nM!
A 1$ 1
I n
%
&'
(
)*
. (21)
Доказательство. M+%$5* /$27, -*;0>@92-3 < )%('%8% 2. ";: (+%$52 /<%' 0,
2/ &(;@7,%8 $%'*<%$/)<( Y%$/%$* .;: ! :
ATn ! = ! ATn(x)( ) dx " ! ATn(x) dx
T
#
$
%
&
'
(
)
T
# =
= ! ATn 1( ) " ! A 1# 1 Tn 1( ) . (22)
H7 <C&,5;(/)2 <<%'0 U /;%.,%), 3)( 4,$5+2:
! (x)
x
,AC<*9D*:. G(E) ( 8,
! Tn(x)( )
Tn(x)
"
! Tn C( )
Tn C
"
! I n Tn 1( )
I n Tn 1
,
Tn ! = ! Tn(x)( ) dx "
T
# Tn(x)
! I n Tn 1( )
I n Tn 1
dx =
T
# I n
$1! I n Tn 1( ) .
(23)
H7 (22) 2 (23) /;%.,%) (21):
ATn !
Tn !
"
! A 1# 1 Tn 1( )
I n
$1! I n Tn 1( )
,
A ! , n " I n sup
0<s<#
! A 1$ 1 s( )
! (I ns)
= I nM!
A 1$ 1
I n
%
&'
(
)*
.
Q%('%8* 5 .(5*7*$*.
O/;2 ! 27 ! $% :<;:%)/: <C&,5;(6 <<%'0, )( .;: $*28%$@?%6 <C&,5;(6
<<%'0 8*-('*$)C ! &( ;%88% I)%352$* (/8., $*&'9:0>, [11])
! (x) " ! (x) " 2! (x) .
Q(=.* &(/;% (3%<2.$C0 278%$%$26 < .(5*7*)%;@/)<% &(;,328 .;: /;,3*: &'(27-
<(;@$(6 ! 27 ! /;%.,9D,9 (+%$5 , /<%'0,:
A !, n " 4InM!
A 1#1
In
$
%&
'
()
. (24)
B<%.%8 5;*// &'(/)'*$/)<, .;: 5()('(=( 8C /8(-%8 ,)(3$2)@ $%'*<%$/)<* (21).
4@?;<;A;=$;. F(86$ 1040#3&>, ;&0 ! 2#3)%8,6'3& ",%55( ! 1 , 65,3
! : R+ " R+ Ñ 4<2(",<@ 446#7 $08(,> )62#6#<4)05&3 3 4<20,)-6&5- %53$2&0-
&3;65"06 #%46)5&40
!(x) " x 2#3 x ! 0 . (25)
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1667
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
.;(?;FD 6. G5,3 ! " # 1 , &0 4<20,)-.&5- )6#%46)5&4%
max
|k|! n
" k ! sup
Tn# !T 2n+1,Tn$0
ATn %
Tn %
! max In; A 1& 1{ } . (26)
E ;%5&)05&3, 8,- ,./010 r ! 1 ()6 0/-=%&6,>)0 B6,010) 2#3 4567 n ! 1
sup
Tn! !T 2n+1,Tn" 0
Tn
(r )
#
Tn #
= (2$n)r . (27)
Доказательство. "(5*-%8 /$*3*;* (+%$5, /$27,. "*-7*.87,
! (x)
x
" , )(
!(x)
x
" lim
x#0
!(x)
x
= 1,
). e. !(x) " x . G(E)(8, Tn ! " Tn 1.
P*//8()'28 &(;2$(8C ! pk (x) = ! cos(2" kx), k = 0,1, É , n , ! > 0. Q(=.*
A(! pk ) "
! pk "
# cos(2$x) 1
%1 " ! &k cos(2$x)( ) dx
!T
' ,
A ! , n " cos(2#x) 1
$1 lim
%& 0
! %' k cos(2#x)( ) dx
%T
( .
H/&(;@7o/1/ )%('%8, R%A%=* ( 8*-('2'(<*$$(6 /0(.28(/)2, (/,D%/)<28 &'% .%;@-
$C6 &%'%0(. &(. 7$*5(8 2$)%='*;*. !32)C<*: (25), &(;,3ae8 (+%$5, /$27,.
Q%&%'@ &(5*-%8, 3)( .;: ;9A(6 ! 27 !1
M ! (y) = y " y # 1 . (28)
Q(=.* (+%$5* /<%'0, A,.%) /;%.(<*)@ 27 (21). H7 (25) /;%.,%), 3)(
M ! (y) = sup
s>0
! (sy)
! (s)
" lim
s# 0
! (sy)
! (s)
= y . (29)
I .',=(6 /)('($C, )*5 5*5
! (x)
x
" , )( &'2 y ! 1
! (sy)
! (s)
= y
! (sy) / sy
! (s) / s
" y ,
&(E)(8, M ! (y) " y . M)/9.* 2 27 (29) /;%.,%) (28).
Q%('%8* 6 .(5*7*$*.
6. *;?D);=I>)D <AB @(A$=(F() ) ?DL=HM F;>?$GDM. G(':.(5 '(/)* < %;232-
$C
C(n; r; X,Y) := sup
Tn! !T 2n+1,Tn" 0
Tn
(r )
X
Tn Y
1668 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
&'2 7*.*$$(8 r = 0,1,É 2 n ! " < /;,3*% X = Lp(T ) , Y = Lq(T) , ! " p >
> q ! 1, 2/ /;%.(<*; I. F. N25(;@/526 [10]. "*;@$%6?2% '%7,;@)*)C /8. < [12].
FC '*//8()'28 *$*;(=23$,9 7*.*3, < /;,3*% X = L1(T ) , Y = L! (T) , ! " # .
.;(?;FD 7. 1. +,- ,./0@ ! " # )%@86&5- "0)5&%)&% C! &%"%-, ;&0
4<!"#$%&'(% )6#%46)5&4%
! n Tn 1( ) " C! n Tn ! (30)
2#3 4567 n 3 Tn .
2. G5,3 ! " > 0 , &0 )%@86&5- "0)5&%)&% C! ,1 > 0 &%"%-, ;&0 2#3 4567 n
)*+&' *+('" )6#%46)5&4%
C! ,1nM !
1
n
"
#$
%
&'
( sup
Tn ) !T 2n+1,Tn * 0
! Tn 1( )
Tn !
( C! nM !
1
n
"
#$
%
&'
(31)
(=865> C! Ñ &% '6, ;&0 3 4 (30)).
Доказательство. H/&(;@7,%8 4('8,;, (7):
Tn(x + t) !
1
3n
Tn(t + xj ) Vn(x " xj )
j =1
3n
# .
G'(2$)%='2',%8 (A% 3*/)2 &( &%'%8%$) (6 x :
n Tn 1 !
1
3
Vn 1 Tn(t + xj )
j =1
3n
" .
M)/9.* &(;,3*%8
! n Tn 1( ) " M!
1
3
Vn 1
#
$%
&
'(
! Tn(t + xj )( )
j=1
3n
) .
Q%&%'@ &'(2$)%='2',%8 &( &%'%8%$) (6 t 2 &(;,328 $%'*<%$/)<(
! n Tn 1( ) " M!
1
3
Vn 1
#
$%
&
'(
3n Tn ! , (32)
7*2*>*0 /6?*.)302-3 .;: ;9A(=( :.'* Vn <2.* (4). B 3*/)$(/)2, &,/)@ Vn Ñ
5;*// 23%/5(% :.'( B*;;% G,//%$*. H7<%/)$( [3], 3)( sup Vn 1 ; n ! N{ } = K < ! .
Q(=.* 27 (32) &(;,3*%8 (30) / 5($/)*$)(6 C! := 3M! (K /3) .
H7 (30) /;%.,%) <%'0$:: (+%$5* < (31):
! Tn 1( ) = ! n
1
n
Tn
1
"
#$
%
&'
( C! n
1
n
Tn
!
( C! nM!
1
n
"
#$
%
&'
Tn ! .
Q*528 (A'*7(8, <%'0$:: (+%$5* < (31) /&'*<%.;2<* 2 < /;,3*% ! " = 0 .
G,/)@ )%&%'@ ! " > 0 2 :.'* Vm (&'%.%;:9)/: 4,$5+2%6 ! 27 P! ! C" (R) .
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1669
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
Q(=.* .;: ;9A(=( c > 0 / &(8(D@9 4('8,;C /,882'(<*$2: G,*//($* &(; , 3*%8
cVm ! " ! cm #̂ (my)( ) dy =
1
m
! cm #̂ (y)( ) dy "
R
$
R
$
!
1
m
" (cm) M" ö#(y)( ) dy = K1
1
m
" (cm)
R
$ , (33)
=.% K1 := M! ö" (y)( ) dy < #
R$ .
";: (+%$52 /$27, < (31) .(/)*)(3$( (='*$232)@/: /;,3*%8 n ! 3 . G(; ( -28
Tn = cV n/2[ ] , 2/&(;@7,%8 (33) 2 )() 4*5), 3)( Vm 1 > 1:
sup
Tn! !T 2n+1,Tn" 0
# Tn 1( )
Tn #
$ sup
c>0
# c V n / 2[ ] 1( )
cV n / 2[ ] #
$ sup
c>0
# (c)
K1 n / 2[ ]%1 # (c n / 2[ ] )
$
! n
3K1
sup
c>0
"(c)
"(cn/2)
=
n
3K1
M"
2
n
#
$%
&
'( ! 1
3K1
nM"
1
n
#
$%
&
'( .
Q%('%8* 7 .(5*7*$*.
.;(?;FD 8. 1. H%@8(&5- "0)5&%)&< C! < " 3 C! , 2 > 0 &%"36, ;&0 8,-
, . /010 026#%&0#% A 5 $)0'3&6,-$3 ! k; k " n{ } 4<!"#$%&'(% )6#%46)5&4%
C! , 2nM !
1
n
max
|k| " n / 2
#k
$
%&
'
()
" sup
Tn * !T 2n+1,Tn+0
! ATn 1( )
Tn !
" C! nM !
1
n
A 1, 1
$
%&
'
()
.
(34)
?#3 D&0$ 2#%406 )6#%46)5&40 4<20,)-6&5- ,#% #&-"." ! "# , % ,6406 Ñ 2#3
(5, 0433 ! " > 0 .
2. G5,3 ! " # 1 , &0
C! ,2 max
|k| " n / 2
#k " sup
Tn$ !T 2n+1,Tn%0
! ATn 1( )
Tn !
" C! max n; A 1& 1{ } . (35)
Доказательство. G'*<(% $%'*<%$/)<( < (34) /;%.,%) 27 (30) (/ )(6 -% 5($-
/)*$ ) (6 C! ):
! ATn 1( ) " ! A 1# 1 Tn 1( ) = !
1
n
A 1# 1
$
%&
'
()
n Tn 1( )$
%&
'
()
"
! M"
1
n
A 1# 1
$
%&
'
()
" n Tn 1( ) ! M"
1
n
A 1# 1
$
%&
'
()
C" n Tn " .
";: (+%$52 /$27, < (34) .(/)*)(3$( (='*$232)@/: /;,3*%8 n ! 3 . G(; ( -28
Tn = cV n / 2[ ] , c > 0 , 2 ,3)%8, 3)( &'2 k ! n / 2[ ]
1670 I. J. GHK!LMB
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
AV n / 2[ ] 1
! AV n / 2[ ] (x)ei2" kxdx
T
# = $k .
#'(8% )(=(, %/;2 ! " > 0 , )( 8(-$( 2/&(;@7(<*)@ (33). B '%7,;@)*)% &(;, 328 ;%-
<,9 3*/)@ (34):
sup
Tn! !T 2n+1,Tn" 0
# ATn 1( )
Tn #
$ sup
c>0
# AcV n / 2[ ] 1( )
cV n / 2[ ] 1
$
! sup
c>0
" c max
k # n / 2[ ]
$k
%
&'
(
)*
K1 n / 2[ ]+1 " c n / 2[ ]( )
! C" nM "
1
n
max
k # n / 2[ ]
$k
%
&'
(
)*
.
O/;2 -% ! " # 1, )( ! (x) " x, &(E)(8,
cV n / 2[ ] !
" cV n / 2[ ] 1
" cK ,
sup
Tn! !T 2n+1,Tn" 0
# ATn 1( )
Tn #
$ sup
c
# AcV n / 2[ ] 1( )
cV n / 2[ ] 1
$
! lim
c" 0
# c max
k $ n / 2[ ]
%k
&
'(
)
*+
cK
=
1
K
max
k $ n / 2
%k .
G'*<*: 3*/)@ (35) /;%.,%) 27 )(=(, 3)( M! (y) " max(1; y) .
Q%('%8* 8 .(5*7*$*.
Следствие 4. 1. G5,3 ! " > 0 , &0 8,- r ! [0, " )
!C" ,2 (r )nM" (nr #1) $ sup
Tn%!T 2n+1,Tn&0
" Tn
(r )
1( )
Tn "
$ !C" (r )nM" (nr #1) .
2. G5,3 ! " # 1 , &0 8,- r ! [1, " )
!C" ,2(r )nr # sup
Tn$ !T 2n+1,Tn%0
" Tn
(r )
1( )
Tn "
# !C" (r )nr .
3. +,- ,./0@ ! " # 2#3 4567 k, r = 0,1, 2,É 3 4567 h ! (0,1] , n ! N
sup
Tn! !T 2n+1,Tn"0
# h$k %h
kTn
(r )
1( )
Tn #
& C# (r, k)nM# (nr$1 min(nk, h$k )) .
NOPJBONIQBJ "RS QPHL MNMFOQPHKOI#HT GMRHNMFMB B GPMIQPJNIQBJT É 1671
ISSN 1027-3190. !"#. $%&. '(#). , 2011, &. 63, * 12
1. ?3;(104 I. J . M )%('%8% "-%5/($* .;: &%'2(.23%/520 4,$5+26 < &'( /)'*$/)<*0 / 2$)%='*;@$(6
8%)'25(6 // !5'. 8*). -,'$. Ð 2000. Ð 52, Z 1. Ð I. 122 Ð 133.
2. ?3;(104 I. J . M )%('%8% "-%5/($* .;: &%'2(.23%/520 4,$5+26 < 8%) '23%/520 &'(/)'*$/)<*0 /
2$)%='*;@$(6 8%)'25(6. II // !5'. 8*). -,'$. Ð 2011. Ð 63, A 11. Ð C. 1524 Ð 1533.
3. C3$%) J. K. Q%('2: &'2A;2-%$26 4,$5+26 .%6/)<2)%;@$(=( &%'%8%$$(=(. Ð F. : V278*)=27, 1960.
Ð 624 /.
4. L0#)6@;(" H. ?. , F%/6)"0 E. K. , M31() J. J. X5/)'%8*;@$C% /<(6/)<* &(;2$(8(< 2 /&;*6$(<. Ð
#2%<: N*,5. .,85*, 1992. Ð 304 /.
5. I&0#0'6)"0 N. J., L#0&04 E. O., P5 4%,>8 ?. G':8C% 2 (A'*)$C% )%( '%8C )2&* "-%5/($* < &'( -
/)'*$/)<*0 Lp , 0 < p < 1 // F*). /A. Ð 1975. Ð 98, Z 3. Ð I. 395 Ð 415.
6. Q4%)04 E. Q. N%5()('C% $%'*<%$/)<* .;: )'2=($(8%)'23%/520 &(;2$( 8(< 2 20 &'(27<(.$C0 < '*7 -
$C0 8%)'25*0 // F*). 7*8%)52. Ð 1975. Ð 18, Z 4. Ð I. 489 Ð 498.
7. J#65&04 E. E. MA 2$)%='*;@$C0 $%'*<%$/)<*0 .;: )'2=($(8%)'23%/520 &(;2$(8(< 2 20 &'(27<(. -
$C0 // H7<. JN IIIP. I%'. 8*). Ð 1982. Ð 45, Z 1. Ð I. 3 Ð 22.
8. L#6@) I. O., ?6&()3) R. Q., I6$6)04 G. S . H$)%'&(;:+2: ;2$%6$C0 (&%'*)('(<. Ð F. : N*,5*,
1978. Ð 400 /.
9. I&6@) Q., E6@5 O. B<%.%$2% < =*'8($23%/526 *$*;27 $* %<5;2.(<C0 &'(/)'*$/)<*0. Ð F. : F2',
1974. Ð 330 /.
10. H3"0,>5"3@ I. S. N%'*<%$/)<* .;: +%;C0 4,$5+26 8$(=20 &%'%8%$$C0 // Q',.C F*). 9$-)* JN
IIIP. Ð 1951. Ð 38. Ð I. 244 Ð 278.
11. L0#)6@;(" H. ? . Q(3$C% 5($/)*$)C < )%('22 &'2A;2-%$2:. Ð F.: N*,5*, 1987. Ð 424 /.
12. J#65&04 E. E. M $%'*<%$/)<% '*7$C0 8%)'25 .;: )'2=($(8%)'23%/520 &(;2$(8(< // F*). 7*8%)52.
Ð 1980. Ð 27, Z 4. Ð I. 539 Ð 547.
"*.,B0 $( 11.10.10
|
| id | nasplib_isofts_kiev_ua-123456789-166404 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | Russian |
| last_indexed | 2025-12-07T17:19:28Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Пичугов, С.А. 2020-02-19T05:45:23Z 2020-02-19T05:45:23Z 2011 Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой / С.А. Пичугов // Український математичний журнал. — 2011. — Т. 63, № 12. — С. 1657–1671. — Бібліогр.: 12 назв. — рос. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166404 517.5 ru Інститут математики НАН України Український математичний журнал Статті Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой Inequalities for trigonometric polynomials in spaces with integral metric Article published earlier |
| spellingShingle | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой Пичугов, С.А. Статті |
| title | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой |
| title_alt | Inequalities for trigonometric polynomials in spaces with integral metric |
| title_full | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой |
| title_fullStr | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой |
| title_full_unstemmed | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой |
| title_short | Неравенства для тригонометрических полиномов в пространствах с интегральной метрикой |
| title_sort | неравенства для тригонометрических полиномов в пространствах с интегральной метрикой |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/166404 |
| work_keys_str_mv | AT pičugovsa neravenstvadlâtrigonometričeskihpolinomovvprostranstvahsintegralʹnoimetrikoi AT pičugovsa inequalitiesfortrigonometricpolynomialsinspaceswithintegralmetric |