Comonotone approximation of twice differentiable periodic functions
In the case where a $2π$-periodic function $f$ is twice continuously differentiable on the real axis $ℝ$ and changes its monotonicity at different fixed points $y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ $(i.e., on $ℝ$, there exists a set $Y := {y_i } i∈ℤ$ of points $y_i = y_{i+2s} + 2π$ such that the funct...
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| Datum: | 2009 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2009
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3032 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509053602496512 |
|---|---|
| author | Dzyubenko, H. A. Дзюбенко, Г. А. |
| author_facet | Dzyubenko, H. A. Дзюбенко, Г. А. |
| author_sort | Dzyubenko, H. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:50Z |
| description | In the case where a $2π$-periodic function $f$ is twice continuously differentiable on the real axis $ℝ$ and changes its monotonicity at different fixed points $y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ $(i.e., on $ℝ$, there exists a set $Y := {y_i } i∈ℤ$ of points $y_i = y_{i+2s} + 2π$ such that the function $f$ does not decrease on $[y_i , y_{i−1}]$ if $i$ is odd and does not increase if $i$ is even), for any natural $k$ and $n, n ≥ N(Y, k) = const$, we construct a trigonometric polynomial $T_n$ of order $≤n$ that changes its monotonicity at the same points $y_i ∈ Y$ as $f$ and is such that
$$∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n)$$
$$(∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/
n),f ∈ C^{(r)},\; r ≥ 2),$$
where $N(Y, k)$ depends only on $Y$ and $k, c(k, s)$ is a constant depending only on $k$ and $s, ω k (f, ⋅)$ is the modulus of smoothness of order $k$ for the function $f$, and $‖⋅‖$ is the max-norm. |
| first_indexed | 2026-03-24T02:34:59Z |
| format | Article |
| fulltext |
UDK 517.5
H. A. Dzgbenko (MiΩnar. mat. centr NAN Ukra]ny, Ky]v)
KOMONOTONNE NABLYÛENNQ
DVIÇI DYFERENCIJOVNYX PERIODYÇNYX FUNKCIJ
We consider the case where a 2π-periodic function f is twice continuously differentiable on the real
axis R and changes the monotonicity at various fixed points yi ∈ −[ , )π π , i s= …1 2, , , s ∈N (i.e.,
on R , there exists the set Y yi i: { }= ∈Z of points y yi i s= ++2 2π such that, on [ , ]y yi i−1 , f does
not decrease if i is odd and does not increase if i is even). In this case, for every natural k and n,
n N Y k≥ =( , ) const , we construct a trigonometric polynomial Tn of order ≤ n, which changes its
monotonicity at the same points y Yi ∈ as f and is such that
f T
c k s
n
f nn k− ≤ ′′
( , )
( , )/2
1ω
f T
c r k s
n
f n f C rn r k
r r− ≤
+
∈ ≥
( , )
( , ), ,( ) ( )
/ω 1 2 ,
where N Y k( , ) depends only on Y and k, c k s( , ) is a constant depending only on k and s ,
ωk f( , )⋅ is a module of smoothness of order k of the function f, and ⋅ is a max-norm.
V sluçae, kohda dvaΩd¥ neprer¥vno dyfferencyruemaq na dejstvytel\noj osy R 2π-peryody-
çeskaq funkcyq f yzmenqet monotonnost\ v razlyçn¥x fyksyrovann¥x toçkax yi ∈ −[ , )π π ,
i s= …1 2, , , s ∈N (t. e. na R est\ mnoΩestvo Y yi i: { }= ∈Z toçek y yi i s= ++2 2π takyx,
çto na [ , ]y yi i−1 f ne ub¥vaet, esly i neçetnoe, y ne vozrastaet, esly i çetnoe), dlq kaΩd¥x
natural\n¥x k y n, n N Y k≥ =( , ) const , postroen tryhonometryçeskyj polynom Tn porqdka
≤ n, kotor¥j yzmenqet svog monotonnost\ v tex Ωe toçkax y Yi ∈ , çto y f, y takoj, çto
f T
c k s
n
f nn k− ≤ ′′
( , )
( , )/2
1ω
f T
c r k s
n
f n f C rn r k
r r− ≤
+
∈ ≥
( , )
( , ), ,( ) ( )
/ω 1 2 ,
hde N Y k( , ) zavysyt tol\ko ot Y y k ; c k s( , ) � postoqnnaq, zavysqwaq tol\ko ot k y s ;
ωk f( , )⋅ � modul\ hladkosty porqdka k funkcyy f y ⋅ � max-norma.
1. Vstup. Nexaj C � prostir neperervnyx 2 π -periodyçnyx funkcij f : R →
→ R , f : = f R : =
max ( )
x
f x
∈R
i Tn, n ∈N , � prostir tryhonometryçnyx
polinomiv t xn( ) = a a jx b jxj jj
n
0 1
+ +=∑ ( cos sin ) porqdku ≤ n , de aj , bj ∈R .
Nahada[mo klasyçnu teoremu DΩeksona � Zyhmunda � Axi[zera � St[çkina:
pry koΩnyx natural\nyx k i n dlq bud\-qko] funkci] f C∈ znajdet\sq po-
linom σn n∈T takyj, wo
f n− σ ≤ c k f nk( ) ( , )/ω 1 , (1)
© H. A. DZGBENKO, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4 435
436 H. A. DZGBENKO
de c k( ) � stala, qka zaleΩyt\ lyße vid k , i ωk f( , )⋅ � modul\ hladkosti
porqdku k funkci] f .
Krim toho, qkwo f C r∈ ( ) : = { }: ( )f f Cr ∈ , r ∈N , to z (1) vyplyva[ neriv-
nist\
f n− σ ≤
c r k
n
f nr k
r( )
( , )( ) /
+ ω 1 , n ∈N . (2)
( Detal\niße dyv., napryklad, [1, c. 204 – 212]. )
U danij roboti navedeno komonotonni analohy nerivnostej (1) i (2). A same,
nexaj na [ )– ,π π zafiksovano 2s , s ∈N , toçok yi :
– π ≤ y s2 < y s2 1− < … < y1 < π,
a dlq reßty indeksiv i ∈Z toçky yi vyznaçagt\sq rivnistg yi = yi s+ +2 2π
( tobto y0 = y s2 2+ π , … , y s2 1+ = y1 2− π , … ). Poznaçymo Y : = { }yi i∈Z,
∆( )( )1 Y � mnoΩyna vsix funkcij f , qki ne spadagt\ na [ ],y y1 0 , ne zrostagt\
na [ ],y y2 1 , ne spadagt\ na [ ],y y3 2 i t. d. ZauvaΩymo, wo qkwo periodyçna
funkciq f dyferencijovna, to
f Y∈∆( )( )1 ⇔ ′f x x( ) ( )Π ≥ 0, x ∈R,
de
Π( )x : = Π( , )x Y : = sin
x yi
i
s −
=
∏ 21
2
, Π( )x > 0, x y y∈( , )1 0 .
U robotax [2, 3] dlq bud\-qko] funkci] f C Y∈ I ∆( )( )1 oznaçeno vidpovidno
polinomy Tn i Pn z Tn YI ∆( )( )1 taki, wo
f Tn− ≤ c s f n( ) ( , )/ω1 1 , n ∈N , (3)
f Pn− ≤ c s f n( ) ( , )/ω2 1 , n N Y≥ ( ), (4)
f Pn− ≤ C Y f n( ) ( , )/ω2 1 , n ∈N , ( 4 ′ )
de c s( ) � stala, qka zaleΩyt\ lyße vid s, a N Y( ) i C Y( ) � stali, qki zale-
Ωat\ lyße vid Y, tobto vid min
, ,
{ }
i s
i iy y
= …
+−
1 2
1 . (Polinom Pn v (4 ′ ) pry 1 ≤ n <
< N Y( ) �lyße� isnu[, bo [ (4) i nerivnist\ Uitni [4] f f− ( )0 ≤ k f kkω π( , ),
k ∈N . ) U robotax [5, c. 64 – 83; 6] navedeno kontrpryklady, qki vkazugt\ na
nemoΩlyvist\ zaminy ω2 v (4 ′ ) (a otΩe, i v (4)) na ωk z k ≥ 3.
Dovedemo nastupnu teoremu.
Teorema 1. Pry koΩnyx natural\nyx k i n, n ≥ N Y k( , ) = const, dlq bud\-
qko] funkci] f C Y∈ ( ) ( )( )2 1I ∆ znajdet\sq polinom R Yn n∈T I ∆( )( )1 takyj,
wo
f Rn− ≤
c k s
n
f nk
( , )
( , )/2 1ω ′′ (5)
f R
c r k s
n
f n f C rn r k
r r− ≤ + ∈ ≥
( , )
( , ), ,( ) ( )/ω 1 2 ,
de N Y k( , ) zaleΩyt\ lyße vid Y i k , a c k s( , ) � stala, qka zaleΩyt\
lyße vid k i s .
Naslidkom teoremy 1 i nerivnosti Uitni [4] f f− ( )0 ≤ k f kkω π( , ), k ∈N ,
[ taka teorema.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 437
Teorema 1 ′′′′. Pry koΩnyx natural\nyx k i n dlq bud\-qko] funkci] f ∈
∈ C Y( ) ( )( )2 1I ∆ znajdet\sq polinom R Yn n∈T I ∆( )( )1 takyj, wo
f Rn− ≤
C k Y
n
f nk
( , )
( , )/2 1ω ′′ ( 5 ′ )
f R
C r k Y
n
f n f C rn r k
r r− ≤ + ∈ ≥
( , )
( , ), ,( ) ( )/ω 1 2 ,
de C k Y( , ) � stala, qka zaleΩyt\ lyße vid k i Y .
ZauvaΩennq 1. My prypuska[mo, wo stali N Y( ) v (4) i N Y k( , ) v teore-
mi 1, a takoΩ stali C Y( ) i C k Y( , ) v nerivnostqx ( 4 ′ ) i ( 5 ′ ) nemoΩlyvo zaminy-
ty stalymy, qki ne zaleΩat\ vid Y, a zaleΩat\, skaΩimo, vid s . Ce prypuwennq
ne rozhlqda[mo v danij roboti.
V [5] (rozdil 2) dovedeno okremyj vypadok teoremy 1 ′: qkwo f W Yr∈ ( ) ( )( )I∆1
(de W r( ) � mnoΩyna funkcij g z absolgtno neperervnymy g r( )−1 i g xr( )( ) ≤
≤ 1 majΩe skriz\ na R ), to znajdet\sq T Yn n∈T I ∆( )( )1 takyj, wo
f Tn− ≤
C r Y
nr
( , )
, n ∈ N , r ≥ 2,
C r Y( , ) � stala, qka zaleΩyt\ lyße vid r i Y . Dlq r = 1 ce tverdΩennq [
okremym vypadkom nerivnosti (3).
2. DopomiΩni fakty. Nahada[mo, wo modulem hladkosti porqdku k ∈ N
obmeΩeno] na [ a, b ] funkci] g = g ( x ) nazyvagt\ funkcig
ωk g t a b( ), , [ , ] : = sup sup ( )
[ ],h t x a k h b k h
h
k g x
≤ ∈ + −
∆ , t ∈ [ / ], ( ) ( )0 2b a k− ,
de
∆h
k g x( ) : = ( ) ( )−
+−
=
∑ 1
0
k i
i
k k
i
g x ih
� k -ta riznycq funkci] g v toçci x iz krokom h . Dlq t > ( ) ( )/b a k− 2 pokla-
demo ωk g t a b( ), , [ , ] : = ωk g b a k a b( / ), ( ) ( ), [ , ]− 2 . U vypadku 2π -periodyçno] g
poklademo ωk g t( , ) : =
sup , , [ , ]( )
a
k g t a a
∈
+
R
ω π2 , tobto
ωk g t( , ) = sup ( )
h t
h
k g
≤
⋅∆ , 0 ≤ t ≤ π .
Dali çerez cν , ν = 1, … , 37, budemo poznaçaty dodatni çysla, qki moΩut\ za-
leΩaty lyße vid fiksovanyx r, k, l ∈ N , i p ∈ Z+ . Dovedemo lemu 1, qka dewo
utoçng[ ocinku (1), a otΩe i (2), a takoΩ vidpovidni ocinky dlq odnoçasnoho na-
blyΩennq funkci] ta ]] poxidnyx. Nexaj
J xn l, ( ) : =
sin( )
sin( )
/
/
nx
x
l
2
2
2
, K xn l, ( ) : = J x J x dxn l n l, ,( ) ( )
−
−
∫
π
π 1
� parne i nevid�[mne qdro typu DΩeksona, de n ∈N i l ∈N , i
σn l f x, ( , ) : = ( ) ( ) ( ) ( ),− −
++
−
−
=
∫ ∑1 11
1
k
n l
k i
i
k
K t
k
i
f x it dt
π
π
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
438 H. A. DZGBENKO
� polinom z Tl n( )−1 , zaproponovanyj St[çkinym [7] dlq dovedennq nerivnosti
(1), de f C∈ i k ∈N .
Lema 1. Pry koΩnyx natural\nyx ( r + 1 ) , k, l, 2 l ≥ k + r + 2, i n dlq bud\-
qko] funkci] f C r∈ ( ) polinom σn l l n, ( )∈ −T 1 [ takym, wo pry bud\-qkyx x i
δ > 0
f x f xp
n l
p( )
,
( )( ) ( , )− σ ≤
c
n
f n x x
n
f nr p k
r
l k r
k
r1
2 1
1
1
1−
− − −
− + +
ω δ δ
δ
ω( / ) ( / )( ) ( ), , [ , ] ,
(6)
≤
−
c
n
f nr p k
r1 1ω ( / )( ), , p = 0, 1, … , r .
Dovedennq. Bez vtraty zahal\nosti budemo vvaΩaty, wo δ π∈[ / ],1 n . Oci-
nymo f x f xn l( ) ( , ),− σ . Oskil\ky
f x f xn l( ) ( , ),− σ = ( ) ( ) ( ) ( ),− −
+
−
−
=
∫ ∑1 1
0
k
n l
k i
i
k
K t
k
i
f x it dt
π
π
=
= ( ) ( ) ( ),−
−
∫1 k
n l t
kK t f x dt
π
π
∆ , (7)
to
f x f xn l( ) ( , ),− σ ≤ K t f x dtn l t
k
, ( ) ( )
−
∫
π
π
∆ =
−
−
−
∫ ∫ ∫+ +
π
δ
δ
δ
δ
π/
/
/
/
k
k
k
k
= : I I I1 2 3+ + . (8)
Ocinymo I2 :
I2 ≤
− ≤∫
δ
δ
/
/
, ( ) max ( )
k
k
n l
h t
t
kK t f x dt∆ ≤
− ≤ − + + −[ ]∫ ⋅
δ
δ
δ δ
/
/
, ,
( ) sup ( )
k
k
n l
h t
t
k
x k h x k h
K t f dt∆ ≤
≤
−
∫ − +
δ
δ
ω δ δ
/
/
, ( ) , , [ , ]( )
k
k
n l kK t f t x x dt .
Dlq ocinky ostann\oho intehrala skorysta[mosq vlastyvistg
ωk f n t n a b( ), , [ , ]−1 ≤ n t n f n a bk k
k( ) ( ), , [ , ]+ − −1 1ω (9)
i nerivnistg [7] (lema 8)
K t t n dtn l
k
, ( )( )
−
−∫ +
π
π
1 ≤ c n k
2
− .
OtΩe, I2 ≤ c f n x xk2
1ω δ δ( ), , [ , ]− − + . Z dvox analohiçnyx intehraliv I1 i I3
ocinymo lyße I3. Pry c\omu vraxu[mo (9) i vlastyvosti Kn l, (dyv., napryklad,
[1, c. 131]). Nexaj δ/k ≤ n−1 ≤ δ . Todi
I3 = K t f x dtn l
k
t
k
,
/
( ) ( )
δ
π
∫ ∆ ≤ K t f dtn l
k
t
k
,
/
( ) ( )
δ
π
∫ ⋅∆ ≤
≤ K t f t dtn l
k
k,
/
( ) ( , )
δ
π
ω∫ ≤ n f n t n K t dtk
k
k
n l
k
ω
δ
π
( / ) ( ), ( ),
/
1 1+ −∫ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 439
≤ n f n
n
K t dt t K t dtk
k
k
k n l
k
n
k k
n l
n
ω
δ
π
( / ), ( ) ( ),
/
/
,
/
1
2
2
1
1
∫ ∫+
≤
≤ n f n
n
K t dt
c n
t
nt
t
dtk
k
k
k n l
k
k
l
k
l
n
ω
δ
π π
( / ) /
/
, ( )
sin( )
sin( ),
/ /
1
2 2 2
22
2 1
2
1
∫ ∫+
− ≤
≤ n f n
c n n
dt
t
dt
t
k
k
k
l k l
k
l
l k
k
ω
π
π
δ δ
( / )
/
,
( )
/ /
1
2 1
3
2 1 2
2
2−
∞
−
∞
∫ ∫+
=
= ω π
δ δk
k l
l k
l
k l
l k
l kf n
c n
k
n l
k
l k
( / ),
( ) ( )
1
2
2 1 2 1
2
3
2 1
2 1
2 1
2 1
2 1− −
−
−
− −
− −−
+
− −
≤
≤
c
n
f nl k k
4
2 1 1
( )
,( / )
δ
ω− − .
Dlq δ/k > 1/n analohiçno
I3 ≤ n f n
c n
dt
t
k
k
k l
l l k
k
ω π
δ
( / ),
/
1
2 2
3
2 1 2− −
∞
∫ ≤
c
n
f nl k k
4
2 1 1
( )
,( / )
δ
ω− − .
Zbyragçy u (8) ocinky I1 2 3, , , oderΩu[mo nerivnist\ (6) dlq vypadku r = p = 0.
Reßta vypadkiv lemy 1 vyplyvagt\ z rivnostej (7), dovedennq nerivnosti (8) i
ocinok
ωk i f n a b+
−( ), , [ , ]1 ≤ n f n a bi
k
i− −ω ( )( ), , [ , ]1 , i = 0, 1, … , r .
Lemu 1 dovedeno.
Skriz\ dali çysla cν moΩut\ zaleΩaty we j vid fiksovanyx s, b ∈ N . Dlq
fiksovanyx n ∈ N i Y = { }yi i∈Z poznaçymo
h : = hn : =
π
n
, x j : = x j n, : = – j h , Ij : = I j n, : = [ ],x xj j−1 , j ∈ Z ,
Oi : = O Y ni( , ) : = ( ),x xj j+ −5 5 , qkwo y x xi j j∈ −[ ), 1 , O : = O Y n( , ) : =
Oi
i∈Z
U .
Budemo pysaty j ∈ H : = H Y n( , ) , qkwo x Oj ∈R \ . Vyberemo N Y( ) ∈N take,
wo koΩen vidrizok [ ],y yi i−1 , i = 1, … , 2 s , mistyt\ prynajmni 10 riznyx
vidrizkiv Ij dlq vsix n ≥ N Y( ). Poznaçymo
χ( , )x a : =
0
1
, ,
, ,
qkwo
qkwo
x a
x a
≤
>
a ∈R , χ j x( ) : = χ( , )x x j ,
˘ ( )Γn x : = min ,
sin( )/
1
1
2n x
, x ∈ R , n ∈ N ,
Γj x( ) : = Γj n x, ( ) : = ˘ ( ( ))/Γn jx x h− + 2 , j ∈ Z ,
i zauvaΩymo, wo
Γj
j n
n
2
1= −
∑ < 6 (10)
(detal\niße dyv. [2]).
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
440 H. A. DZGBENKO
Skriz\ dali n > N Y( ) i bez vtraty zahal\nosti budemo vvaΩaty, wo y s2 =
= – π . Poznaçymo
J xj ( ) : = J x x n J x x nn j n j
b
2 1 2 14 3 4, ,( / ) ( / )( ( )) ( ( ))− + + − +( )π π , b ∈ N .
Dlq j ∈ H i b ≥ s + 4 poklademo
t xj ( ) : = t x b Yj n, ( , , ) : =
1
d
J u u du
j
j
x
x
j
( ) ( )Π
−
∫
π
∈ Tc n5
, (11)
(
t xj ( ) : =
(
t x b Yj n, ( , , ) : =
1(
d
u J u u du
j
j j
x
x
j
Π Π( ) ( ) ( )
−
∫
π
∈ Tc n5
, (12)
de
d j : = J u u duj
x
x
j
j
( ) ( )Π
−
+
∫
π
π
,
(
d j : = Π Πj j
x
x
u J u u du
j
j
( ) ( ) ( )
−
+
∫
π
π
,
Π j x( ) : = – Π( { }), ,x x xj j−1 ,
zokrema, d j ≠ 0 i
(
d j ≠ 0 (dlq vkazanyx j i b ) (dyv. detal\nu ocinku analo-
hiçno] velyçyny v [2], lema 1). U nastupnij lemi zberemo u zruçnij dlq nas for-
mi spivvidnoßennq (13) � (16) z roboty [2] i analohy nerivnostej (5.22) i (5.27) z
roboty [8]. ZauvaΩymo, wo spivvidnoßennq v [2] opysugt\ nevid�[mne qdro
J x xn l j, ( )− , a ]x analohy v lemi 2 � stroho dodatne J xj ( ) qk sumu dvox �susid-
nix� nevid�[mnyx.
Lema 2. Qkwo j ∈ H i b ≥ s + 4, to
′t x x xj j( ) ( ) ( )Π Π ≥ 0, x ∈ R , (13)
( ′t x x xj j( ) ( ) ( )Π Π ≤ 0, x ∈ [ ]\,x h x Ij j j− + +2 2π π , (14)
t xj j( )± π =
(
t xj j( )± π = χ πj ( )± , (15)
χ j jx t x( ) ( )− ≤ c xj
b s
6
2 1( )( )Γ − − , x ∈ [ ],x xj j− +2 2π π , (16)
χ j jx t x( ) ( )−
(
≤ c xj
b s
6
2 1( )( )Γ − − , x ∈ [ ],x xj j− +2 2π π , (17)
′t xj ( ) ≤ c
h
xj
b s
7
21 ( )( )Γ − , x ∈ R , (18)
(′t xj ( ) ≤ c
h
xj
b s
7
21 ( )( )Γ − , x ∈ R , (19)
′t xj ( ) ≥ c
h
xj
b s
8
2 21 ( )( )Γ + , x O Y n∈R \ ( , ) , (20)
′t xj ( ) ≥ c
h
x
x y
x yj
b s i
j i
8
2 21 ( )( )Γ + −
−
, x ∈ O Y ni( , ) , i ∈ Z , (21)
t xj ( ) =
1
2π
x R xj+ ( ),
(
t xj ( ) =
1
2π
x R xj+
(
( ), x ∈ R , (22)
de Rj i
(
Rj � deqki polinomy z Tc n5
.
Zaznaçymo, wo lema 2 dovodyt\sq za dopomohog rivnostej (15) i nerivnostej
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KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 441
1
9
2
c h
x
x
xj
b
j
Γ Π
Π
( )
( )
( )
≤ ′t xj ( ) ≤
c
h
x
x
xj
b
j
9 2Γ Π
Π
( )
( )
( )
,
Π
Π
( )
( )
x
x j
≤ Γj
s x− ( ), x ∈ R , j ∈ H ,
Γj
a
x
x
u du
j
( )
+
∫
π
≤ c h xj
a
10
1Γ − ( ), a ∈ N , x ∈ [ ],x xj j + 2π ,
Γj
a
x
x
u du
j
( )
−
∫
π
≤ c h xj
a
10
1Γ − ( ), a ∈ N , x ∈ [ ],x xj j− 2π .
Çerez Φk , k ∈ N , poznaçymo mnoΩynu vsix k - maΩorant, tobto neperervnyx
i nespadnyx na [ 0, ∞ ) funkcij φ( )t takyx, wo φ( )0 = 0 i t tk− φ( ) ne zrosta[ pry
t > 0. Vidomo (dyv., napryklad, [1, c. 167]), wo dlq bud\-qkoho modulq ωk g t( , )
mnoΩyna Φk ma[ funkcig φ( )t taku, wo ωk g t( , ) ≤ φ( )t ≤ 2k
k g tω ( , ), t ≥ 0.
Vyberemo ϕ ∈Φk tak, wo
ωk f t( , )′ ≤ ϕ( )t ≤ 2k
k f tω ( , )′ , t ≥ 0.
Poznaçymo
H0 : = H Y n0( , ) : = j H Y n j n∈ <{ }( , ) :
i
Z : = { }zq q
n
=
∗
0 : = x j H yj i i
s: { }∈{ } =0 0
2U ,
de n∗ : = 2 1 8 2 1n s+ − +( ) i toçky zq uporqdkovano za spadannqm. Nexaj
j q( ) : = j, qkwo zq = x j ( z j H∈ 0 ), i j q( ) : = j q( )− 1 , qkwo zq = yi . Po-
klademo
b1 = s + 4.
Lema 3. Qkwo ′f [ 2π -periodyçnog, ′f ≤ ϕ( )h i ′f x x( ) ( )Π ≥ 0,
x ∈ R , to funkciq
τn f x( , ) : = f f z f z t x b Yq q
q
n
j q n( ) ( ) ( ) ( , , )( ),− + −( )−
=
∗
∑π 1
1
1
zadovol\nq[ nerivnosti
f fn− ⋅τ ( , ) ≤ c h h11 ϕ( ) , (23)
′τn f x x( , ) ( )Π ≥ 0, x ∈ R . (24)
Krim toho, qkwo dlq A = const f x Ax( ) − [ periodyçnog, to τn f x Ax( , ) − ∈
∈ Tc n5
.
Dovedennq. Nerivnosti ( )( ) ( ) ( )f x f x xj j j− −1 Π ≥ 0, j H∈ 0, i (13) porodΩu-
gt\ (24). Ocinka (23) dovodyt\sq za dopomohog nerivnostej (16), (10) i rivnosti
f x f xn( ) ( , )− τ = f x S x S x f xn( ) ( ) ( ) ( , )− + − τ , x ∈ −[ ],π π ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
442 H. A. DZGBENKO
de S x( ) : = f f z f z xq q j qq
n
( ) ( ) ( ) ( )( )− + −( )−=
∗
∑π χ11
. Vklgçennq τn f x Ax( , ) − ∈
∈ Tc n5
dovodyt\sq analohiçno podibnomu vklgçenng v [3, c. 207] (suma alhebra-
]çnyx dodankiv z (22) dorivng[ Ax , oskil\ky τn f x( , ) = τn S x( , ) , a na Oi , i =
= 0, … , 2 s , S x( ) = const, tomu ]x suma po q = 1, … , n∗ dorivng[ sumi po j =
= 1 – n , … , n , i dorivng[ Ax ).
Lemu 3 dovedeno.
Dlq koΩnoho i ∈ Z poznaçymo
( ),y yi i : = ( )( ) ( ),x xj i j i : = Oi ,
tobto livyj i pravyj kinci promiΩku Oi . Nexaj dlq x ∈ R
d x O( , ) : =
min ,( / ) ( / )
i i ix y h x y h
∈
− + − +{ }
Z
2 2 ,
˜ ( ),t xi n : = t x b Y y t x b Y yj i n i j i n i( ), ( ),( , , ) ( ) ( , , ) ( )1 1sign signΠ Π+ .
Lema 4. Funkciq
U xn( ) : = h h t xi n
i
s
ϕ( ) ˜ ( ),
=
∑
1
2
(25)
zadovol\nq[ spivvidnoßennq
U Yn c n∈T
5
1I ∆( )( ), (26)
Un ≤ c h h12 ϕ( ), (27)
′U xn( ) ≥ c h13 ϕ( ) ˘ ( , )( ) ( )Γn
sd x O( ) +4 2 , x O∈R \ , (28)
′U xn( ) ≥
c
h
h x yi
13 ϕ( ) − , x Oi∈ , i ∈Z. (29)
Dovedennq. Z (13) vyplyva[, wo (26) � (29) [ vidpovidno naslidkamy (22)
(dodanky v (25) magt\ poparno protyleΩni znaky), (16) i rivnosti Un =
= U S Sn − + , de S � kuskovo-stala funkciq u formi (25), (20) ( U xn( ) doriv-
ng[ sumi moduliv dodankiv ) i (21).
Lemu 4 dovedeno.
Dlq k ∈N çerez L g x a bk ( , , [ , ]) poznaçymo mnohoçlen LahranΩa stepenq
≤ k , qkyj na [ , ]a b interpolg[ funkcig g = g x( ) u rivnoviddalenyx toçkax
a b a k+ −ν( )/ , ν = 0, … , k ; L g x a b0( , , [ , ]) : = g a( ). Nam bude potribna vidoma
nerivnist\ Uitni [4]
g L g a bk a b− ⋅−1( , , [ , ]) [ , ] ≤ c g b a k a bk14 ω ( , ( ) , [ , ])/− , g ∈ C a b([ , ]) , (30)
i lema 4.2 ′ z [9]: qkwo g C a bp∈ ( )([ , ]), p ∈N , p k< , to
g L g a bp
k
p
a b
( ) ( )
[ , ]
( , , [ , ])− ⋅−1 ≤ c g b a k a bk p
p
15 ω − −( / )( ), ( ) , [ , ] . (31)
Poznaçymo
l1 : =
k r
s
+ +
+ + +1
2
2 2 1( ) , Ji : = Ji n, : = [ ],y yi ih h− + ,
Yi : =
Y y yi i\{ } { }+( ) +∈ ∈2 2πν πνν νZ ZU ,
ˆ ( ),t xi n : =
t x b Y t x b Yj i n i j i n i( ), ( ),( , , ) ( , , )1 1−
(
,
de [ ]⋅ � cila çastyna i i ∈Z.
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Lema 5. Qkwo f C∈ ( )1 i ′f yi( ) = A dlq vsix i ∈Z, de A = const, to po-
linom
ˆ ( , )σn f x : =
σ
σ
n l
n l i
i n i
i n c n
i
s
f x
f y A
t y
t x,
,
,
,( , )
( , )
ˆ ( )
ˆ ( )
1
1
5
1
2
−
′ −
′
∈
=
∑ T
pry bud\-qkyx δ > 0 zadovol\nq[ nerivnosti
f fn− ⋅ˆ ( , )σ ≤ c h h16 ϕ( ) , (32)
′ − ′f x f xn( ) ˆ ( , )σ ≤ c f h x x
n
hk
s
17
4 2 1
1ω δ δ
δ
ϕ( , , [ , ]) ( )
( )
′ − + +
+ +
, x ∈R , (33)
′ − ′ ⋅f fn
ˆ ( , )σ ≤ c h17 ϕ( ) , (34)
L f x J L f y J f x Ak i k i i n− −′ − ′ − ′ +1 1( , , ) ( , , ) ˆ ( , )σ ≤
c
h
h x yi
18 ϕ( ) − , x Ji∈ , i ∈Z,
(35)
zokrema ˆ ( , )′σn if y = A .
Dovedennq. Vklgçennq σ̂n c n∈T
5
[ naslidkom (22), (13) i (14). Vykorysto-
vugçy (6), (20), (13) i (14), a takoΩ (16) i (17), dlq x ∈ −[ , ]π π zapysu[mo
f x f xn( ) ˆ ( , )− σ ≤ c h h1 ϕ( ) +
+
′ − ′
′
− + −
=
∑
σ
χ χn l i i
j i n i ii
s
j i n i j i j i j i n i
f y f y
t y b Y
t x b Y x x t x b Y
,
( ),
( ), ( ) ( ) ( ),
( , ) ( )
( , , )
( , , ) ( ) ( ) ( , , )1
11
2
1 1
(
≤
≤ c h h s
c h h
c y
c x
j
b s j
b s
1
1
8 1 1
2 2 6 1
2 12 2
1
1ϕ ϕ
( )
( )
( )
( )
( )
( )
( )
( )+ +
− −
Γ
Γ ≤ c h h16 ϕ( ),
tobto ocinka (32) [ pravyl\nog. Nerivnist\ (33), a otΩe i (34), dovodyt\sq ana-
lohiçno, z vykorystannqm (18) i (19) zamist\ (16) i (17). Oskil\ky za oznaçennq-
my ′f yi( ) = ˆ ( , )′σn if y = A , i ∈Z, to nerivnist\ (35) vykonu[t\sq, qkwo na Ji
spravdΩu[t\sq nerivnist\
′B x( ) : = ′ ′ − ′′−L f x J f xk i n1( , , ) ˆ ( , )σ ≤
c
h
h18 ϕ( ). (36)
Z (30) i (34) vyplyva[ ocinka
B Ji
= L f J f f fk i n Ji− ′ ⋅ − ′ + ′ − ′ ⋅1( , , ) ˆ ( , )σ ≤ c f J k J c hk i i14 17ω ϕ( / ), , ( )′ + ≤
≤ c f h c hk19 17ω ϕ( ), ( )′ + ≤ c h20 ϕ( ).
Z [7] vidomo, wo qkwo polinom zadovol\nq[ (34), to ˆ ( , )( )σn
k f+ ⋅1 ≤ c h hk
21ϕ( )/ .
Tomu
B k
Ji
( ) ≤ c h hk
21ϕ( )/ .
Teper poxidni B p( ), p k< , zadovol\nqgt\ nerivnist\ typu Kolmohorova (dyv.
[9, c. 35])
B p
Ji
( ) ≤ c J
c
h
h
J
c hi
k p
k
i
p22
21
20
1− +
ϕ ϕ( ) ( ) ≤
c
h
hp
18 ϕ( ).
Lemu 5 dovedeno.
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444 H. A. DZGBENKO
3. Dovedennq teoremy 1. 1°. Dlq j ∈Z budemo pysaty j V∈ , qkwo isnu[
toçka x I j∈ taka, wo
′f x( ) ≤ 2 17c hϕ( ). (37)
Poznaçymo c23 : = 96 17 8k c c[ ]/ + i c : = c s23 20 15+ + . Bez vtraty zahal\nosti bu-
demo vvaΩaty, wo n dilyt\sq na c, tobto n = pc, de p ∈N . Poklademo νp =
= n + 8 i ν− p = 8 − n . Nexaj dlq koΩnoho q = p – 1, … , 0, … , 1 – p νq po-
znaça[ najmenße cile sered cilyx j ≥ cq, dlq qkyx [ , ]x x Oj j+ −3 3 I = ∅ . Po-
znaçymo
Eq : = [ , ]x x
q qν ν −1
= …( )− +−
I I I
q q qν ν νU U U1 11
, q = 1− p p, .
OtΩe, cq +15 ≥ νq ≥ cq i koΩen vidrizok Eq sklada[t\sq prynajmni z
c s23 20+ i ne bil\ße niΩ z c s23 20 30+ + riznyx vidrizkiv I j .
Dali budemo vvaΩaty, wo q ∈Z ( f [ periodyçnog ) . Budemo pysaty q W∈ ,
qkwo Eq mistyt\ prynajmni 2 1k − promiΩkiv I j takyx, wo j V∈ . ZauvaΩy-
mo, wo qkwo q W∈ , to z nerivnostej (37) i (30) vyplyva[ nerivnist\
′f x( ) ≤ c h24 ϕ( ), x Eq∈ . (38)
V oznaçenni 1 funkcig ′f x( ) zapyßemo u vyhlqdi sumy �malen\ko]� g x1( ) i
�velyko]� g x2( ) funkcij tak, wob na mnoΩyni
E : =
Eq
q W∉
U
g x2( ) ≡ ′f x( ), a na R \ E (za vynqtkom okoliv kinciv E ) g x2( ) ≡ 0. Na kincqx
E mnoΩennqm na funkcig S j zabezpeçymo neperervnist\.
Dlq koΩnoho j ∈Z poznaçymo
S xj( ) : = ( ) ( ) ( ) ( )u x x u du u x x u duj
k
j
k
x
x
j
k
j
k
x
x
j j
j
− − − −
− −
−
∫ ∫
−
1 1
1
1
.
Dlq dovil\no] neporoΩn\o] mnoΩyny E ⊂ R çerez E∗ poznaçymo ob�[dnannq
vsix I j , j ∈Z, takyx, wo I j I E ≠ ∅ . Analohiçno E∗∗ : = ( )E∗ ∗ i t. d.
( )E E E⊂ ⊂ ⊂ …∗ ∗∗ .
Oznaçennq 1. Dlq x I j∈ poklademo
g x1( ) : =
0
1
, ,
( ), ,
( ) ( ), ,
( )( ( )), ,
\
\
\
qkwo
qkwo
qkwo
qkwo
I E
f x I E
f x S x I E E i x E
f x S x I E E i x E
j
j
j j j
j j j
⊂
′ ⊂
′ ⊂ ∈
′ − ⊂ ∉
∗
∗∗
∗∗ ∗ ∗
∗∗ ∗ ∗
R
i g x2( ) : = ′f x( ) – g x1( ).
Lema 6. Magt\ misce nerivnosti
g1 ≤ c h24 ϕ( ), ωk g t( , )1 ≤ c t25 ϕ( ), ωk g t( , )2 ≤ ( ) ( )c t25 1+ ϕ .
Lema 6 � ce faktyçno lema 17.4 z [9]. ZauvaΩymo, wo ]] perßa nerivnist\
vyplyva[ z (38); druha � z perßo], ocinky ωk f t( , )′ ≤ ϕ( )t i nerivnosti
S xj
( )( )ν ≤ c h26 / ν, x I j∈ , ν ∈N; tretq � z druho].
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KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 445
Poznaçymo c27 : = [ ]c25 2+ . Bez vtraty zahal\nosti budemo vvaΩaty, wo p ≥
≥ 4 27c . Podamo mnoΩynu E ≠ R u vyhlqdi ob�[dnannq vidrizkiv F a bm m m: [ , ]= ,
m ∈Z , wo ne peretynagt\sq. Budemo pysaty m X∈ , qkwo Fm sklada[t\sq ne
bil\ße niΩ z c27 riznyx vidrizkiv Eq (abo, wo te same, ne bil\ße niΩ z
c c27 15+ riznyx vidrizkiv I j ). Qkwo m X∉ , to Fm mistyt\ prynajmni
c c c27 23+ riznyx I j .
Oznaçennq 2. Poklademo
g x3( ) : =
g x x F
x F
mm X
mm X
2
0
( ), ,
, ,\
∈( )
∈ ( )
∈
∗∗
∈
∗∗
U
UR
i g x4( ) : = g x2( ) – g x3( ).
Lema 7. Magt\ misce nerivnosti
g3 ≤ c h28 ϕ( ), ωk g t( , )3 ≤ c t29 ϕ( ), ωk g t( , )4 ≤ ( ) ( )c c t25 291+ + ϕ .
Lema 7 dovodyt\sq analohiçno lemi 6 z uraxuvannqm samo] lemy 6.
Poznaçymo
f x1( ) : = f g u g u A du
x
( ) ( ( ) ( ) )0 1 3
0
+ + −∫ , f x2( ) : = ( ( ) )g u A du
x
4
0
+∫ ,
tak wo f x( ) = f x f x1 2( ) ( )+ i dijsne çyslo A , A ≤ ϕ( ) max{ , }/h c c24 28 2 , vy-
brano z umovy f1 0( ) = f1 2( )π (abo, wo te same, f2 0( ) = f2 2( )π ). Qkwo f x2( ) ≡
≡ Ax , to f x1( ) ≡ f x( ) ( )A = 0 i teorema 1 [ naslidkom lem 6, 7 i 3.
2°. Zadaça zvelasq do nablyΩennq funkci] f x2( ). Nexaj dlq vyznaçenosti
A ≥ 0. Poznaçymo
F : = Fm
m X∉
U .
Nahada[mo, wo za pobudovog
′f x2( ) =
′ + ∈
∉
∗
∗∗
f x A x F
A x F
( ) , ,
, ,
i na �bil\ßij� çastyni mnoΩyny F ma[mo ′ −f x A2( ) > 2 17c hϕ( ). Tomu zhidno z
(34) ˆ ( , )′ −σn f x A
1 2 > c hn17 ϕ( ) pry n1 > n. Odnak moΩut\ isnuvaty i �pohani�
toçky x (zokrema, na F ), v qkyx ( )ˆ ( , ) ( )′ −σn f x A x
1 2 Π < 0. V usix �pohanyx�
toçkax x ∈ F O\ , x ∈ ( \ )\R F O i x ∈ O my �vypravymo� polinom ˆ ( , )′σn f x
1 2 za
dopomohog polinomiv ′Q x( ) (lema 8), ′M x( ) (lema 9) i ′U xn( ) (lema 4) vidpo-
vidno.
Nexaj
δ j : = signΠ( )x j , t xj( ) = t x b Yj n, ( , , )1 ,
(
t xj( ) =
(
t x b Yj n, ( , , )1 .
Dlq koΩnoho E Fq ⊂ , q = 1− p p, , takoho, wo E Oq I ≠ ∅ , çerez νq
+ i νq
−
poznaçymo najbil\ße j H∈ , dlq qkoho I Ej q⊂ i δ j > 0 abo δ j < 0 vidpo-
vidno.
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446 H. A. DZGBENKO
Oznaçennq 3. Dlq koΩnoho q = 1− p p, poklademo Q xq( ) :≡ 0, qkwo
E Fq /⊂ abo E Fq ⊂ i Eq ne mistyt\ vidrizkiv I j z j V H∈ I . Dlq reßty
Eq , tobto dlq E Fq ⊂ i Eq , wo mistyt\ I j z j V H∈ I , poklademo
Q xq( ) : =
2
2
39
2
17
8 1
17
7 1
17
8 1
1 1
1
c
c
t x
c
c
t x E O
c
c
t x t x t
j j
j j V
q j j
j j V
q
j j
j j V H
q q
q
q
q
q
q
q
q q
( ) ( ) , , ( )
( ) ( )
, ,
,
δ α δ
δ α α
ν
ν
ν
ν
ν
ν
ν ν
= + ∈ = + ∉
= + ∈
+ −
− −
−
+ −
∑ ∑
∑
− = ∅
+ −
(
I
I
(( ) , , ( )x E Oq
≠ ∅
I 40
de çysla αq > 0, αq
+ ≥ 0 i αq
− ≥ 0 vybrano tak, wo Qq( )− π = Qq( )π i
α αq q
+ − = 0.
Poznaçymo
F1 : = I j
j I F j V Hj: ,⊂ ∈ I
U , F2 : = I j
j I F j V j Hj: , ,⊂ ∉ ∈
U ,
tak wo F \ ( )F F1 2U ⊂ O.
Lema 8. Funkciq
Q x( ) : = h h Q xq
q p
p
ϕ( ) ( )
= −
∑
1
zadovol\nq[ spivvidnoßennq
Q ∈ Tc n5
, (41)
Q ≤ c h h30 ϕ( ) , (42)
′Q x( ) ≥ 2 17c hϕ( ), x ∈F1, (43)
′Q x x( ) ( )signΠ ≥ –
c
h17
2
ϕ( ), x ∈F2 , (44)
′Q x x( ) ( )Π ≥ 0, x ∈R \F2 . (45)
Dovedennq. Nerivnosti (13), (14), oznaçennq δ j i vybir νq
± harantugt\
strohu dodatnist\ αq i, vidpovidno, strohu dodatnist\ αq
+ pry αq
− = 0 abo
αq
− > 0 pry αq
+ = 0. Tomu Qq( )− π = Qq( )π . Razom z (22) ce porodΩu[ (41).
Wob dovesty (42), pokaΩemo, wo αq < 1 i αq k± < 2 . Dijsno, qkwo E Fq ⊂ , to,
zokrema, q W∉ , tobto perßi sumy v (39) i (40) mistqt\ ne bil\ße niΩ 2 2k −
dodanky koΩna, a druha suma v (39) � prynajmni c23 2/ dodankiv. Tomu, vraxo-
vugçy dodatnist\ αq , (22) i umovu Qq( )− π = Qq( )π , zapysu[mo
αq ≤ 2 1
2
2 2
17
8
23 17
7
1
( )k
c
c
c c
c
−
−
<
16 7
8 23
kc
c c
< 1.
Analohiçno αq
± ≤ 2 1 1( )/k − < 2k . Poklademo S xq( ) := 0, qkwo Q xq( ) = 0,
inakße
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 447
S xq( ) : =
2
2
2
17
8 1
17
7 1
17
8 1
1 1
1
c
c
x
c
c
x E O
c
c
x x x
j j
j j V
q j j
j j V
q
j j
j j V H
q q
q
q
q
q
q
q
q q
χ δ α χ δ
χ δ α χ α χ
ν
ν
ν
ν
ν
ν
ν ν
( ) ( ) , ,
( ) ( ) ( )
, ,
,
= + ∈ = + ∉
= + ∈
+ −
− −
−
+ −
∑ ∑
∑
− = ∅
+ −
I
I
≠ ∅
, .E Oq I
Vykorystovugçy (16) i (17), zapysu[mo nerivnist\ dlq x ∈ [ ],− − −π π8 8h h =
= [ ],x x
p pν ν−
= : I :
Q x S xq q( ) ( )− ≤
2
2
4
217
8
17
7
17
8
6
2 1
1
1
1
c
c
c
c
k
c
c
c xj
b s
j q
q
+ +
− −
= +−
∑ ( )( )Γ
ν
ν
= : c B xq31 ( ).
Teper qkwo x Eq∈R \ , to χ j x( ) = χνq
x( ) dlq vsix j = ν νq q− +1 1, , i tomu vna-
slidok periodyçnosti Q xq( ) spravdΩu[t\sq rivnist\
S xq( ) = 0, x Eq∈R \ ,
z qko] vyplyva[ ocinka
S xq( ) ≤ c B xq32 ( ), x ∈ I .
OtΩe, z uraxuvannqm (10) zapyßemo nerivnist\
Q x( ) ≤ h h Q x S x S xq q q
q p
p
ϕ( ) ( ) ( ) ( )− +
= −
∑ ≤
≤ ( ) ( ) ( )c c h h B xq
q p
p
31 32+
= −
∑ϕ ≤ c h h30 ϕ( ) , x ∈ I ,
z qko] vnaslidok periodyçnosti Q x( ) vyplyva[ (42). Dovedemo (43) � (45). Zob-
razymo Q x( ) u vyhlqdi
Q xq( ) =
2
2
217
8
17
7
17
8 61 2 0
c
c
t x
c
c
t x
c
c
t xj j
j I I
j j j
j I I
j j j
j H nj j
( ) ( ) ( )
: : : { }
δ β δ γ δ
⊂ ⊂ +
∑ ∑ ∑+ +
F FI I U
(
,
de − < ≤1 0β j i 0 2≤ <γ j k dlq vsix j . Dlq koΩnoho j H n n∈ + +0 6 7U { , ,
n + 8} z nerivnostej (13), (14), (20) i (19) otrymu[mo ocinky
′t x xj j( ) ( )Π δ ≥ 0, x ∈ R ,
β δj j jt x x′
(
( ) ( )Π ≥ 0, x ∈ I I j\ ,
2 17
8
c
c
t x xj j′ ( ) ( )signΠ δ ≥
2 17c
h
, x ∈ Ij ,
c
c
t xj
17
72
′
(
( ) ≤
c
h
17
2
, x ∈ Ij .
Teper z uraxuvannqm (41) z perßyx dvox ocinok vyplyva[ (45), z perßyx tr\ox �
(43) i z perßyx dvox i çetverto] � (44).
Lemu 8 dovedeno.
Nahada[mo, wo F =
Fmm X∉U , de Fm = [ ],a bm m ne peretynagt\sq i koΩen
Fm mistyt\ prynajmni c c c27 23+ = : c33 riznyx I j ( c27 1+ riznyx Eq ). Bude-
mo pysaty m X∈ 0 , qkwo m X∉ i F a am I [ ],0 0 2+ π = Fm. Dlq koΩnoho
m X∈ 0 poznaçymo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
448 H. A. DZGBENKO
[ ]
, ,
,x xj ja m b m
: = [ ],a bm m = Fm,
Fa m, : = [ ]
, ,
,x xj j ca m a m − 32
, Fb m, : = [ ]
, ,
,x xj c jb m b m+ 32
.
Dlq koΩnoho m X∈ 0 takoho, wo
F Oa m, I ≠ ∅ (
F Ob m, I ≠ ∅ ) , çerez νa m,
+ i
νa m,
− ( ν νb m b mi, ,
+ −
) poznaçymo dva najbil\ßi cili j H∈ , dlq qkyx I Fj a m⊂ ,
( I Fj b m⊂ , ) i δ j > 0 abo δ j < 0 vidpovidno.
Oznaçennq 4. Dlq koΩnoho m X∈ 0 poklademo
M xa m, ( ) : =
: =
2 1
2
2 1
17 25
8 1
3
17
7 1
17 25
8 1
3
33
c c
c
t x
c
c
t x F O
c c
c
t x
j j
j j
j
a m j j
j j c j V
j
a m
j j
j j
j
a
a m
a m
a m
a m
a m
a m
( )
( ) ( ) , ,
( )
( )
,
,
,
,
,
,
,
,
,
+ − = ∅
+ +
= +
+
= − + ∉
= +
+
∑ ∑
∑
δ µ δ
δ µ
(
I
,, , ,
, ,
( ) ( ) , ,m a m a mt x t x F O
a m a m
+ −
+ −−
≠ ∅
3 3ν νµ I
M xb m, ( ) : =
: =
2 1
2
2 1
3
17 25
8 2
17
7 1
17 25
8 2
33c c
c
t x
c
c
t x F O
c c
c
t x
j j
j j
j
b m j j
j j j V
j c
b m
j j
j j
j
b m
b m
b m
b m
b m
b m
b m
( )
( ) ( ) , ,
( )
( )
,
,
,
,
,
,
,
,
,
,
+ − = ∅
+ +
= − = + ∉
+
= −
+
∑ ∑
∑
δ µ δ
δ µ
(
I
tt x t x F O
b m b m
b m b mν νµ
, ,
( ) ( ) , ,, ,+ −−
≠ ∅
− 3 I
de çysla µa m, > 0, µa m,
± ≥ 0, µb m, > 0 i µb m,
± ≥ 0 vybrano tak, wo Ma m, ( )− π =
= Ma m, ( )π , µ µa m a m, ,
+ − = 0, Mb m, ( )− π = Mb m, ( )π i µ µb m b m, ,
+ − = 0.
Poznaçymo
F3 : = F F∗∗∗ \ .
Lema 9. Funkciq
M x( ) : = h h M x M xa m b m
m X
ϕ( ) ( ) ( )( ), ,+
∈
∑
0
zadovol\nq[ spivvidnoßennq
M ∈ Tc n5
, (46)
M ≤ c h h34 ϕ( ), (47)
′M x( ) ≥ 2 117 25c c h( ) ( )+ ϕ
˘ ( ( , )) ( )Γn
sxdist F3
4 2( ) + , x F O∈R \ ( )U , (48)
′M x x( ) ( )signΠ ≥ –
c
h17
4
ϕ( ), x ∈F2 , (49)
′M x x( ) ( )Π ≥ 0, x ∈R \F2 . (50)
Dovedennq lemy 9 [ analohiçnym dovedenng lemy 8. Dovedemo lyße neriv-
nist\ µa m, < 1 4/ . Za pobudovog koΩen vidrizok E Fq ⊂ mistyt\ ne bil\ße niΩ
2 2k − riznyx I j z j V∈ , tomu suma pry µa m, mistyt\ prynajmni
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 449
c c k33 27 1 2 2− + −( )( ) >
c
c33
272
1 2+ +( ) >
c33
2
dodankiv. Tomu
µa m, ≤ 3
2 1
2 2
17 25
8
33 17
7
1
c c
c
c c
c
( )+
−
<
24 7
8
c
c c
<
1
4
.
Lemu 9 dovedeno.
3°. Poznaçymo
c35 : = 4 125 29π ( )c c+ + , n1 : = c n35 , h1 : =
π
n1
, (51)
c36 : =
max{ , }( )c c c c c
c
17 18 25 29 35
13
1+ +
,
R xn1
( ) : = τ σn n nf Ax x Ax f x Q x M x c
c
c
U x( , ) ˆ ( , ) ( ) ( ) ( )1 2 36
15
13
1
+ − + + + + +
∈ Tc n5 1
.
PokaΩemo, wo Rn1
� ßukanyj v teoremi 1 polinom. Vraxu[mo lemy 6 i 7 i
zberemo (23), (32), (42), (47) i (27) v ocinku
f Rn−
1
= f f Rn1 2 1
+ − ≤ f f Ax A f fn n1 1 2 21
− + ⋅ + ⋅ + − ⋅τ σ( , ) ˆ ( , ) +
+ Q M c
c
c
Un+ + +
36
15
13
≤
≤ c h c c c c h c h c c h11 24 28 25 29 16 1 25 29 11max{ , }( ) ( ) ( ) ( )+ + + +ϕ ϕ +
+ c c c
c
c
c h h30 34 36
15
13
12+ + +
ϕ( ) ≤ c h h37 ϕ( ) . (52)
Perevirymo nerivnist\
′R x xn1
( ) ( )Π ≥ 0, x ∈ R , (53)
vykorystavßy rivnist\
′R x xn1
( ) ( )signΠ = ′ + + ′ −τn f Ax x x f x A x( , ) ( ) ( ( ) ) ( )1 2sign signΠ Π +
+ ( )ˆ ( , ) ( ) ( )′ − ′σn f x f x x
1 2 2 signΠ +
+ ( )( ) ( ) ( ) ( ) ( ) ( ) ( )′ + ′ + ′ + ′Q x M x x c U x x
c
c
U x xn nsign sign signΠ Π Π36
15
13
= : Ψν
ν
( )x
=
∏
1
6
.
Z (24), pobudovy f2 i (26) vydno, wo
Ψ1( )x ≥ 0, Ψ2( )x ≥ 0, Ψ5( )x ≥ 0, Ψ6( )x ≥ 0, x ∈ R .
Nexaj F4 : = R \ ( )F OU UF3 , tak wo F F F F1 2 3 4U U U U O = R . Rozhlqnemo
p�qt\ vypadkiv.
1) x ∈F1. Dlq u ∈ ∗F1 funkciq ′f u2( ) = ′ +f u A( ) . Beruçy do uvahy neriv-
nist\ (33) z δ = h, (45), (43), (50), lemu 7 i (51), zapysu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
450 H. A. DZGBENKO
Ψ Ψ3 4( ) ( )x x+ ≥ – c f A h c f h
n h
c hk k
s
17 1 17 2 1
1
4 2 1
17
1
2 0ω ω ϕ( , ) ( , ) ( )
( )
′ + − ′
+ +
+ +
≥
≥ ϕ π( ) ( )h c c c
n
n17 25 29
1
1 1− + +
≥ 0.
2) x ∈F2 . Dlq u ∈ ∗F2 funkciq ′f u2( ) = ′ +f u A( ) . Bil\ß toho, ′ −f x A2( ) ≥
≥ 2 17c hϕ( ), x ∈F2 . Teper iz nerivnosti (33) z δ = h, (44), (49), lemy 7 i (51)
otrymu[mo nerivnist\
Ψ Ψ Ψ2 3 4( ) ( ) ( )x x x+ + ≥ 2
1
17 17 1 17 2 1
1
4 2 1
c h c f A h c f h
n hk k
s
ϕ ω ω( ) ( , ) ( , )
( )
− ′ + − ′
+ +
–
–
c
h
c
h17 17
2 4
ϕ ϕ( ) ( )− ≥ ϕ π( ) ( )h c c c
n
n17 25 29
1
1
4
1− + +
≥ 0.
3) x ∈F3. Dlq u ∈ ∗F3 z funkci] ′f u2( ) = g u A2( ) + , nerivnostej (33) z δ =
= h, (45), (50), (48), lem 6, 7 i (51) oderΩu[mo nerivnist\
Ψ Ψ3 4( ) ( )x x+ ≥
≥ – c g A h c c c h
n h
c c hk
s
17 2 1 17 25 29 1
1
4 2 1
17 251
1
0 2 1ω ϕ ϕ( , ) ( ) ( ) ( ) ( )
( )
+ − + +
+ + +
+ +
≥
≥ ϕ π( ) ( )h c c c c
n
n17 25 25 29
1
1 1+ − + +
≥ 0.
4) x ∈F4 . Dlq u ∈ ∗F4 funkciq ′f u2( ) = A . Tomu ωk f t( , , )′2 4F ≡ 0. Vyko-
rystovugçy (33) z δ = dist ( , )x F∗∗ , lemu 7, (45), (50), (48), (51) i nerivnist\
1
1n x Fdist ( , )∗∗ < Γ̆n x( ( , ))dist F3 ,
zapysu[mo
Ψ Ψ3 4( ) ( )x x+ ≥ – c c c c h
n x F
s
17 17 25 29 1
1
4 2 1
0 1
1⋅ − + +
∗∗
+ +
( ) ( )
( , )
( )
ϕ
dist
+
+
0 2 117 25 3
4 2
+ +
+
c c h xn
s
( ) ( ) ( ( , ))
( )
ϕ Γ̆ dist F ≥
≥
ϕ π( ) ( ( , )) ( ) ( )
( )
h c x c c c
n
n
n
s
17 3
4 2
25 25 29
1
2 1 1Γ̆ dist F
+ − + +
+
≥ 0.
5) x O∈ . Zhidno z (45) i (50) ma[mo Ψ4( )x ≥ 0. Dlq x Oi∈ tako], wo
O Fi I = ∅, funkciq ′f x2( ) = A , tomu z (26), (29), (51), lemy 7 i vidpovidno (35)
i (34) ma[mo
Ψ Ψ5 3( ) ( )x x+ ≥
c c
h
h x y
c
h
c c h x yi i
36 13 18
1
25 29 11ϕ ϕ( ) ( ) ( )− − + + − ≥ 0, x Ji n∈ , 1
,
(54)
Ψ Ψ5 3( ) ( )x x+ ≥
c c
h
h x y c c c hi
36 13
17 25 29 11ϕ ϕ( ) ( ) ( )− − + + ≥ 0, x O Ji i n∈ \ , 1
.
(55)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
KOMONOTONNE NABLYÛENNQ DVIÇI DYFERENCIJOVNYX PERIODYÇNYX … 451
Dlq reßty O Oi ⊂ , tobto dlq x Oi∈ : O Fi I ≠ ∅, funkciq ′f x2( ) = ′ +f x A( ) .
V c\omu vypadku nerivnist\ (55) vykonu[t\sq, a nerivnist\
Ψ Ψ Ψ5 3 2( ) ( ) ( )x x x+ + = Ψ5 2 1 2 1 21 1 1
( ) ˆ ( , ) ( , , ) ( , , ), ,x f x L f x J L f y Jn k i n k i i n+ ′ − ′ + ′( − −σ +
+ L f x J L f y J f x x xk i n k i i n− −′ − ′ − ′ ) +1 2 1 2 2 21 1
( , , ) ( , , ) ( ) ( ) ( ), , signΠ Ψ =
= Ψ Π5 2 1 2 1 21 1 1
( ) ˆ ( , ) ( , , ) ( , , ) ( ), ,x A f x L f x J L f y J xn k i n k i i n+ − + ′ − ′ + ′( )− −σ sign +
+ L f x J L f y J xk i n k i i n− −′ − ′( )1 11 1
( , , ) ( , , ) ( ), , signΠ = : Ψ Ψ Ψ5 31 3 2( ) ( ) ( , ), ,x x x k+ + ≥ 0,
x Ji n∈ , 1
,
spravdΩu[t\sq analohiçno (54), qkwo
Ψ3 2, ( , )x k ≥ 0, x Ji n∈ , 1
. (56)
Takym çynom, dovedennq nerivnosti (53) zvelosq do dovedennq nerivnos-
ti (56). Dlq f C Y∈ ( ) ( )( )2 1I ∆ vyberemo ω ∈Φk taku, wo ωk f t( , )′′ ≤ ω( )t ≤
≤ 2k
k f tω ( , )′′ . Nexaj teper ϕ( )t : = t tω( ) (tobto ϕ ∈ +Φk 1). Zapyßemo
Ψ3 2 1, ( , )x k + = ′ ′ − ′′ + ′
∫ L f u J f u du f x xk i n
y
x
i
( , , ) ( ) ( ) ( ), 1
signΠ .
Todi z (26), (29), (31) ( p = 1 ) (i ′f x x( ) ( )Π ≥ 0 ) oderΩu[mo nerivnist\
Ψ Ψ6 3 2 1( ) ( , ),x x k+ + ≥ c h x y c h x yi i15 15 1 0ω ω( ) ( )− − − + ≥ 0, x Ji n∈ , 1
,
i (53) spravdΩu[t\sq. Pry c\omu ocinka (5) [ naslidkom (52).
Teoremu 1 dovedeno.
1. Dzqd¥k V. K. Vvedenye v teoryg ravnomernoho pryblyΩenyq funkcyj polynomamy. � M.:
Nauka, 1977. � 512 s.
2. Pleshakov M. G. Comonotone Jackson’s inequality // J. Approxim. Theory. – 1999. – 99. –
P. 409 – 421.
3. Dzgbenko H. A., Pleßakov M. H. Komonotonnoe pryblyΩenye peryodyçeskyx funkcyj //
Mat. zametky. � 2008. � 83, v¥p. 2. � S. 199 � 209.
4. Whitney H. On functions with bounded n-th diferences // J. math. pures et appl. – 1957. – 6(9). –
P. 67 – 95.
5. Pleßakov M. H. Komonotonnoe pryblyΩenye peryodyçeskyx funkcyj klassov Soboleva:
Dys. … kand. fyz.-mat. nauk. � Saratov, 1997.
6. Dzgbenko H. A. Kontrpryklad v komonotonnomu nablyΩenni periodyçnyx funkcij // Zb.
prac\ In-tu matematyky NAN Ukra]ny. � 2008. � 5, # 1. � S. 113 � 123.
7. Steçkyn S. B. O porqdke nayluçßyx pryblyΩenyj neprer¥vn¥x funkcyj // Yzv. AN SSSR.
Ser. mat. � 1951. � 15, # 3. � S. 219 � 242.
8. Dzyubenko G. A., Gilewicz J., Shevcuk I. A. Piecewise monotone pointwise approximation //
Constr. Approxim. – 1998. – 14. – P. 311 – 348.
9. Íevçuk Y. A. PryblyΩenye mnohoçlenamy y sled¥ neprer¥vn¥x na otrezke funkcyj. �
Kyev: Nauk. dumka, 1992. � 225 s.
OderΩano 10.06.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
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| id | umjimathkievua-article-3032 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:59Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c5/f2443b744aef63d9d20308e0b0e8a6c5.pdf |
| spelling | umjimathkievua-article-30322020-03-18T19:43:50Z Comonotone approximation of twice differentiable periodic functions Комонотонне наближення двічі диференційовних періодичних функцій Dzyubenko, H. A. Дзюбенко, Г. А. In the case where a $2π$-periodic function $f$ is twice continuously differentiable on the real axis $ℝ$ and changes its monotonicity at different fixed points $y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ $(i.e., on $ℝ$, there exists a set $Y := {y_i } i∈ℤ$ of points $y_i = y_{i+2s} + 2π$ such that the function $f$ does not decrease on $[y_i , y_{i−1}]$ if $i$ is odd and does not increase if $i$ is even), for any natural $k$ and $n, n ≥ N(Y, k) = const$, we construct a trigonometric polynomial $T_n$ of order $≤n$ that changes its monotonicity at the same points $y_i ∈ Y$ as $f$ and is such that $$∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n)$$ $$(∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/ n),f ∈ C^{(r)},\; r ≥ 2),$$ where $N(Y, k)$ depends only on $Y$ and $k, c(k, s)$ is a constant depending only on $k$ and $s, ω k (f, ⋅)$ is the modulus of smoothness of order $k$ for the function $f$, and $‖⋅‖$ is the max-norm. В случае, когда дважды непрерывно дифференцируемая на действительной оси $ℝ$ $2π$-периоди-ческая функция $f$ изменяет монотонность в различных фиксированных точках $y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ $ (т. е. на $ℝ$ есть множество $Y := {y_i } i∈ℤ$ точек yі = $y_i = y_{i+2s} + 2π$ таких, что на $[y_i , y_{i−1}]$ $f$ не убывает, если $i$ нечетное, и не возрастает, если $i$ четное), для каждых натуральных $k$ и $n, n ≥ N(Y, k) = const$, построен тригонометрический полином $T_n$ порядка $≤n$, который изменяет свою монотонность в тех же точках $y_i ∈ Y$ , что и $f$, и такой, что $$∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n)$$ $$(∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/ n),f ∈ C^{(r)},\; r ≥ 2),$$ где $N(Y, k)$ зависит только от $Y$ и $k, c(k, s)$ — постоянная, зависящая только от $k$ и $s, ω k (f, ⋅)$ — модуль гладкости порядка $k$ функции $f$ и $‖⋅‖$ — max-норма. Institute of Mathematics, NAS of Ukraine 2009-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3032 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 4 (2009); 435-451 Український математичний журнал; Том 61 № 4 (2009); 435-451 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3032/2813 https://umj.imath.kiev.ua/index.php/umj/article/view/3032/2814 Copyright (c) 2009 Dzyubenko H. A. |
| spellingShingle | Dzyubenko, H. A. Дзюбенко, Г. А. Comonotone approximation of twice differentiable periodic functions |
| title | Comonotone approximation of twice differentiable periodic functions |
| title_alt | Комонотонне наближення двічі диференційовних періодичних функцій |
| title_full | Comonotone approximation of twice differentiable periodic functions |
| title_fullStr | Comonotone approximation of twice differentiable periodic functions |
| title_full_unstemmed | Comonotone approximation of twice differentiable periodic functions |
| title_short | Comonotone approximation of twice differentiable periodic functions |
| title_sort | comonotone approximation of twice differentiable periodic functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3032 |
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