Degree of comonotone approximation of periodic functions
If a continuously differentiable on a real axe $\Bbb R\ \ 2\pi - $periodic function $f$ changes its monotonicity at different fixed points $y_i\in [-\pi,\pi),\ i=1,...,2s,\ s\in\Bbb N ,$ (i.e., on $\Bbb R$ there is a set $Y:=\{y_i\}_{i\in\Bbb Z}$ of points $y_i=y_{i+2s}+2\pi $ such that on $[y_i,y_{...
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| Дата: | 2013 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська |
| Опубліковано: |
Інститут математики НАН України
2013
|
| Онлайн доступ: | https://trim.imath.kiev.ua/index.php/trim/article/view/142 |
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| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552659603423232 |
|---|---|
| author | Dzyubenko, G. A. Дзюбенко, Г. А. |
| author_facet | Dzyubenko, G. A. Дзюбенко, Г. А. |
| author_institution_txt_mv | [
{
"author": "Г. А. Дзюбенко",
"institution": "Міжнародний математичний центр ім. Ю. О. Митропольського НАН України"
}
] |
| author_sort | Dzyubenko, G. A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-29T14:44:27Z |
| description | If a continuously differentiable on a real axe $\Bbb R\ \ 2\pi - $periodic function $f$ changes its monotonicity at different fixed points $y_i\in [-\pi,\pi),\ i=1,...,2s,\ s\in\Bbb N ,$ (i.e., on $\Bbb R$ there is a set $Y:=\{y_i\}_{i\in\Bbb Z}$ of points $y_i=y_{i+2s}+2\pi $ such that on $[y_i,y_{i-1}]\ f$ is nondecreasing if $i$ is odd, and nonincreasing if $i$ is even), then for each natural number $n,\ n\ge N(Y)=const,$ in the article a trigonometric polynomial $T_n $ of order $\le n,$ which changes its monotonicity at the same points $y_i\in Y,$ like $f,$ is found such that $$\left\Vert f-T_n \right \Vert \le \frac{c(s)}{n}\, \omega_3\left(f',1/n\right),$$ where $N(Y)$ depends only on $Y,$ $c(s)-$ constant which is depending only on $s,\ \omega_3 \left(f,\cdot\right)-$ modulus of smoothness of order $3$ of the function $f$ and $\Vert \cdot \Vert -\max $-norm. Also the other estimates that are possible in this kind of approximation are listed. |
| first_indexed | 2026-08-04T01:02:57Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Ií-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2013, Ò.10, �1, 110�125
ÓÄÊ 517.5
Ã.À. Äçþáåíêî (Ìiæíàðîäíèé ìàòåìàòè÷íèé öåíòð iì. Þ. Î. Ìèòðî-
ïîëüñüêîãî ÍÀÍ Óêðà¨íè, Êè¨â)
ÏÎÐßÄÊÈ ÊÎÌÎÍÎÒÎÍÍÎÃÎ ÍÀÁËÈÆÅÍÍß ÏÅ-
ÐIÎÄÈ×ÍÈÕ ÔÓÍÊÖIÉ
If a continuously di�erentiable on a real axe R 2π−periodic function f changes
its monotonicity at di�erent �xed points yi ∈ [−π, π), i = 1, ..., 2s, s ∈ N,
(i.e., on R there is a set Y := {yi}i∈Z of points yi = yi+2s + 2π such that on
[yi, yi−1] f is nondecreasing if i is odd, and nonincreasing if i is even), then
for each natural number n, n ≥ N(Y ) = const, in the article a trigonometric
polynomial Tn of order ≤ n, which changes its monotonicity at the same points
yi ∈ Y, like f, is found such that
‖f − Tn‖ ≤
c(s)
n
ω3
(
f ′, 1/n
)
,
where N(Y ) depends only on Y, c(s)− constant which is depending only on
s, ω3 (f, ·)− modulus of smoothness of order 3 of the function f and ‖·‖−max-
norm. Also the other estimates that are possible in this kind of approximation
are listed.
ßêùî íåïåðåðâíî äèôåðåíöiéîâíà íà äiéñíié îñi R 2π−ïåðiîäè÷íà ôóíê-
öiÿ f çìiíþ¹ ìîíîòîííiñòü â ðiçíèõ ôiêñîâàíèõ òî÷êàõ yi ∈ [−π, π),
i = 1, ..., 2s, s ∈ N, (òîáòî, íà R ¹ ìíîæèíà Y := {yi}i∈Z òî-
÷îê yi = yi+2s + 2π òàêèõ, ùî íà [yi, yi−1] f íå ñïàäà¹, ÿêùî i íå
ïàðíå, i íå çðîñòà¹, ÿêùî i ïàðíå), òî äëÿ êîæíîãî íàòóðàëüíîãî
n, n ≥ N(Y ) = const, â ñòàòòi çíàéäåíî òðèãîíîìåòðè÷íèé ïîëiíîì Tn
ïîðÿäêó ≤ n, ÿêèé çìiíþ¹ ñâîþ ìîíîòîííiñòü â òèõ ñàìèõ òî÷êàõ yi ∈ Y,
ùî i f, i òàêèé, ùî
‖f − Tn‖ ≤
c(s)
n
ω3
(
f ′, 1/n
)
,
äå N(Y ) çàëåæèòü òiëüêè âiä Y, c(s)− ñòàëà, ÿêà çàëåæàòü òiëüêè âiä
s, ω3 (f, ·)− ìîäóëü ãëàäêîñòi ïîðÿäêó 3 ôóíêöi¨ f i ‖ · ‖ − max-íîðìà.
Òàêîæ íàâåäåíî iíøi îöiíêè, ÿêi ìîæëèâi ïðè òàêîìó íàáëèæåííi.
1. Âñòóï. Íåõàé C −ïðîñòið íåïåðåðâíèõ 2π−ïåðiîäè÷íèõ ôóíê-
öié f : R → R, ‖f‖ := ‖f‖R := max
x∈R
|f(x)| , i Tn, n ∈ N, −ïðîñòið
c© Ã.À. Äçþáåíêî, 2013
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 111
òðèãîíîìåòðè÷íèõ ïîëiíîìiâ tn(x) = a0 +
∑n
j=1(aj cos jx + bj sin jx)
ïîðÿäêó ≤ n, äå aj , bj ∈ R. Íàãàäà¹ìî êëàñè÷íó òåîðåìó Äæåêñîíà-
Çiãìóíäà-Àõi¹çåðà-Ñòå÷êiíà (äèâ., íàïðèêëàä, [1], ñ. 204 � 212): ïðè
êîæíèõ íàòóðàëüíèõ k i n äëÿ áóäü-ÿêî¨ ôóíêöi¨ f ∈ C çíàéäåòüñÿ
ïîëiíîì σn ∈ Tn òàêèé, ùî
‖f − σn‖ ≤ c(k) ωk (f, 1/n) , (1)
äå c(k)− ñòàëà, ÿêà çàëåæèòü ëèøå âiä k i ωk (f, ·)− ìî-
äóëü ãëàäêîñòi ïîðÿäêó k ôóíêöi¨ f. Êðiì òîãî, ÿêùî
f ∈ C(r) :=
{
f : f (r) ∈ C
}
, r ∈ N, òî íàñëiäêîì (1) ¹ íåðiâíiñòü
‖f − σn‖ ≤
c(r + k)
nr
ωk
(
f (r), 1/n
)
, n ∈ N. (2)
 öié ñòàòòi, â Òåîðåìi 1′ íàâåäåíî êîìîíîòîííi àíàëîãè íåðiâíî-
ñòåé (1) i (2). Äëÿ òî÷íîãî ¨õ ôîðìóëþâàííÿ äàìî íåîáõiäíi ïîçíà-
÷åííÿ. Íåõàé íà [−π, π) çàôiêñîâàíî 2s, s ∈ N, òî÷îê yi :
−π ≤ y2s < y2s−1 < · · · < y1 < π,
à äëÿ ðåøòè iíäåêñiâ i ∈ Z, òî÷êè yi âèçíà÷àþòüñÿ ðiâíiñòþ
yi = yi+2s + 2π (òîáòî, y0 = y2s + 2π, ..., y2s+1 = y1 − 2π, ...). Ïî-
çíà÷èìî Y := {yi}i∈Z, ∆(1)(Y )− ìíîæèíà âñiõ ôóíêöié f, ÿêi íå
ñïàäàþòü íà [y1, y0], íå çðîñòàþòü íà [y2, y1], íå ñïàäàþòü íà [y3, y2]
i ò.ä.. Çàìiòèìî, ÿêùî ïåðiîäè÷íà ôóíêöiÿ f äèôåðåíöiéîâíà, òî
f ∈ ∆(1)(Y ) ⇐⇒ f ′(x)Π(x) ≥ 0, x ∈ R, äå
Π(x) := Π(x, Y ) :=
2s∏
i=1
sin
x− yi
2
, (Π(x) > 0, x ∈ (y1, y0)) .
 öié ñòàòòi ìè äîâîäèìî íåðiâíiñòü (4) íàñòóïíî¨ Òåîðåìè 1,
íåðiâíiñòü (3) äîâåäåíî â [2], à (5) − â [3].
Òåîðåìà 1. Äëÿ áóäü-ÿêî¨ ôóíêöi¨ f ∈ ∆(1)(Y ) çíàéäóòüñÿ ïîëiíîìè
Tn, Pn i Rn ç Tn ∩∆(1)(Y ) òàêi, ùî
‖f − Tn‖ ≤ c(2, s) ω2 (f, 1/n) , f ∈ C, n ≥ N(2, Y ), (3)
112 Ã.À. Äçþáåíêî
‖f − Pn‖ ≤
c(3, s)
n
ω3 (f ′, 1/n) , f ∈ C(1), n ≥ N(3, Y ), (4)
‖f −Rn‖ ≤
c(k, s)
n2
ωk (f ′′, 1/n) , f ∈ C(2), n ≥ N(k, Y ), (5)(
‖f −Rn‖ ≤
c(r+k, s)
nr
ωk(f (r), 1/n), f ∈ C(r), r ≥ 2, n ≥ N(r+k, Y )
)
äå k ∈ N, c(k, s)− ñòàëi, ÿêi çàëåæàòü òiëüêè âiä k i s, N(k, Y )−
ñòàëi, ÿêi çàëåæàòü òiëüêè âiä k i Y (òîáòî âiä min
i=1,...,2s
{yi+1−yi}).
Íàñëiäêîì Òåîðåìè 1 i íåðiâíîñòi Óiòíi [4] ||f − f(0)|| ≤
≤ k ωk(f, kπ), k ∈ N, ¹
Òåîðåìà 1′. Ïðè êîæíîìó íàòóðàëüíîìó n äëÿ áóäü-ÿêî¨ ôóíêöi¨
f ∈ ∆(1)(Y ) çíàéäóòüñÿ ïîëiíîìè Tn, Pn i Rn ç Tn ∩∆(1)(Y ) òàêi,
ùî
‖f − Tn‖ ≤ C(2, Y ) ω2 (f, 1/n) , f ∈ C, (3′)
‖f − Pn‖ ≤
C(3, Y )
n
ω3 (f ′, 1/n) , f ∈ C(1), (4′)
‖f −Rn‖ ≤
C(k, Y )
n2
ωk (f ′′, 1/n) , f ∈ C(2), (5′)(
‖f −Rn‖ ≤
C(r+k, Y )
nr
ωk(f (r), 1/n), f ∈ C(r), r ≥ 2
)
äå k ∈ N, à C(k, Y )− ñòàëi, ÿêi çàëåæàòü òiëüêè âiä k i Y.
 [5, ñ. 64 � 83] i [6] ïîáóäîâàíî êîíòðïðèêëàäè, ÿêi âêàçóþòü íà
òå, ùî ω2 â (3′) (i (3)) i ω3 â (4′) (i (4)) íåìîæëèâî çàìiíèòè íà ωk ç
k > 2 i k > 3, âiäïîâiäíî.
Çàóâàæåííÿ 1. Ìè ïðèïóñêà¹ìî, ùî ñòàëi N(k, Y ) â (3) � (5), à
òàêîæ ñòàëi C(k, Y ) â (3′) � (5′), íåìîæëèâî çàìiíèòè ñòàëèìè, ÿêi
íå çàëåæàòü âiä Y, à çàëåæàòü, ñêàæiìî, âiä s. Öå ïðèïóùåííÿ íå
ðîçãëÿäà¹òüñÿ â öié ñòàòòi.
 [7] ïðè êîæíîìó n ∈ N äëÿ áóäü-ÿêî¨ f ∈ ∆(1)(Y ) îçíà÷åíî
Tn ∈ Tn ∩∆(1)(Y ) òàêèé, ùî
‖f − Tn‖ ≤ c(1, s) ω1 (f, 1/n) , f ∈ C, (6)
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 113
i â [5, ðîçäië 2] äîâåäåíî îêðåìèé âèïàäîê íåðiâíîñòi (5′): ÿêùî
f ∈ W (r) ∩ ∆(1)(Y ) (äå W (r)− ìíîæèíà ôóíêöié g ç àáñîëþòíî
íåïåðåðâíèìè g(r−1) i ç |g(r)(x)| ≤ 1 ì. ñ. íà R), òî çíàéäåòüñÿ
Tn ∈ Tn ∩∆(1)(Y ) òàêèé, ùî
‖f − Tn‖ ≤
C(r, Y )
nr
n ∈ N, r ≥ 2,
äå C(r, Y )− ñòàëà, ÿêà çàëåæàòü òiëüêè âiä r i Y. Äëÿ r = 1 öå òâåð-
äæåíÿ ¹ îêðåìèì âèïàäêîì íåðiâíîñòi (6).
2. Äîâåäåííÿ íåðiâíîñòi (4). Íåðiâíiñòü (4) äîâåäåì ó íàñòóïíèé
ñïîñiá: ôóíêöiþ f ïðåäñòàâèìî ñóìîþ f = f1 + f2, äå ‖f ′1‖ áóäå "ìà-
ëåíüêîþ" ñêðiçü íà R, à |f ′2(x)| áóäå "âåëèêèì" íà "áiëüøié" ÷àñòèíi
äåÿêî¨ ìíîæèíè F. Òîäi, îçíà÷èìî ïîëiíîì Pn, ÿê ñóìó ï'ÿòè ïîëiíî-
ìiâ, ïåðøèé ç ÿêèõ τn áóäå íàáëèæàòè f1 (ÿê òðåáà) i áóäå êîìîíîòîí-
íèì, à äðóãèé σ̂n áóäå (òåæ ÿê òðåáà) íàáëèæàòè f2 i, ùî âàæëèâî, f
′
2
(ñâî¹þ ïîõiäíîþ, çâiñíî), àëå âií íå áóäå êîìîíîòîííèì. Íàòîìiñòü
éîãî ïîõiäíà (çàâäÿêè ñïiëüíîìó íàáëèæåííþ) áóäå "âåëèêîþ" òàì
äå "âåëèêà" f ′2. Òðè iíøi ïîëiíîìè Un, Q i M áóäóòü ìàòè "ìàëåíüêi"
íîðìè (ðiâíi çà ïîðÿäêîì îöiíöi (4)) i áóäóòü "âèïðàâëÿòè" ïîëiíîì
σ̂′n (êîæåí íà ñâî¨é ìíîæèíi) òàê, ùîá ¨õ ñïiëüíà ç σ̂n ñóìà âæå áóëà
êîìîíîòîííèì ïîëiíîìîì. Öå ìîæëèâî çàâäÿêè iñíóâàííþ äiëÿíîê,
äå |σ̂′n(x)| "âåëèêèé" i òîìó íà öèõ äiëÿíêàõ, âèïðàâëÿþ÷i ïîëiíîìè
ìîæóòü ïîðóøóâàòè ñâîþ îñîáèñòó êîìîíîòîííiñòü (íà âåëè÷èíó íå
áiëüøó íiæ |σ̂′n(x)| ðàçîì). Áåç òàêîãî ïîðóøåííÿ, çðîáèòè ¨õ íîðìè
ìàëåíüêèìè íå ìîæëèâî, ïðèíöèïîâî. Ùå çàóâàæåìî, ùî äîâåäåííÿ
îöiíêè (4) ãðóíòó¹òüñÿ íà ôàêòàõ i íà ñõåìi ñòàòòi [3], äå äîâåäåíî
îöiíêó (5).
1◦. Íåõàé
Jn,l(x) :=
(
sin(nx/2)
sin(x/2)
)2l
, Kn,l(x) := Jn,l(x)
π∫
−π
Jn,l(x)dx
−1
−ïàðíå i íåâiä'¹ìíå ÿäðî òèïó Äæåêñîíà, n ∈ N, l ∈ N, i
σn,l(f, x) := (−1)k+1
π∫
−π
Kn,l(t)
k∑
i=1
(−1)k−i
(
k
i
)
f(x + it)dt (6)
114 Ã.À. Äçþáåíêî
−ïîëiíîì ç Tl(n−1), çàïðîïîíîâàíèé Ñò¹÷êiíèì [8] äëÿ äîâåäåííÿ
íåðiâíîñòi (1) ç f ∈ C i k ∈ N. Ïîçíà÷èìî
h := hn :=
π
n
, xj := xj,n := −j h, Ij := Ij,n := [xj , xj−1], j ∈ Z,
Jj(x) :=
(
J2n,1(x− (xj + π/(4n))) + J2n,1(x− (xj + 3π/(4n)))
)b
−ñòðîãî äîäàòí¹ ÿäðî, ÿê ñóìà äâîõ "ñóñiäíiõ" íåâiä'¹ìíèõ ç b ∈ N, i
+tj(x) :=+ tj,n(x, b, Y ) :=
x∫
xj−π
Jj(u)Π(u)du
xj+π∫
xj−π
Jj(u)Π(u)du
−1
,
−tj(x) :=− tj,n(x, b, Y ) :=
x∫
xj−π
Πj(u)Jj(u)du
xj+π∫
xj−π
Πj(u)Jj(u)du
−1
,
−äâi ôóíêöi¨ âiãëÿäó 1
2π x +±Rj(x) ç ±Rj ∈ Tc1n i Πj(x) := −Π(x, Y ∪
∪{xj , xj−1}). Òóò i äàëi cν , ν = 1, ..., 23, ïîçíà÷àòèìóòü äîäàòíi ñòàëi,
ÿêi ìîæóòü çàëåæèòè òiëüêè âiä k, s i b ∈ N. Ïîçíà÷èìî
Oi := (xj+5, xj−5), ÿêùî yi ∈ [xj , xj−1), O :=
⋃
i∈Z
Oi (7)
i áóäåìî ïèñàòè j ∈ H := H(Y, n), ÿêùî xj ∈ R \ O. Âèáåðåìî
N(Y ) ∈ N òàêå, ùî êîæåí âiäðiçîê [yi, yi−1], i = 1, · · · , 2s, ìiñòèòü
ïðèíàéìíi 10 ðiçíèõ âiäðiçêiâ Ij , äëÿ âñiõ n ≥ N(Y ). Äàëi n ≥ N(Y )
i äëÿ çðó÷íîñòi y2s = −π. Äëÿ x, a ∈ R íåõàé
χ(x, a) :=
{
0, ÿêùî x ≤ a,
1, ÿêùî x > a,
χj(x) := χ(x, xj), (8)
Γ̌n(x) := min
{
1,
1
n |sin(x/2)|
}
, Γj(x) := Γ̌n (x− (xj + h/2)) , j ∈ Z.
Ëåìà 1 [3]. ßêùî j ∈ H i b ≥ s + 4, òî
+t′j(x) Π(x) Π(xj) ≥ 0, x ∈ [xj−1 − 2π, xj + 2π], (9)
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 115
−t′j(x) Π(x) Π(xj) ≤ 0, x ∈ [xj−1 − 2π, xj + 2π] \ Ij , (10)∣∣χj(x)−±tj(x)
∣∣ ≤ c2 (Γj(x))2b−s−1
, x ∈ [xj−1 − 2π, xj + 2π], (11)∣∣±t′j(x)
∣∣ ≤ c3
1
h
(Γj(x))2b−s
, x ∈ R, (12)
∣∣+t′j(x)
∣∣ ≥ c4
1
h
(Γj(x))2b+2s
, x ∈ R \O, (13)
∣∣−t′j(x)
∣∣ ≥ c4
1
h
(Γj(x))2b+2s
∣∣∣∣ x− yi
xj − yi
∣∣∣∣ , x ∈ Oi, i ∈ Z. (14)
Äàëi b = s + 4. Íåõàé
ωk(f ′, t) ≤ ϕ(t) ≤ 2kωk(f ′, t), t ≥ 0,
òîáòî, ϕ − k-ìàæîðàíòà. Ïîçíà÷èìî {zq}n∗
q=0 := {xj : j ∈ H, |j| <
< n} ∪ {yi}2s
i=0, n∗ := 2n + 1− 8(2s + 1) i òî÷êè zq óïîðÿäêîâàíî çà
ñïàäàííÿì. Íåõàé j(q) := j, ÿêùî zq = xj . Ïîêëàäåì j(q) = j(q − 1),
ÿêùî zq = yi.
Ëåìà 2 [3]. ßêùî f ′ ¹ 2π-ïåðiîäè÷íîþ, ‖f ′‖ ≤ ϕ(h) i f ′(x)Π(x) ≥ 0,
x ∈ R, òî ôóíêöiÿ
τn(f, x) := f(−π) +
n∗∑
q=1
(f(zq−1)− f(zq)) +tj(q)(x)
çàäîâîëüíÿ¹ íåðiâíîñòi
‖f − τn(f, ·)‖ ≤ c5hϕ(h), (15)
τ ′n(f, x)Π(x) ≥ 0, x ∈ R. (16)
Êðiì òîãî, ÿêùî äëÿ A = const, f(x) − Ax ¹ ïåðiîäè÷íîþ, òî
τn(f, x)−Ax ∈ Tc1n.
Ëiâèé i ïðàâèé êiíöi ïðîìiæêó Oi ïîçíà÷èìî ÷åðåç
(y
i
, yi) =: (xj(i), xj(i)).
116 Ã.À. Äçþáåíêî
Ëåìà 3 [3]. Ôóíêöiÿ
Un(x) := h ϕ(h)
2s∑
i=1
(
+tj(i)(x) sgnΠ(y
i
) ++tj(i)(x) sgnΠ(yi)
)
¹ ïîëiíîìîì ç Tc1n ∩∆(1)(Y ) òàêèì, ùî
‖Un‖ ≤ c6hϕ(h), (17)
|U ′
n(x)| ≥ c7ϕ(h)
(
Γ̌n (dist(x,O))
)4(s+2)
, x ∈ R \O, (18)
|U ′
n(x)| ≥ c7
h
ϕ(h) |x− yi| , x ∈ Oi, i ∈ Z. (19)
Íåõàé Lk(g, x, [a, b]) ïîçíà÷๠ìíîãî÷ëåí Ëàãðàíæà ñòåïåíÿ ≤ k,
ÿêèé íà [a, b] iíòåðïîëþ¹ ôóíêöiþ g = g(x) ó ðiâíîâiääàëåíèõ òî÷êàõ
a + ν(b− a)/k, ν = 0, ..., k. Äëÿ i ∈ Z ïîêëàäåìî
Ji := [yi − h, yi + h], Yi :=
(
Y \ {yi + 2πν}ν∈Z
)
∪ {y
i
+ 2πν}ν∈Z,
t̂i(x) := +tj(i)(x, b, Yi)−−tj(i)(x, b, Yi).
Ëåìà 4 [3]. ßêùî f ∈ C(1) i äëÿ âñiõ i ∈ Z f ′(yi) = A = const, òî
ïîëiíîì
σ̂n(f, x) := σn,l(f, x)−
2s∑
i=1
σ′n,l(f, yi)−A
t̂′i(yi)
t̂i(x),
l =
[
k+2
2
]
+2(s+2)+1, ïîðÿäêó c1n ïðè áóäü-ÿêèõ δ > 0 çàäîâîëüíÿ¹
íåðiâíîñòi
‖f − σ̂n(f, ·)‖ ≤ c8hϕ(h), (20)
|f ′(x)− σ̂′n(f, x)| ≤
≤ c9
(
ωk(f ′, h, [x− δ, x + δ]) +
(
1
nδ
)4(s+2)+1
ϕ(h)
)
, x ∈ R, (21)
‖f ′ − σ̂′n(f, ·)‖ ≤ c9ϕ(h), (22)
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 117
|Lk−1(f ′, x, Ji)− Lk−1(f ′, yi, Ji)− σ̂′n(f, x) + A| ≤
≤ c10
h
ϕ(h)|x− yi|, x ∈ Ji, (23)
çîêðåìà, σ̂′n(f, yi) = A, i ∈ Z.
2◦. Äëÿ j ∈ Z áóäåìî ïèñàòè j ∈ V, ÿêùî íà Ij iñíó¹ òî÷êà x òàêà,
ùî
|f ′(x)| ≤ 2c9ϕ(h). (24)
Ïîçíà÷èìî c11 := 96k[c3/c4 + 1] i c := c11 + 20s + 15. Áåç âòðà-
òè çàãàëüíîñòi áóäåìî ââàæàòè, ùî n äiëåòüñÿ íà c, òîáòî n = pc
ç p ∈ N. Ïîêëàäåìî νp = n + 8 i ν−p = 8 − n. Äëÿ êîæíîãî
q = p− 1, ..., 0, ..., 1− p íåõàé νq ïîçíà÷๠íàéìåíøå öiëå ñåðåä öiëèõ
j ≥ cq äëÿ ÿêèõ [xj+3, xj−3] ∩O = ∅. Ïîçíà÷èìî
Eq := [xνq
, xνq−1 ]
(
= Iνq
∪ Iνq−1 ∪ ... ∪ Iνq−1+1
)
, q = 1− p, p.
Îòæå, êiíöi êîæíîãî âiäðiçêó Eq âiäñòîÿòü âiä O ïðèíàéìíi íà òðè
ðiçíèõ Ij i êîæåí Eq ñêëàäà¹òüñÿ ïðèíàéìíi ç c11 + 20s i íå áiëüøå
íiæ ç c11 + 20s + 30 ðiçíèõ Ij (cq + 15 ≥ νq ≥ cq).
Äàëi áóäåìî ââàæàòè, ùî q ∈ Z (îñêiëüêè f ¹ ïåðiîäè÷íîþ). Áó-
äåìî ïèñàòè q ∈ W, ÿêùî Eq ìiñòèòü ïðèíàéìíi 2k − 1 ïðîìiæêiâ Ij
òàêèõ, ùî j ∈ V. Çàóâàæèìî, ùî ÿêùî q ∈ W, òî ç (24) i íåðiâíîñòi
Óiòíi âèïëèâàå îöiíêà
|f ′(x)| ≤ c12ϕ(h), x ∈ Eq. (25)
Äëÿ äîâiëüíî¨ íåïîðîæíüî¨ ìíîæèíè E ⊂ R, ÷åðåç E∗ ïîçíà÷è-
ìî îá'¹äíåííÿ âñiõ Ij , j ∈ Z, òàêèõ, ùî Ij ∩ E 6= ∅. Àíàëîãi÷íî,
E∗∗ := (E∗)∗ i ò.ä. (E ⊂ E∗ ⊂ E∗∗ ⊂ ...). Òåïåð íåõàé
E := ∪q/∈W Eq
i äëÿ x ∈ Ij , j ∈ Z, ïîêëàäåìî
g1(x) :=
0, ÿêùî Ij ⊂ E∗,
f ′(x), ÿêùî Ij ⊂ R \ E∗∗,
f ′(x)Sj(x), ÿêùî Ij ⊂ E∗∗ \ E∗ i xj ∈ E∗,
f ′(x)(1− Sj(x)), ÿêùî Ij ⊂ E∗∗ \ E∗ i xj /∈ E∗,
118 Ã.À. Äçþáåíêî
äå Sj(x) :=
x∫
xj
(u−xj)k(xj−1−u)kdu
(
xj−1∫
xj
(u− xj)k(xj−1 − u)kdu
)−1
.
Íåõàé
g2(x) := f ′(x)− g1(x), x ∈ R.
Ëåìà 5 [3]. Ìàþòü ìiñöå íåðiâíîñòi
‖g1‖ ≤ c12ϕ(h), ωk(g1, t) ≤ c13ϕ(t), ωk(g2, t) ≤ c14ϕ(t).
Ïîçíà÷èìî c15 := [c14 + 1]. Áåç âòðàòè çàãàëüíîñòi áóäåìî ââà-
æàòè, ùî p ≥ 4c15. Ïîäàìî ìíîæèíó E 6= R ó âèãëÿäi îá'¹äíåííÿ
âiäðiçêiâ Fm := [am, bm], m ∈ Z, ùî íå ïåðåòèíàþòüñÿ. Áóäåìî ïèñà-
òè m ∈ X, ÿêùî Fm ñêëàäà¹òüñÿ íå áiëüøå íiæ ç c15 ðiçíèõ âiäðiçêiâ
Eq (àáî, ùî òå ñàìå, íå áiëüøå íiæ ç c15c + 15 ðiçíèõ âiäðiçêiâ Ij).
ßêùî m /∈ X, òî Fm ìiñòèòü ïðèíàéìíi c15c + c11 ðiçíèõ Ij . Ïîêëà-
äåìî
g3(x) :=
{
g2(x), x ∈ (∪m∈XFm)∗∗ ,
0, x ∈ R \ (∪m∈XFm)∗∗,
g4(x) := g2(x)− g3(x), x ∈ R.
Ëåìà 6 [3]. Ìàþòü ìiñöå íåðiâíîñòi
‖g3‖ ≤ c16ϕ(h), ωk(g3, t) ≤ c17ϕ(t), ωk(g4, t) ≤ (c14 + c17)ϕ(t).
Ïîçíà÷èìî
f1(x) := f(0) +
x∫
0
(g1(u) + g3(u)−A) du, f2(x) :=
x∫
0
(g4(u) + A) du,
òàê ùî f(x) = f1(x) + f2(x) i äå äiéñíå ÷èñëî A,
|A| ≤ ϕ(h) max{c12, c16}/2, âèáðàíî ç óìîâè f1(0) = f1(2π) (àáî,
ùî òå ñàìå, f2(0) = f2(2π)). ßêùî g4(x) ≡ 0, òî f1(x) ≡ f(x) (A = 0)
i òîäi íåðiâíiñòü (4) ¹ íàñëiäêîì Ëåì 5, 6 i 2.
3◦. Çàäà÷à çâåëàñÿ äî íàáëèæåííÿ ôóíêöi¨ f2(x). Íåõàé äëÿ
âèçíà÷åíîñòi A ≥ 0. Ïîçíà÷èìî
F := ∪m/∈XFm.
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 119
Íàãàäà¹ìî, ùî çà ïîáóäîâîþ
f ′2(x) =
{
f ′(x) + A, x ∈ F ∗,
A, x /∈ F ∗∗,
i íà áiëüøié ÷àñòèíi ìíîæèíè F ìà¹ìî |f ′2(x) − A| > 2c9ϕ(h).
Òîìó, çãiäíî (22), |σ̂′n1
(f2, x) − A| > c9ϕ(hn) ïðè n1 > n. Îä-
íàê ìîæóòü iñíóâàòè i "ïîãàíi" òî÷êè x (íà F çîêðåìà) â ÿêèõ
(σ̂′n1
(f2, x)−A)Π(x) < 0.  óñiõ "ïîãàíèõ" òî÷êàõ x ∈ F \ O,
x ∈ (R \ F ) \O i x ∈ O ìè "âèïðàâèìî" ïîëiíîì σ̂′n1
(f2, x) çà äî-
ïîìîãîþ ïîëiíîìiâ Q′(x) (Ëåìà 7), M ′(x) (Ëåìà 8) i U ′
n(x) (Ëåìà 3),
âiäïîâiäíî.
Íåõàé δj := sgnΠ(xj). Äëÿ êîæíîãî Eq ⊂ F, q = 1− p, p, òàêîãî,
ùî Eq ∩ O 6= ∅, ÷åðåç ν+
q i ν−q ïîçíà÷èìî íàéáiëüøå j ∈ H ç Ij ⊂ Eq
äëÿ ÿêîãî δj > 0 i δj < 0, âiäïîâiäíî.
Îçíà÷åííÿ 1. Äëÿ êîæíîãî q = 1− p, p, ïîêëàäåìî Qq(x) :≡ 0,
ÿêùî Eq 6⊂ F, àáî Eq ⊂ F i Eq íå ìiñòèòü âiäðiçêiâ Ij ç j ∈ V ∩H. Äëÿ
ðåøòè Eq, òîáòî äëÿ Eq ⊂ F i Eq ìiñòèòü Ij ç j ∈ V ∩H, ïîêëàäåìî
Qq(x) :=
2c9
c4
νq∑
j=νq−1+1, j∈V
+tj(x)δj − αq
c9
2c3
νq∑
j=νq−1+1, j /∈V
−tj(x)δj , Eq ∩O = ∅,
2c9
c4
νq∑
j=νq−1+1, j∈V ∩H
+tj(x)δj + βq
+tν+
q
(x)− γq
+tν−q (x)
, Eq ∩O 6= ∅,
äå ÷èñëà αq > 0, βq ≥ 0 i γq ≥ 0 âèáðàíî òàê, ùî Qq(−π) = Qq(π) i
βqγq = 0.
Ïîçíà÷èìî
F1 :=
⋃
j: Ij⊂F,j∈V ∩H
Ij , F2 :=
⋃
j: Ij⊂F,j /∈V,j∈H
Ij ,
òàê, ùî F \ (F1 ∪ F2) ⊂ O.
Ëåìà 7 [3]. Ôóíêöiÿ
Q(x) := h ϕ(h)
p∑
q=1−p
Qq(x)
120 Ã.À. Äçþáåíêî
¹ ïîëiíîìîì ïîðÿäêó c1n òàêèì, ùî
‖Q‖ ≤ c18hϕ(h), (28)
|Q′(x)| ≥ 2c9ϕ(h), x ∈ F1, (27)
Q′(x) sgnΠ(x) ≥ −c9
2
ϕ(h), x ∈ F2, (28)
Q′(x)Π(x) ≥ 0, x ∈ R \ F2. (29)
Íàãàäà¹ìî, ùî F = ∪m/∈XFm, äå Fm = [am, bm] íå ïåðåòèíàþòüñÿ
i êîæåí Fm ìiñòèòü ïðèíàéìíi c15c+c11 =: c19 ðiçíèõ Ij (c15+1 ðiçíèõ
Eq). Áóäåìî ïèñàòè m ∈ X0, ÿêùî m /∈ X i Fm ∩ [a0, a0 + 2π] = Fm.
Äëÿ êîæíîãî m ∈ X0 ïîçíà÷èìî
[xja,m , xjb,m
] := [am, bm] = Fm,
Fa,m := [xja,m , xja,m−c19 ], Fb,m := [xjb,m+c19 , xjb,m
],
(òîáòî äâi ïà÷êè ðiçíèõ Ij-õ, ïî c18 øòóê, íà êiíöÿõ êîæíîãî Fm,
ÿêèé íå âèõîäèòü çà ìåæi âiäðiçêó äîâæèíîþ 2π âiä ïî÷àòêó F0).
Äëÿ êîæíîãî m ∈ X0 òàêîãî, ùî Fa,m ∩ O 6= ∅ (Fb,m ∩ O 6= ∅),
÷åðåç ν+
a,m i ν−a,m (ν+
b,m i ν−b,m) ïîçíà÷èìî äâà íàéáiëüøi öiëi j ∈ H ç
Ij ⊂ Fa,m (Ij ⊂ Fb,m) äëÿ ÿêèõ δj > 0 i δj < 0, âiäïîâiäíî.
Îçíà÷åííÿ 2. Äëÿ êîæíîãî m ∈ X0 ïîêëàäåìî
Ma,m(x) :=
2c9c14
c4
ja,m+3∑
j=ja,m+1
+tj(x)δj − µa,m
c9
2c3
ja,m∑
j=ja,m−c19+1,j /∈V
−tj(x)δj ,
Fa,m ∩O = ∅,
2c9c14
c4
ja,m+3∑
j=ja,m+1
+tj(x)δj + µ+
a,m3+tν+
a,m
(x)− µ−a,m3+tν−a,m
(x)
,
Fa,m ∩O 6= ∅,
Mb,m(x) :=
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 121
2c9c14
c4
jb,m∑
j=jb,m−2
+tj(x)δj − µb,m
c9
2c3
jb,m+c19∑
j=jb,m+1,j /∈V
−tj(x)δj ,
Fb,m ∩O = ∅,
2c9c14
c4
jb,m∑
j=jb,m−2
+tj(x)δj + µ+
b,m3+tν+
b,m
(x)− µ−b,m3+tν−b,m
(x)
,
Fb,m ∩O 6= ∅,
äå ÷èñëà µa,m > 0, µ±a,m ≥ 0, µb,m > 0 i µ±b,m ≥ 0 âèáðàíî òàê,
ùî Ma,m(−π) = Ma,m(π), µ+
a,mµ−a,m = 0, Mb,m(−π) = Mb,m(π) i
µ+
b,mµ−b,m = 0.
Ïîçíà÷èìî
F3 := F ∗∗∗ \ F .
Ëåìà 8 [3]. Ôóíêöiÿ
M(x) := h ϕ(h)
∑
m∈X0
(Ma,m(x) + Mb,m(x))
¹ ïîëiíîìîì ïîðÿäêó c1n òàêèì, ùî
‖M‖ ≤ c20hϕ(h), (30)
|M ′(x)| ≥ 2c9c14ϕ(h)
(
Γ̌n(dist(x,F3))
)4(s+2)
, x ∈ R \ (F ∪O), (31)
M ′(x) sgnΠ(x) ≥ −c9
4
ϕ(h), x ∈ F2, (32)
M ′(x)Π(x) ≥ 0, x ∈ R \ F2. (33)
4◦. Ïîçíà÷èìî
c21 := 4π(c14 + c17), n1 := c21n, h1 :=
π
n1
, (34)
c22 :=
max{c9, c10}(c14 + c17)c21
c7
,
Pn1(x) := τn(f1 + Ax, x)−Ax + σ̂n1(f2, x) + Q(x) + M(x) + c22Un(x).
122 Ã.À. Äçþáåíêî
Ïîêàæåìî, ùî Pn1 −øóêàíèé ïîëiíîì. Âðàõó¹ìî Ëåìè 5 i 6 i çáåðåìî
(15), (20), (26), (30) i (17) â îöiíêó
‖f−Pn1‖ = ‖f1+f2−Pn1‖ ≤ ‖f1−τn(f1+Ax, ·)+A·‖+‖f2−σ̂n1(f2, ·)‖+
+‖Q‖+ ‖M‖+ c22‖Un‖ ≤
≤ c5h max{c12, c16}(c13 + c17)ϕ(h) + c8h1(c14 + c17)ϕ(h1)+
+ (c18 + c20 + c22c6) hϕ(h) ≤ c23hϕ(h). (35)
Ïåðåâiðåìî íåðiâíiñòü
P ′
n1
(x)Π(x) ≥ 0, x ∈ R, (36)
âèêîðèñòàâøè ðiâíiñòü
P ′
n1
(x) sgnΠ(x) = τ ′n(f1 + Ax, x) sgnΠ(x) + (f ′2(x)−A) sgnΠ(x)+
+
(
σ̂′n1
(f2, x)− f ′2(x)
)
sgnΠ(x)+
+ (Q′(x) + M ′(x)) sgnΠ(x) + c22U
′
n(x) sgnΠ(x) =:
5∑
ν=1
Ψν(x).
Ç (16), ïîáóäîâè f2 i Un âèäíî, ùî
Ψ1(x) ≥ 0, Ψ2(x) ≥ 0, Ψ5(x) ≥ 0, x ∈ R.
Íåõàé F4 := R \ (F ∪ F3 ∪O), òàê ùî F1 ∪ F2 ∪ F3 ∪ F4 ∪ O = R.
Ðîçãëÿíåìî ï'ÿòü âèïàäêiâ.
1) x ∈ F1. Äëÿ u ∈ F∗
1 ôóíêöiÿ f ′2(u) = f ′(u)+A. Áåðó÷è äî óâàãè
(21) ç δ = h, (29), (27), (33), Ëåìó 6 i (34), çàïèñó¹ìî
Ψ3(x) + Ψ4(x) ≥ −c9ωk(f ′ + A, h1)− c9ωk(f ′2, h1)
(
1
n1h
)4(s+2)+1
+
+2c9ϕ(h) + 0 ≥ ϕ(h)c9
(
1− π(c14 + c17)
n
n1
)
≥ 0.
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 123
2) x ∈ F2. Äëÿ u ∈ F∗
2 ôóíêöiÿ f ′2(u) = f ′(u) + A. Áiëüø òîãî,
|f ′2(x) − A| ≥ 2c9ϕ(h), x ∈ F2. Òåïåð iç íåðiâíîñòi (21) ç δ = h, (28),
(32), Ëåìè 6 i (34) îòðèìó¹ìî íåðiâíiñòü
Ψ2(x) + Ψ3(x) + Ψ4(x) ≥ 2c9ϕ(h)− c9ωk(f ′ + A, h1)− c9ωk(f ′2, h1)×
×
(
1
n1h
)4(s+2)+1
− c9
2
ϕ(h)− c9
4
ϕ(h)≥ϕ(h)c9
(
1
4
− π(c14 + c17)
n
n1
)
≥0.
3) x ∈ F3. Äëÿ u ∈ F∗
3 ôóíêöiÿ f ′2(u) = g2(u) + A i íåðiâíiñòü (21)
ç δ = h, (29), (33), (31), Ëåìè 5, 6 i (34) ïîðîäæóþòü íåðiâíiñòü
Ψ3(x)+Ψ4(x) ≥ −c9ωk(g2+A, h1)−c9(c14+c17)ϕ(h1)
(
1
n1h
)4(s+2)+1
+
+0 + 2c9c14ϕ(h) ≥ ϕ(h)c9
(
c14 − π(c14 + c17)
n
n1
)
≥ 0.
4) x ∈ F4. Äëÿ u ∈ F∗
4 ôóíêöiÿ f ′2(u) = A. Òîìó ωk(f ′2, t,F4) ≡ 0.
Âèêîðèñòîâóþ÷è (21) ç δ = dist(x, F ∗∗), Ëåìó 6, (29), (33), (31), (34)
i íåðiâíiñòü
1
n1 dist(x, F ∗∗)
< Γ̌n(dist(x,F3)),
çàïèñó¹ìî
Ψ3(x)+Ψ4(x) ≥ −c9 ·0−c9(c14+c17)ϕ(h1)
(
1
n1 dist(x, F ∗∗)
)4(s+2)+1
+
+0 + 2c9c14ϕ(h)
(
Γ̌n(dist(x,F3))
)4(s+2)
≥
≥ ϕ(h)c9
(
Γ̌n(dist(x,F3))
)4(s+2)
(
2c14 − π(c14 + c17)
n
n1
)
≥ 0.
5) x ∈ O. Çãiäíî (29) i (33) ìà¹ìî Ψ4(x) ≥ 0. Äëÿ x ∈ Oi òàêî¨, ùî
Oi ∩ F = ∅, ôóíêöiÿ f ′2(x) = A, òîìó êîìîíîòîííiñòü Un, (19), (34),
Ëåìà 6 i, âiäïîâiäíî, (23) i (22) ïîðîäæóþòü íåðiâíîñòi
Ψ5(x) + Ψ3(x) ≥ c22c7
h
ϕ(h)|x− yi|−
124 Ã.À. Äçþáåíêî
−c10
h1
(c14 + c17)ϕ(h1)|x− yi| ≥ 0, x ∈ Ji,n1 , (37)
Ψ5(x) + Ψ3(x) ≥ c22c7
h
ϕ(h)|x− yi|−
−c9(c14 + c17)ϕ(h1) ≥ 0, x ∈ Oi \ Ji,n1 . (38)
Äëÿ ðåøòè Oi ⊂ O, òîáòî äëÿ x ∈ Oi : Oi ∩ F 6= ∅, ôóíêöiÿ
f ′2(x) = f ′(x) + A.  öüîìó âèïàäêó íåðiâíiñòü (38) çàëèøà¹òüñÿ âið-
íîþ, à íåðiâíiñòü
Ψ5(x) + Ψ3(x) + Ψ2(x) = Ψ5(x)+
+
(
σ̂′n1
(f2, x)− Lk−1(f ′2, x, Ji,n1) + Lk−1(f ′2, yi, Ji,n1)+
+Lk−1(f ′2, x, Ji,n1)− Lk−1(f ′2, yi, Ji,n1)− f ′2(x)
)
sgnΠ(x) + Ψ2(x) =
= Ψ5(x) +
(
−A + σ̂′n1
(f2, x)− Lk−1(f ′2, x, Ji,n1) + Lk−1(f ′2, yi, Ji,n1)
)
×
× sgnΠ(x) +
(
Lk−1(f ′, x, Ji,n1)− Lk−1(f ′, yi, Ji,n1)
)
sgnΠ(x) =:
=: Ψ5(x) + Ψ3,1(x) + Ψ3,2(x, k) ≥ 0, x ∈ Ji,n1 ,
ñïðàâäæó¹òüñÿ àíàëîãi÷íî (37), ÿêùî
Ψ3,2(x, k) ≥ 0, x ∈ Ji,n1 . (39)
Îñêiëüêè, ïðè f ′(x)Π(x) ≥ 0,
Ψ3,2(x, 3) =
(
L2(f ′, x, Ji,n1)− L2(f ′, yi, Ji,n1)
)
sgnΠ(x) =
= L2(f ′, x, Ji,n1) sgnΠ(x) ≥ 0, x ∈ Ji,n1 ,
òî (39) ¹ âiðíîþ çàâæäè òiëüêè ïðè k = 3 (ïðî áiëüøi k òàêîãî,
âçàãàëi êàæó÷è, ñêàçàòè íå ìîæíà).
Íåðiâíiñòü (36), òîáòî êîìîíîòîííiñòü Pn1 , äîâåäåíî. À (4) ¹ íà-
ñëiäêîì (35).
1. Äçÿäûê Â. Ê. Ââåäåíèå â òåîðèþ ðàâíîìåðíîãî ïðèáëèæåíèÿ ôóíêöèé
ïîëèíîìàìè. � Ì.: Íàóêà, 1977. � 512 ñ.
Ïîðÿäêè êîìîíîòîííîãî íàáëèæåííÿ ïåðiîäè÷íèõ ôóíêöié . . . 125
2. Äçþáåíêî Ã. À., Ïëåøàêîâ Ì. Ã. Êîìîíîòîííîå ïðèáëèæåíèå ïåðèîäè÷å-
ñêèõ ôóíêöèé // Ìàò. çàìåòêè. � 2008. � 83, � 2. � Ñ. 199 � 209.
3. Äçþáåíêî Ã. À. Êîìîíîòîííå íàáëèæåííÿ äâi÷i äèôåðåíöiéîâíèõ ïåðiî-
äè÷íèõ ôóíêöié // Óêð. ìàòåì. æóðí. � 2009. � 61, � 4. � Ñ. 435 � 451.
4. Whitney H. On Functions with Bouded n-th Di�erences // J. Math. Pures
Appl. � 1957. � 6, � 9. � P. 67 � 95.
5. Ïëåøàêîâ Ì. Ã. Êîìîíîòîííîå ïðèáëèæåíèå ïåðèîäè÷åñêèõ ôóíêöèé
êëàññîâ Ñîáîëåâà: Äèññ. ... êàíä. ôèç.-ìàò. íàóê. � Ñàðàòîâ, ÑÃÓ, 1997.
6. Äçþáåíêî Ã. À. Êîíòðïðèêëàä â êîìîíîòîííîìó íàáëèæåííi ïåðiîäè÷íèõ
ôóíêöié // Çáiðíèê ïðàöü Ií-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè. � 2008. � 5,
� 1. � Ñ. 113 � 123.
7. Pleshakov M. G. Comonotone Jackson's Inequality // J. Approx. Theory. �
1999. � 99. � P. 409 � 421.
8. Ñòå÷êèí Ñ. Á. Î ïîðÿäêå íàèëó÷øèõ ïðèáëèæåíèé íåïðåðûâíèõ ôóíê-
öèé // Èçâ. ÀÍ ÑÑÑÐ. Ñåð. ìàòåì. � 1951. � 15, � 3. � Ñ. 219 � 242.
|
| id | oai:trim.imath.kiev.ua:article-142 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:02:57Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/c5/f0f3269163089d335990e01a527b53c5.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1422018-01-29T14:44:27Z Degree of comonotone approximation of periodic functions Порядки комонотонного наближення періодичних функцій Dzyubenko, G. A. Дзюбенко, Г. А. If a continuously differentiable on a real axe $\Bbb R\ \ 2\pi - $periodic function $f$ changes its monotonicity at different fixed points $y_i\in [-\pi,\pi),\ i=1,...,2s,\ s\in\Bbb N ,$ (i.e., on $\Bbb R$ there is a set $Y:=\{y_i\}_{i\in\Bbb Z}$ of points $y_i=y_{i+2s}+2\pi $ such that on $[y_i,y_{i-1}]\ f$ is nondecreasing if $i$ is odd, and nonincreasing if $i$ is even), then for each natural number $n,\ n\ge N(Y)=const,$ in the article a trigonometric polynomial $T_n $ of order $\le n,$ which changes its monotonicity at the same points $y_i\in Y,$ like $f,$ is found such that $$\left\Vert f-T_n \right \Vert \le \frac{c(s)}{n}\, \omega_3\left(f',1/n\right),$$ where $N(Y)$ depends only on $Y,$ $c(s)-$ constant which is depending only on $s,\ \omega_3 \left(f,\cdot\right)-$ modulus of smoothness of order $3$ of the function $f$ and $\Vert \cdot \Vert -\max $-norm. Also the other estimates that are possible in this kind of approximation are listed. If a continuously differentiable on a real axe $\Bbb R\ \ 2\pi - $periodic function $f$ changes its monotonicity at different fixed points $y_i\in [-\pi,\pi),\ i=1,...,2s,\ s\in\Bbb N ,$ (i.e., on $\Bbb R$ there is a set $Y:=\{y_i\}_{i\in\Bbb Z}$ of points $y_i=y_{i+2s}+2\pi $ such that on $[y_i,y_{i-1}]\ f$ is nondecreasing if $i$ is odd, and nonincreasing if $i$ is even), then for each natural number $n,\ n\ge N(Y)=const,$ in the article a trigonometric polynomial $T_n $ of order $\le n,$ which changes its monotonicity at the same points $y_i\in Y,$ like $f,$ is found such that $$\left\Vert f-T_n \right \Vert \le \frac{c(s)}{n}\, \omega_3\left(f',1/n\right),$$ where $N(Y)$ depends only on $Y,$ $c(s)-$ constant which is depending only on $s,\ \omega_3 \left(f,\cdot\right)-$ modulus of smoothness of order $3$ of the function $f$ and $\Vert \cdot \Vert -\max $-norm. Also the other estimates that are possible in this kind of approximation are listed. Інститут математики НАН України 2013-07-15 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/142 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 1 (2013): Approximation Theory of Functions and Related Problems; 110-125 Сборник Трудов Института математики НАН Украины; Том 10 № 1 (2013): Tеорiя наближення функцiй та сумiжнi питання; 110-125 Збірник Праць Інституту математики НАН України; Том 10 № 1 (2013): Tеорiя наближення функцiй та сумiжнi питання; 110-125 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/142/117 Авторське право (c) 2013 Інститут математики НАН України |
| spellingShingle | Dzyubenko, G. A. Дзюбенко, Г. А. Degree of comonotone approximation of periodic functions |
| title | Degree of comonotone approximation of periodic functions |
| title_alt | Порядки комонотонного наближення періодичних функцій |
| title_full | Degree of comonotone approximation of periodic functions |
| title_fullStr | Degree of comonotone approximation of periodic functions |
| title_full_unstemmed | Degree of comonotone approximation of periodic functions |
| title_short | Degree of comonotone approximation of periodic functions |
| title_sort | degree of comonotone approximation of periodic functions |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/142 |
| work_keys_str_mv | AT dzyubenkoga degreeofcomonotoneapproximationofperiodicfunctions AT dzûbenkoga degreeofcomonotoneapproximationofperiodicfunctions AT dzyubenkoga porâdkikomonotonnogonabližennâperíodičnihfunkcíj AT dzûbenkoga porâdkikomonotonnogonabližennâperíodičnihfunkcíj |