Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions
We obtain the estimates of norm of deviations of the de Vall\'{e}e Poussin sums and interpolation analogues of sums of Vall\'{e}e Poussin from the functions that belong to the space $C_{\bar{\beta}}^\psi L_s, \ 1\leq s\leq\infty$ and are represented by the...
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Інститут математики НАН України
2013
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| author | Voĭtovych, V. A. Musienko, A. P. Войтович, В. А. Мусієнко, А. П. |
| author_facet | Voĭtovych, V. A. Musienko, A. P. Войтович, В. А. Мусієнко, А. П. |
| author_institution_txt_mv | [
{
"author": "В. А. Войтович",
"institution": "Інститут математики НАН України"
},
{
"author": "А. П. Мусієнко",
"institution": "Інститут математики НАН України"
}
] |
| author_sort | Voĭtovych, V. A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-29T14:44:27Z |
| description | We obtain the estimates of norm of deviations of the de Vall\'{e}e Poussin sums and interpolation analogues of sums of Vall\'{e}e Poussin from the functions that belong to the space $C_{\bar{\beta}}^\psi L_s, \ 1\leq s\leq\infty$ and are represented by the best approximations of $(\psi,\bar{\beta})$-differentiable functions of this sort by trigonometric polynomials in the metric $L_s$ |
| first_indexed | 2026-08-04T01:02:51Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Ií-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2013, Ò.10, �1, 39�58
ÓÄÊ 517.5
Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî (Ií�ò ìàòåìàòèêè ÍÀÍ Óêðà¨íè,
Êè¨â)
ÍÅÐIÂÍÎÑÒI ÒÈÏÓ ËÅÁÅÃÀ ÄËß ÑÓÌ ÂÀËËÅ ÏÓÑ-
ÑÅÍÀ ÒÀ �Õ IÍÒÅÐÏÎËßÖIÉÍÈÕ ÀÍÀËÎÃIÂ ÍÀ ÊËÀ-
ÑÀÕ (ψ, β̄)-ÄÈÔÅÐÅÍÖIÉÎÂÍÈÕ ÔÓÍÊÖIÉ
We obtain the estimates of norm of deviations of the de Vall�ee Poussin sums
and interpolation analogues of sums of Vall�ee Poussin from the functions
that belong to the space Cψ
β̄
Ls, 1 ≤ s ≤ ∞ and are represented by the best
approximations of (ψ, β̄)-di�erentiable functions of this sort by trigonometric
polynomials in the metric Ls.
Îäåðæàíî îöiíêè íîðì âiäõèëåíü ñóì Âàëëå Ïóññåíà òà ¨õ iíòåðïîëÿöié-
íèõ àíàëîãiâ âiä ôóíêöié ç ìíîæèí Cψ
β̄
Ls, 1 ≤ s ≤ ∞, ÿêi âèðàæàþòüñÿ
÷åðåç íàéêðàùi íàáëèæåííÿ (ψ, β̄)-ïîõiäíèõ òàêèõ ôóíêöié òðèãîíîìåò-
ðè÷íèìè ïîëiíîìàìè â ìåòðèöi ïðîñòîðó Ls.
Ðîáîòà ¹ ïðîäîâæåííÿì äîñëiäæåíü [1 � 9] ïî âèâ÷åííþ àïðîê-
ñèìàòèâíèõ âëàñòèâîñòåé ñóì Âàëëå Ïóññåíà àáî ¨õ iíòåðïîëÿöiéíèõ
àíàëîãiâ [10 � 13] íà êëàñàõ (ψ, β̄)-äèôåðåíöiéîâíèõ ôóíêöié.
Íåõàé Ls, 1 ≤ s <∞, � ïðîñòið 2π-ïåðiîäè÷íèõ ñóìîâíèõ â s�ìó
ñòåïåíi íà (0, 2π) ôóíêöié f(t) ç íîðìîþ ‖f‖s =
( π∫
−π
|f(t)|s dt
)1/s
.
L∞ � ïðîñòið âèìiðíèõ i iñòîòíî îáìåæåíèõ 2π-ïåðiîäè÷íèõ ôóíê-
öié f(t) ç íîðìîþ ‖f‖∞ = ess sup
t
|f(t)|. C � ïðîñòið íåïåðåðâíèõ
2π-ïåðiîäè÷íèõ ôóíêöié f(t), â ÿêîìó íîðìà çàäà¹òüñÿ ðiâíiñòþ
‖f‖C = max
t
|f(t)|.
Íåõàé f � 2π-ïåðiîäè÷íà, ñóìîâíà ôóíêöiÿ (f ∈ L1) i
a0
2
+
∞∑
k=1
(ak cos kx+ bk sin kx)
� ¨¨ ðÿä Ôóð'¹. Íåõàé, äàëi ψ = ψ(k) i β̄ = βk, k ∈ N, � äîâiëüíi
c© Â.À. Âîéòîâè÷, À.Ï.Ìóñi¹íêî, 2013
40 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
ïîñëiäîâíîñòi äiéñíèõ ÷èñåë. ßêùî ðÿä
∞∑
k=1
1
ψ(k)
(
ak cos
(
kx+
βkπ
2
)
+ bk sin
(
kx+
βkπ
2
))
¹ ðÿäîì Ôóð'¹ äåÿêî¨ ñóìîâíî¨ ôóíêöi¨ ϕ, òî öþ ôóíêöiþ íàçèâàþòü
(ψ, β̄)-ïîõiäíîþ ôóíêöi¨ f i ïîçíà÷àþòü ÷åðåç fψ
β̄
[14, c. 33]. Ìíî-
æèíó âñiõ ôóíêöié f , ÿêi ìàþòü (ψ, β̄)�ïîõiäíó, ïîçíà÷àþòü ÷åðåç
Lψ
β̄
. ßêùî f ∈ Lψ
β̄
i â òîé æå ÷àñ fψ
β̄
∈ N, äå N � äåÿêà ïiäìíî-
æèíà ç L0
1 = {ϕ ∈ L1 :
π∫
−π
ϕ(t)dt = 0}, òî çàïèñóþòü f ∈ Lψ
β̄
N. ßê-
ùî Fψ
β̄
= f , òî ôóíêöiþ F íàçèâàþòü (ψ, β̄)-iíòåãðàëîì ôóíêöi¨ f ,
ïðè öüîìó çàïèñóþòü F (x) = J ψ
β̄
(f ;x). Ïîêëàäåìî Cψ
β̄
= Lψ
β̄
⋂
C,
Cψ
β̄
N = Lψ
β̄
N
⋂
C. Îçíà÷åííÿ (ψ, β̄)�ïîõiäíèõ òà (ψ, β̄)�iíòåãðàëiâ òà
êëàñiâ Lψ
β̄
N òà Cψ
β̄
N íàëåæàòü Î.I. Ñòåïàíöþ [15, c. 112].
Íàäàëi áóäåìî ââàæàòè, ùî ïîñëiäîâíiñòü ψ(k), ÿêà ïîðîäæó¹
êëàñè Lψ
β̄
N i Cψ
β̄
N, çàäîâîëüíÿ¹ óìîâó D0 (ψ ∈ D0), òîáòî ψ(k) äî-
äàòíà i òàêà, ùî
lim
k→∞
ψ(k + 1)
ψ(k)
= 0. (1)
Ó âèïàäêó, êîëè ψ ∈ D0, ìíîæèíè Cψ
β̄
N ñêëàäàþòüñÿ ç ôóíêöié,
ðåãóëÿðíèõ â óñié êîìïëåêñíié ïëîùèíi, òîáòî ç öiëèõ ôóíêöié.
Âiäîìî [15, c. 144], ùî êëàñè Lψ
β̄
N ñêëàäàþòüñÿ ç ôóíêöié, ÿêi
ìàéæå ïðè âñiõ x ∈ R ìîæíà çîáðàçèòè ó âèãëÿäi çãîðòêè
f(x) =
a0
2
+
1
π
π∫
−π
ϕ(x− t)Ψβ̄(t)dt, ϕ ∈ N, ϕ ⊥ 1, (2)
ç ñóìîâíèì ÿäðîì Ψβ̄(t), ðÿä Ôóð'¹ ÿêîãî ì๠âèãëÿä
Ψβ̄(t) ∼
∞∑
k=1
ψ(k) cos
(
kt− βkπ
2
)
, ψ(k) > 0, βk ∈ R, k ∈ N.
ßêùî æ f ∈ Cψ
β̄
N, òî ðiâíiñòü (2) âèêîíó¹òüñÿ äëÿ âñiõ x ∈ R.
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 41
Îäèíè÷íó êóëþ ïðîñòîðó L0
s, 1 ≤ s ≤ ∞, ïîçíà÷èìî ÷åðåç U0
s i
ïîêëàäåìî Cψ
β̄
U0
s = Cψ
β̄,s
, Lψ
β̄
U0
s = Lψ
β̄,s
.
Íåõàé T2m−1 ïiäïðîñòið òðèãîíîìåòðè÷íèõ ïîëiíîìiâ
tm−1(x) =
m−1∑
k=0
(αk cos kx+ γk sin kx), αk, γk ∈ R, ïîðÿäîê ÿêèõ
íå ïåðåâèùó¹ m− 1. Âåëè÷èíà
Em(f)X = inf
tm−1∈T2m−1
‖f − tm−1‖X
¹ íàéêðàùèì íàáëèæåííÿì ôóíêöi¨ f ∈ X ⊂ L1 â ìåòðèöi ïðîñòîðó
X òðèãîíîìåòðè÷íèìè ïîëiíîìàìè ïîðÿäêó m − 1. Äàëi â ðîëi X
âèñòóïàòèìóòü ïðîñòîðè C àáî Ls, 1 ≤ s ≤ ∞.
Ïîçíà÷èìî ÷åðåç Vn,p(f) ñóìè Âàëëå Ïóññåíà ôóíêöi¨ f ∈ L1,
òîáòî ïîëiíîìè âèãëÿäó
Vn,p(f) = Vn,p(f ;x) =
1
p
n−1∑
k=n−p
Sk(f ;x),
äå Sk(f) = Sk(f ;x) � ÷àñòèííi ñóìè Ôóð'¹ ïîðÿäêó k ôóíêöi¨ f , à
p = p(n) � ïåâíèé íàòóðàëüíèé ïàðàìåòð, p 6 n. Ïðè p = 1 ñóìè
Âàëëå Ïóññåíà Vn,p(f) ¹ ÷àñòèííèìè ñóìàìè Ôóð'¹ Sn−1(f) ïîðÿäêó
n− 1, ÿêùî æ p = n, òî ñóìè Vn,p(f) ïåðåòâîðþþòüñÿ ó âiäîìi ñóìè
Ôåé¹ðà σn−1(f) ïîðÿäêó n− 1:
σn−1(f) = σn−1(f ;x) =
1
n
n−1∑
k=0
Sk(f ;x).
Êðiì çâè÷àéíèõ ñóì Âàëëå Ïóññåíà Vn,p(f) áóäåìî ðîçãëÿäàòè ¨õ
iíòåðïîëÿöiéíi àíàëîãè Ṽn,p(f).
Íåõàé f ∈ C. ×åðåç S̃n−1(f ;x) áóäåìî ïîçíà÷àòè òðèãîíî-
ìåòðè÷íèé ïîëiíîì ïîðÿäêó n − 1, ùî iíòåðïîëþ¹ f(x) ó òî÷êàõ
x
(n−1)
k = 2kπ
2n−1 , k ∈ Z, òîáòî òàêèé, ùî
S̃n−1(f ;x(n−1)
k ) = f(x(n−1)
k ), k ∈ Z.
42 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
Iíòåðïîëÿöiéíèé òðèãîíîìåòðè÷íèé ïîëiíîì S̃n−1(f ;x), ìîæíà çàïè-
ñàòè â òàêèé ñïîñiá (äèâ., íàïðèêëàä, [18, c. 13�14]):
S̃n−1(f ;x) =
a
(n−1)
0
2
+
n−1∑
k=1
(a(n−1)
k cos kx+ b
(n−1)
k sin kx), (3)
äå
a
(n−1)
k =
2
2n− 1
2n−2∑
j=0
f(x(n−1)
j ) cos kx(n−1)
j , k = 0, 1, ..., n, (4)
b
(n−1)
k =
2
2n− 1
2n−2∑
j=0
f(x(n−1)
j ) sin kx(n−1)
j , k = 1, 2, ..., n. (5)
Ïîëiíîìè
∼
V n,p(f ;x) =
a
(n−1)
0
2
λ
(n)
0 +
n−1∑
k=1
λ
(n)
k (a(n−1)
k cos kx+ b
(n−1)
k sin kx), (6)
äå
λ
(n)
k =
{
1, 0 6 k 6 n− p,
1− k−n+p
p , n− p+ 1 6 k 6 n.
(7)
p ∈ N, 1 ≤ p ≤ n, à a
(n−1)
k i b
(n−1)
k îçíà÷åíi ôîðìóëàìè (4) òà (5)
âiäïîâiäíî, íàçèâàþòü iíòåðïîëÿöiéíèìè àíàëîãàìè ñóì Âàëëå Ïóñ-
ñåíà ç ïàðàìåòðàìè n òà p. Ïðè p = 1 ñóìè
∼
V n,p(f ;x) ñïiâïàäàþòü ç
iíòåðïîëÿöiéíèìè òðèãîíîìåòðè÷íèìè ïîëiíîìàìè
∼
Sn−1(f ;x). Ó âè-
ïàäêó p = n ñóìè
∼
V n,p(f ;x) ïåðåòâîðþþòüñÿ â iíòåðïîëÿöiéíi ñóìè
Ôåé¹ðà σ̃n−1(f ;x) ïîðÿäêó n− 1:
σ̃n−1(f ;x) =
1
n
n−1∑
k=0
∼
S
(n−1)
k (f ;x),
äå
S̃
(n−1)
k (f ;x) =
a
(n−1)
0
2
+
k∑
j=1
(a(n−1)
j cos jx+ b
(n−1)
j sin jx).
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 43
 çàãàëüíîìó âèïàäêó iíòåðïîëÿöiéíi ñóìè Ṽn,p(f ;x) âèðàæàþòüñÿ
÷åðåç ñóìè S̃
(n−1)
k íàñòóïíèì ÷èíîì
Ṽn,p(f ;x) =
1
p
n−1∑
k=n−p
S̃
(n−1)
k (f ;x).
Ïîçíà÷èìî ÷åðåç ρn,p(f ;x) i ρ̃n,p(f ;x) âåëè÷èíè âèãëÿäó
ρn,p(f ;x) = f(x)− Vn,p(f ;x),
ρ̃n,p(f ;x) = f(x)− Ṽn,p(f ;x).
 ðîáîòi âñòàíîâëåíî àñèìïòîòè÷íî íåïîêðàùóâàíi àíàëîãè
íåðiâíîñòåé òèïó Ëåáåãà äëÿ âiäõèëåíü ñóì Vn,p(f) òà Ṽn,p(f ;x) íà
ìíîæèíàõ Cψ
β̄
Ls ïðè ψ ∈ D0, βk ∈ R, k ∈ N i 1 ≤ s ≤ ∞.
Òåîðåìà 1. Íåõàé ψ ∈ D0, βk ∈ R, k ∈ N. Òîäi äëÿ äîâiëüíèõ
f ∈ Cψ
β̄
Ls, 1 ≤ s <∞, n, p ∈ N, p ≤ n ñïðàâåäëèâà íåðiâíiñòü
‖ρn,p(f ;x)‖C ≤
≤
(‖ cos t‖s′
πp
ψ(n−p+1)+O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(f
ψ
β̄
)Ls
. (8)
Ïðè öüîìó äëÿ áóäü-ÿêî¨ ôóíêöi¨ f ∈ Cψ
β̄
Ls, 1 ≤ s <∞ i äîâiëü-
íèõ n, p ∈ N, p ≤ n â ìíîæèíi Cψ
β̄
Ls, 1 ≤ s <∞, çíàéäåòüñÿ ôóíê-
öiÿ F (x) = F (f ;n; p;x) òàêà, ùî En−p+1(F
ψ
β̄
)
Ls
= En−p+1(f
ψ
β̄
)
Ls
, i
äëÿ íå¨ ïðè n− p→∞ âèêîíó¹òüñÿ ðiâíiñòü
‖ρn,p(F ;x)‖C =
=
(‖ cos t‖s′
πp
ψ(n−p+1)+O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(F
ψ
β̄
)Ls . (9)
Ó (8) i (9) s′ = s
s−1 , êîåôiöi¹íòè τn,p(k) âèçíà÷àþòüñÿ ðiâíiñòþ
τn,p(k) =
{
1− n−k
p , n− p+ 1 ≤ k ≤ n− 1,
1, k ≥ n,
(10)
44 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
à O(1) � âåëè÷èíè, ðiâíîìiðíî îáìåæåíi âiäíîñíî âñiõ ðîçãëÿäóâà-
íèõ ïàðàìåòðiâ.
Äîâåäåííÿ òåîðåìè 1. Äîâåäåííÿ áóäåìî ïðîâîäèòè çà ñõå-
ìîþ, ÿêà çàïðîïîíîâàíà â òåîðåìi 1 ðîáîòè [9]. Íåõàé f ∈ Cψ
β̄
Ls,
1 ≤ s ≤ ∞. Òîäi â êîæíié òî÷öi x ∈ R (äèâ., íàïðèêëàä, [19, c. 810])
ì๠ìiñöå iíòåãðàëüíå çîáðàæåííÿ
ρn,p(f ;x) =
1
π
π∫
−π
fψ
β̄
(x− t)Ψ1,n,p(t)dt =
=
ψ(n− p+ 1)
πp
π∫
−π
fψ
β̄
(x− t) cos
(
(n− p+ 1)t− βn−p+1π
2
)
dt+
+
1
π
π∫
−π
fψ
β̄
(x− t)Ψ2,n,p(t)dt, (11)
äå Ψj,n,p(t) îçíà÷à¹òüñÿ ðiâíiñòþ
Ψj,n,p(t) =
∞∑
k=n−p+j
τn,p(k)ψ(k) cos
(
kt− βkπ
2
)
, j ∈ N. (12)
Ôóíêöi¨ cos
(
(n − p + 1)t − βn−p+1π
2
)
òà Ψ2,n,p(t) îðòîãîíàëüíi äî
áóäü-ÿêîãî òðèãîíîìåòðè÷íîãî ïîëiíîìà tn−p ïîðÿäêó íå âèùîãî
n− p, òîìó â ñèëó (11)
ρn,p(f ;x) =
=
ψ(n− p+ 1)
πp
π∫
−π
δn,p(x− t) cos
(
(n− p+ 1)t− βn−p+1π
2
)
dt+
+
1
π
π∫
−π
δn,p(x− t)Ψ2,n,p(t)dt, (13)
äå
δn,p(·) = fψ
β̄
(·)− tn−p(·). (14)
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 45
Îáðàâøè â (13) ó ðîëi tn−p(·) ïîëiíîì t∗n−p(·) íàéêðàùîãî íàáëèæåí-
íÿ â ïðîñòîði Ls ôóíêöi¨ fψ
β̄
(·) òà âèêîðèñòîâóþ÷è ôîðìóëó (10) i
íåðiâíiñòü Ãåëüäåðà ∥∥∥∥
π∫
−π
K(t− u)ϕ(u)du
∥∥∥∥
C
≤
≤ ‖K‖s′‖ϕ‖s, ϕ ∈ Ls, K ∈ Ls′ , 1 ≤ s ≤ ∞,
1
s
+
1
s′
= 1, (15)
(äèâ., íàïðèêëàä, [20, c. 43]), iíòåãðàëè ðiâíîñòi (13) îöiíèìî íàñòóï-
íèì ÷èíîì: ∥∥ π∫
−π
δn,p(x− t) cos(n− p+ 1)t dt
∥∥
C
≤
≤ ‖δn,p‖s‖ cos t‖s′ ≤ ‖ cos t‖s′En−p+1(f
ψ
β̄
)Ls
, (16)
∥∥ 1
π
π∫
−π
δn,p(x− t)Ψ2,n,p(t)dt
∥∥
C
≤
≤ 1
π
‖δn,p‖s‖Ψ2,n,p‖s′ ≤
21/s′
π1/s
∞∑
k=n−p+2
ψ(k)τn,p(k)En−p+1(f
ψ
β̄
)Ls . (17)
Îá'¹äíóþ÷è (16) i (17) îòðèìó¹ìî (8).
Äîâåäåìî äðóãó ÷àñòèíó òåîðåìè. Ç iíòåãðàëüíîãî çîáðàæåííÿ
(11) i ôàêòó îðòîãîíàëüíîñòi ôóíêöi¨ Ψ2,n,p(t) äî áóäü-ÿêîãî òðèãî-
íîìåòðè÷íîãî ïîëiíîìà tn−p ∈ T2(n−p)+1 âèïëèâà¹, ùî äëÿ äîâiëüíî¨
ôóíêöi¨ f ∈ Cψ
β̄
Ls, 1 ≤ s < ∞, ψ ∈ D0, βk ∈ R, k ∈ N âèêîíó¹òüñÿ
ðiâíiñòü
|ρn,p(f ;x)| =
=
ψ(n− p+ 1)
πp
∣∣∣ π∫
−π
fψ
β̄
(x− t) cos
(
(n− p+ 1)t− βn−p+1π
2
)
dt
∣∣∣+
+O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)En−p+1(f
ψ
β̄
)Ls
. (18)
46 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
Âðàõîâóþ÷è (18), ùîá ïåðåêîíàòèñÿ â ñïðàâåäëèâîñòi (9) äîñèòü ïî-
êàçàòè, ùî ÿêîþ á íå áóëà ôóíêöiÿ ϕ ∈ Ls, 1 ≤ s < ∞ çíàéäåòüñÿ
ôóíêöiÿ Φ(·) = Φ(ϕ; ·), äëÿ ÿêî¨ ïðè âñiõ n, p ∈ N, p ≤ n
En−p+1(Φ)Ls = En−p+1(ϕ)Ls (19)
i, êðiì òîãî, ì๠ìiñöå ðiâíiñòü
∣∣ π∫
−π
Φ(t) cos
(
(n−p+1)t+
βn−p+1π
2
)
dt
∣∣∣ = ‖ cos t‖s′En−p+1(ϕ)Ls . (20)
 ÿêîñòi Φ(·) ðîçãëÿíåìî ôóíêöiþ
Φ(t) = ‖ cos t‖1−s
′
s′
∣∣ cos
(
(n− p+ 1)t+
βn−p+1π
2
)∣∣s′−1×
×sign cos
(
(n− p+ 1)t+
βn−p+1π
2
)
En−p+1(ϕ)Ls . (21)
Äëÿ íå¨
‖Φ(t)‖s = ‖ cos t‖1−s
′
s′
( π∫
−π
∣∣ cos
(
(n− p+ 1)t+
+
βn−p+1π
2
)∣∣(s′−1)s
dt
) 1
sEn−p+1(ϕ)Ls =
= ‖ cos t‖1−s
′
s′
∥∥ cos
(
(n− p+ 1)t− βn−p+1π
2
)∥∥s′−1
s′
En−p+1(ϕ)Ls =
= En−p+1(ϕ)Ls
. (22)
Êðiì òîãî, îñêiëüêè äëÿ äîâiëüíîãî tn−p ∈ T2(n−p)+1
π∫
−π
tn−p(τ)|Φ(τ)|s−1signΦ(τ)dτ =
(
‖ cos t‖1−s
′
s′ En−p+1(ϕ)Ls
)s−1
×
×
π∫
−π
tn−p(τ) cos
(
(n− p+ 1)τ − βn−p+1π
2
)
dτ = 0,
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 47
òî íà ïiäñòàâi òåîðåìè 1.4.5 ðîáîòè [20, c. 28] ðîáèìî âèñíîâîê, ùî
ïîëiíîì t∗n−p ≡ 0 ¹ ïîëiíîìîì íàéêðàùîãî íàáëèæåííÿ ôóíêöi¨ Φ(t)
â ìåòðèöi ïðîñòîðó Ls, 1 ≤ s <∞. Îòæå, ç óðàõóâàííÿì (22),
En−p+1(Φ)Ls
= ‖Φ‖s = En−p+1(ϕ)Ls
. (23)
 ñèëó (21),
∣∣∣ π∫
−π
Φ(t) cos
(
(n− p+ 1)t+
βn−p+1π
2
)
dt
∣∣∣ =
= ‖ cos t‖1−s
′
s′ En−p+1(ϕ)Ls
∣∣∣ π∫
−π
∣∣ cos
(
(n− p+ 1)t+
βn−p+1π
2
)∣∣s′−1×
×sign cos
(
(n− p+ 1)t+
βn−p+1π
2
)
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
dt
∣∣∣ =
= ‖ cos t‖1−s
′
s′ En−p+1(ϕ)Ls
π∫
−π
∣∣ cos
(
(n− p+ 1)t+
βn−p+1π
2
)∣∣s′dt =
= ‖ cos t‖s′En−p+1(ϕ)Ls
.
Ç îñòàííiõ ñïiââiäíîøåíü âèïëèâ๠(20). Òåîðåìó 1 äîâåäåíî.
Òåîðåìà 2. Íåõàé ψ ∈ D0, βk ∈ R, k ∈ N. Òîäi äëÿ äîâiëüíèõ
f ∈ Cψ
β̄
L∞, n, p ∈ N, p ≤ n ñïðàâåäëèâà íåðiâíiñòü
‖ρn,p(f ;x)‖C ≤
1
p
(
4
π
ψ(n− p+ 1) +O(1)
(ψ2(n− p+ 2)
ψ(n− p+ 1)
+
+p
∞∑
k=n−p+3
ψ(k)τn,p(k)
))
En−p+1(f
ψ
β̄
)L∞ . (24)
Ïðè öüîìó äëÿ áóäü-ÿêî¨ ôóíêöi¨ f ∈ Cψ
β̄
L∞ i äîâiëüíèõ n, p ∈ N,
p ≤ n çíàéäåòüñÿ ôóíêöiÿ F (x) = F (f ;n; p;x) ∈ Cψ
β̄
C òàêà, ùî
48 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
En−p+1(F
ψ
β̄
)
C
= En−p+1(f
ψ
β̄
)
L∞
i äëÿ íå¨ ïðè n− p→∞ âèêîíó¹òüñÿ
ðiâíiñòü
‖ρn,p(F ;x)‖C =
1
p
(
4
π
ψ(n− p+ 1) +O(1)
(ψ2(n− p+ 2)
ψ(n− p+ 1)
+
+p
∞∑
k=n−p+3
ψ(k)τn,p(k)
))
En−p+1(F
ψ
β̄
)C . (25)
Ó (24) i (25) êîåôiöi¹íòè τn,p(k) âèçíà÷àþòüñÿ ðiâíiñòþ (10), à
O(1) � âåëè÷èíè, ðiâíîìiðíî îáìåæåíi âiäíîñíî âñiõ ðîçãëÿäóâàíèõ
ïàðàìåòðiâ.
Äîâåäåííÿ òåîðåìè 2 áóäåìî ïðîâîäèòè çà ñõåìîþ, ÿêà çà-
ïðîïîíîâàíà â òåîðåìi 2 ðîáîòè [9]. Íåõàé f ∈ Cψ
β̄
L∞, ψ ∈ D0. Âèõî-
äÿ÷è ç (11), òà âðàõîâóþ÷è ôàêò îðòîãîíàëüíîñòi ôóíêöi¨ Ψ1,n,p(t)
äî áóäü-ÿêîãî ïîëiíîìà tn−p ïîðÿäêó íå âèùîãî çà n − p, ìîæåìî
çàïèñàòè
ρn,p(f ;x) =
1
π
π∫
−π
δn,p(x− t)Ψ1,n,p(t)dt =
=
1
πp
π∫
−π
δn,p(x− t)
(
ψ(n− p+ 1) cos
(
(n− p+ 1)t− βn−p+1π
2
)
+
+2ψ(n− p+ 2) cos
(
(n− p+ 2)t− βn−p+2π
2
)
+
+p
∞∑
k=n−p+3
τn,p(k)ψ(k) cos
(
kt− βkπ
2
))
dt, (26)
äå τn,p(k) i δn,p(·) âèçíà÷àþòüñÿ ðiâíîñòÿìè (10) i (14) âiäïîâiäíî.
Îáðàâøè â (26) ó ðîëi tn−p ïîëiíîì t
∗
n−p íàéêðàùîãî íàáëèæåííÿ
â ïðîñòîði L∞ ôóíêöi¨ fψ
β̄
i çàñòîñóâàâøè íåðiâíiñòü (15) ïðè s = ∞,
îòðèìó¹ìî îöiíêó
‖ρn,p(f ;x)‖C ≤
1
πp
(∥∥ψ(n− p+ 1) cos
(
(n− p+ 1)t− βn−p+1π
2
)
+
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 49
+2ψ(n− p+ 2) cos
(
(n− p+ 2)t− βn−p+2π
2
)∥∥
1
+
+p
∥∥ ∞∑
k=n−p+3
τn,p(k)ψ(k) cos
(
kt− βkπ
2
)∥∥
1
)
En−p+1(f
ψ
β̄
)
L∞
≤
≤ 1
πp
(∥∥ψ(n− p+ 1) cos(n− p+ 1)t+
+2ψ(n− p+ 2) cos
(
(n− p+ 2)t+ αβ̄,n,p
)∥∥
1
+
+O(1)p
∞∑
k=n−p+3
τn,p(k)ψ(k)
)
En−p+1(f
ψ
β̄
)
L∞
, (27)
äå
αβ̄,n,p =
βn−p+2π
2
− n− p+ 2
n− p+ 1
βn−p+1π
2
. (28)
ßê âèïëèâ๠ç ðîáîòè Ñ.Î. Òåëÿêîâñüêîãî [21, c. 512�513],∥∥ψ(n−p+1) cos(n−p+1)t+2ψ(n−p+2) cos
(
(n−p+2)t+αβ̄,n,p
)∥∥
1
+
+O(1)p
∞∑
k=n−p+3
τn,p(k)ψ(k) ≤
≤ 4ψ(n− p+ 1) +O(1)
(ψ2(n− p+ 2)
ψ(n− p+ 1)
+ p
∞∑
k=n−p+3
τn,p(k)ψ(k)
)
. (29)
Ñïiââiäíîøåííÿ (27) i (29) äîâîäÿòü íåðiâíiñòü (24).
Äîâåäåìî äðóãó ÷àñòèíó òåîðåìè. Âèõîäÿ÷è ç iíòåãðàëüíîãî
çîáðàæåííÿ (26) i âèêîðèñòîâóþ÷è ôàêò îðòîãîíàëüíîñòi ôóíêöi¨
∞∑
k=n−p+3
τn,p(k)ψ(k) cos
(
kt− βkπ
2
)
äî áóäü-ÿêîãî òðèãîíîìåòðè÷íîãî
ïîëiíîìà tn−p ïîðÿäêó íå âèùîãî n − p, äëÿ äîâiëüíî¨ ôóíêöi¨ f
ç ìíîæèíè Cψ
β̄
L∞, ψ ∈ D0, βk ∈ R, k ∈ N âèêîíó¹òüñÿ ðiâíiñòü
|ρn,p(f ;x)| =
=
ψ(n− p+ 1)
πp
∣∣∣ π∫
−π
fψ
β̄
(x− t)
(
cos
(
(n− p+ 1)t− βn−p+1π
2
)
+
50 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t− βn−p+2π
2
))
dt
∣∣∣+
+O(1)
∞∑
k=n−p+3
τn,p(k)ψ(k)En−p+1(f
ψ
β̄
)
L∞
. (30)
Äëÿ äîâåäåííÿ (25), ç óðàõóâàííÿì (30), äîñèòü âñòàíîâèòè, ùî äëÿ
äîâiëüíî¨ ϕ ∈ L0
∞ = {ϕ ∈ L∞ : ϕ⊥1} iñíó¹ ôóíêöiÿ Φ(·) = Φ(ϕ; ·) ∈ C
äëÿ ÿêî¨ ïðè âñiõ n, p ∈ N, p ≤ n
En−p+1(Φ)C = En−p+1(ϕ)L∞
i, êðiì òîãî, ïðè n− p→∞ ì๠ìiñöå ðiâíiñòü
∣∣ π∫
−π
Φ(t)
(
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
+
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+
βn−p+2π
2
))
dt
∣∣ =
=
(
4 +O(1)
(ψ(n− p+ 2)
ψ(n− p+ 1)
)2
)
En−p+1(ϕ)L∞ . (31)
Ïîêëàäåìî
ϕ0(t) = sign cos
(
(n− p+ 1)t+
βn−p+1π
2
)
En−p+1(ϕ)L∞
i ÷åðåç ϕδ(t) ïîçíà÷èìî 2π-ïåðiîäè÷íó ôóíêöiþ, ÿêà çáiãà¹òüñÿ
ç ϕ0(t) ñêðiçü, çà âèêëþ÷åííÿì δ-îêîëiâ (0 < δ < π
2(n−p+1) ) òî÷îê
tk = (2k+1−βn−p+1)π
2(n−p+1) , k ∈ Z, äå âîíà ëiíiéíà i ¨¨ ãðàôiê ñïîëó÷๠òî÷-
êè (tk − δ, ϕ0(tk − δ)) i (tk + δ, ϕ0(tk + δ)). Ôóíêöiÿ ϕδ(t) íåïåðå-
ðâíà i ó òî÷êàõ τk = (2k−βn−p+1)π
2(n−p+1) , k = 1, 2, ..., 2(n− p+ 1), ïåðiîäó(
− βn−p+1π
2(n−p+1) , 2π −
βn−p+1π
2(n−p+1)
]
äîñÿã๠ïî àáñîëþòíié âåëè÷èíi ìàêñè-
ìàëüíîãî çíà÷åííÿ, ÿêå äîðiâíþ¹ En−p+1(ϕ)L∞ , ïî÷åðãîâî çìiíþþ÷è
çíàê. Òîìó ¨¨ ïîëiíîì íàéêðàùîãî ðiâíîìiðíîãî íàáëèæåííÿ ïîðÿäêó
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 51
íå âèùîãî n−p, çãiäíî ç êðèòåði¹ì ×åáèøîâà, ¹ ïîëiíîì, ùî òîòîæíî
äîðiâíþ¹ íóëþ i, îòæå,
En−p+1(ϕδ)C = ‖ϕδ‖C = En−p+1(ϕ)L∞ . (32)
Âðàõîâóþ÷è (15) i (32), îäåðæó¹ìî
∣∣ π∫
−π
ϕδ(t)
(
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
+
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+
βn−p+2π
2
))
dt
∣∣ ≤
≤
π∫
−π
∣∣ cos(n− p+ 1)t+ 2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+ αβ̄,n,p
)∣∣dt×
×En−p+1(ϕ)L∞ , (33)
äå αβ̄,n,p âèçíà÷à¹òüñÿ ðiâíiñòþ (28). Iç íåðiâíîñòi (19) ðîáîòè [21],
âèïëèâ๠îöiíêà
π∫
−π
∣∣ cos(n− p+ 1)t+ 2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+ αβ̄,n,p
)∣∣dt ≤
≤ 4 +O(1)
(ψ(n− p+ 2)
ψ(n− p+ 1)
)2
. (34)
Ç iíøîãî áîêó,
∣∣ π∫
−π
ϕδ(t)
(
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
+
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+
βn−p+2π
2
))
dt
∣∣ =
=
∣∣ π∫
−π
ϕ0(t)
(
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
+
52 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+
βn−p+2π
2
))
dt
∣∣ +O(1)rn,p(δ), (35)
äå
rn,p(δ) =
∣∣ π∫
−π
(ϕδ(t)− ϕ0(t))
(
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
+
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+
βn−p+2π
2
))
dt
∣∣. (36)
Îñêiëüêè ψ ∈ D0, òî äëÿ äîñèòü âåëèêèõ íîìåðiâ n−p ñïðàâäæó¹òüñÿ
íåðiâíiñòü ψ(n−p+2)
ψ(n−p+1) < 1 i, îòæå,
rn,p(δ) < 3
π∫
−π
|ϕδ(t)− ϕ0(t)|dt ≤ 6(n− p+ 1)δEn−p+1(ϕ)L∞ , (37)
òî âèáðàâøè δ íàñòiëüêè ìàëèì, ùîá âèêîíóâàëàñü óìîâà
0 < δ <
1
n− p+ 1
(ψ(n− p+ 2)
ψ(n− p+ 1)
)2
, (38)
iç (37) îäåðæèìî îöiíêó
rn,p(δ) = O(1)
(ψ(n− p+ 2)
ψ(n− p+ 1)
)2
En−p+1(ϕ)L∞ . (39)
Îñêiëüêè
π∫
−π
ϕ0(t) cos
(
(n − p + 2)t + βn−p+2π
2
)
dt = 0, òî, âðàõóâàâøè
ðîçêëàä ôóíêöié sign cos(n−p+1)t òà sign sin(n−p+1)t â ðÿä Ôóð'¹,
îòðèìà¹ìî
∣∣ π∫
−π
ϕ0(t)
(
cos
(
(n− p+ 1)t+
βn−p+1π
2
)
+
+2
ψ(n− p+ 2)
ψ(n− p+ 1)
cos
(
(n− p+ 2)t+
βn−p+2π
2
))
dt
∣∣ =
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 53
=
∣∣ π∫
−π
ϕ0(t) cos
(
(n− p+ 1)t+
βn−p+1π
2
)
dt
∣∣ =
=
π∫
−π
∣∣ cos
(
(n−p+1)t+
βn−p+1π
2
)∣∣dtEn−p+1(ϕ)L∞ = 4En−p+1(ϕ)L∞ . (40)
Iç ôîðìóë (33)�(35), (39) i (40) âèïëèâà¹, ùî äëÿ ôóíêöi¨ Φ(t) = ϕδ(t)
ó ÿêié ïàðàìåòð δ çàäîâîëüíÿ¹ óìîâó (38), ïðè n − p + 1 → ∞ ìà¹
ìiñöå ðiâíiñòü (31), à îòæå, i (25). Òåîðåìó 2 äîâåäåíî.
Äàëi, ðîçãëÿíåìî àíàëîãè òåîðåì 1 òà 2 äëÿ âåëè÷èíè ρ̃n,p(f ;x).
Ó âèïàäêó p = 1, òîáòî êîëè ñóìè Ṽn,p(f ;x) ¹ iíòåðïîëÿöiéíè-
ìè ïîëiíîìàìè S̃n−1(f ;x), íåðiâíîñòi òèïó Ëåáåãà íà êëàñàõ öiëèõ
ôóíêöié âñòàíîâëåíi â ðîáîòi [13]. Òîìó ìè ðîçãëÿíåìî ëèøå âèïà-
äîê 2 ≤ p ≤ n.
Òåîðåìà 3. Íåõàé ψ ∈ D0, βk ∈ R, k ∈ N. Òîäi äëÿ äîâiëüíèõ
f ∈ Cψ
β̄
Ls, 1 ≤ s <∞, n, p ∈ N, 2 ≤ p ≤ n ñïðàâåäëèâà íåðiâíiñòü
|ρ̃n,p(f ;x)| ≤
≤
(‖ cos t‖s′
πp
ψ(n−p+1)+O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(f
ψ
β̄
)Ls . (41)
Ïðè öüîìó äëÿ áóäü-ÿêî¨ ôóíêöi¨ f ∈ Cψ
β̄
Ls, 1 ≤ s <∞ i äîâiëüíèõ
n, p ∈ N, p ≤ n â ìíîæèíi Cψ
β̄
Ls, 1 ≤ s <∞, çíàéäåòüñÿ ôóíêöiÿ
F (x) = F (f ;n; p;x) òàêà, ùî En−p+1(F
ψ
β̄
)
Ls
= En−p+1(f
ψ
β̄
)
Ls
, i äëÿ
íå¨ ïðè n− p→∞ âèêîíó¹òüñÿ ðiâíiñòü
|ρ̃n,p(F ;x)| =
=
(‖ cos t‖s′
πp
ψ(n−p+1)+O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(F
ψ
β̄
)Ls
. (42)
Ó (41) òà (42) s′ = s
s−1 , êîåôiöi¹íòè τn,p(k) îçíà÷àþòüñÿ ðiâíiñòþ
(10), à O(1) � âåëè÷èíè, ðiâíîìiðíî îáìåæåíi âiäíîñíî âñiõ ðîçãëÿ-
äóâàíèõ ïàðàìåòðiâ.
54 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
Òåîðåìà 4. Íåõàé ψ ∈ D0, βk ∈ R, k ∈ N. Òîäi äëÿ äîâiëüíèõ
f ∈ Cψ
β̄
L∞ i áóäü�ÿêèõ n, p ∈ N, 2 ≤ p ≤ n ñïðàâåäëèâà íåðiâíiñòü
|ρ̃n,p(f ;x)| ≤
≤
( 4
πp
ψ(n− p+ 1) +O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(f
ψ
β̄
)L∞ . (43)
Ïðè öüîìó äëÿ áóäü-ÿêî¨ ôóíêöi¨ f ∈ Cψ
β̄
L∞ i äîâiëüíèõ n, p ∈ N,
p ≤ n â ìíîæèíi Cψ
β̄
C, çíàéäåòüñÿ ôóíêöiÿ F (x) = F (f ;n; p;x) òà-
êà, ùî En−p+1(F
ψ
β̄
)
C
= En−p+1(f
ψ
β̄
)
L∞
, i äëÿ íå¨ ïðè n − p → ∞ âè-
êîíó¹òüñÿ ðiâíiñòü
|ρ̃n,p(F ;x)| =
=
( 4
πp
ψ(n− p+ 1) +O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(F
ψ
β̄
)C , (44)
äå êîåôiöi¹íòè τn,p(k) îçíà÷àþòüñÿ ðiâíiñòþ (10), à O(1) � âåëè÷è-
íè, ðiâíîìiðíî îáìåæåíi âiäíîñíî âñiõ ðîçãëÿäóâàíèõ ïàðàìåòðiâ.
Îñêiëüêè äîâåäåííÿ òåîðåì 3 òà 4 íå âiäðiçíÿþòüñÿ, òîìó ïðîâå-
äåìî ¨õ ðàçîì.
Äîâåäåííÿ òåîðåì 3 òà 4. Íåõàé f ∈ Cψ
β̄
Ls, 1 ≤ s ≤ ∞,
ψ ∈ D0. Â ëåìi 2 ðîáîòè [22] âñòàíîâëåíî, ùî êîëè ψ(k) > 0,
∞∑
k=1
ψ(k) < ∞, βk ∈ R, k ∈ N, òî äëÿ äîâiëüíî¨ ôóíêöi¨ f ∈ Cψ
β̄
Ls,
1 ≤ s ≤ ∞, â êîæíié òî÷öi x ∈ R ìàþòü ìiñöå ðiâíîñòi
ρ̃n,p(f ;x) = ρn,p(f ;x) +O(1)En−p+1(f
ψ
β̄
)Ls
∞∑
k=n
ψ(k), (45)
äå O(1) � âåëè÷èíà, ðiâíîìiðíî îáìåæåíà âiäíîñíî âñiõ ðîçãëÿäóâà-
íèõ ïàðàìåòðiâ.
Ç óðàõóâàííÿì ôîðìóëè (13) çàïèøåìî (45) â òàêîìó âèãëÿäi:
ρ̃n,p(f ;x) =
Íåðiâíîñòi òèïó Ëåáåãà äëÿ ñóì Âàëëå Ïóññåíà . . . 55
=
ψ(n− p+ 1)
πp
π∫
−π
δn,p(x− t) cos
(
(n− p+ 1)t− βn−p+1π
2
)
dt+
+
1
π
π∫
−π
δn,p(x− t)Ψ2,n,p(t)dt+O(1)En−p+1(f
ψ
β̄
)Ls
∞∑
k=n
ψ(k). (46)
Çàñòîñóâàâøè íåðiâíiñòü
π∫
−π
ϕ(t)K(t)dt≤‖ϕ‖s‖K‖s′ , ϕ ∈ Ls,K ∈ Ls′ ,
1
s
+
1
s′
= 1, 1 ≤s ≤∞, (47)
(äèâ., íàïðèêëàä, [20, ñ. 391]), òà ðiâíîñòi (16) òà (17), îòðèìà¹ìî (41)
òà (43).
Äîâåäåìî òåïåð ðiâíîñòi (42) òà (44). Ðîçãëÿíåìî ñïî÷àòêó âèïà-
äîê 1 ≤ s < ∞. ßê ïîêàçàíî â äîâåäåííi òåîðåìè 1, äëÿ ôóíêöi¨
Φ(·) = Φ(ϕ; ·), îçíà÷åíî¨ ðiâíiñòþ (21), ïðè âñiõ n, p ∈ N, p ≤ n
En−p+1(Φ)Ls
= En−p+1(f
ψ
β̄
)Ls
i, êðiì òîãî, ì๠ìiñöå ðiâíiñòü
|ρn,p(F ;x)| =
=
(‖ cos t‖s′
πp
ψ(n−p+1)+O(1)
∞∑
k=n−p+2
ψ(k)τn,p(k)
)
En−p+1(Φ)Ls , (48)
äå F = J ψ
β̄
Φ. Òîìó ç (45) òà (48) âèïëèâ๠ðiâíiñòü (42).
Ðîçãëÿíåìî âèïàäîê, êîëè s = ∞. ßê ïîêàçàíî â äîâåäåííi òåî-
ðåìè 2, äëÿ ôóíêöi¨ ϕδ(·) = ϕδ(ϕ; ·), îçíà÷åíî¨ â äîâåäåííi äðóãî¨ ÷à-
ñòèíè òåîðåìè 2, ïðè âñiõ n, p ∈ N, p ≤ n
En−p+1(Φ)Ls
= En−p+1(f
ψ
β̄
)Ls
i, êðiì òîãî, ì๠ìiñöå ðiâíiñòü
|ρn,p(F ;x)| = 1
p
(
4
π
ψ(n− p+ 1) +O(1)
(ψ2(n− p+ 2)
ψ(n− p+ 1)
+
56 Â.À. Âîéòîâè÷, À.Ï. Ìóñi¹íêî
+p
∞∑
k=n−p+3
ψ(k)τn,p(k)
))
En−p+1(ϕδ)C , (49)
äå F = J ψ
β̄
ϕδ. Òîìó ç (45) òà (49) âèïëèâ๠ðiâíiñòü (42). Òåîðåìè 3
òà 4 äîâåäåíî.
Îñêiëüêè äëÿ ñóì
∞∑
k=n−p+j
τn,p(k)ψ(k), ÿêi ôiãóðóþòü â òåîðåìàõ
1 � 4, ìàþòü ìiñöå ðiâíîñòi
∞∑
k=n−p+j
τn,p(k)ψ(k) =
=
n−1∑
k=n−p+j
k−n+p
p ψ(k) +
∞∑
k=n
ψ(k), p > j, j ∈ N,
∞∑
k=n−p+j
ψ(k), p ≤ j, j ∈ N,
(50)
òî, ÿê íåâàæêî ïåðåêîíàòèñÿ, äëÿ íèõ ñïðàâåäëèâà íàñòóïíà îöiíêà
çâåðõó:
∞∑
k=n−p+j
τn,p(k)ψ(k) 6
≤ min
{ ∞∑
k=n−p+j
ψ(k),
1
p
∞∑
k=n−p+j
(k − n+ p)ψ(k)
}
, j ∈ N.
Îòæå, â ñïiââiäíîøåííÿõ (8) i (9) òåîðåìè 1, ñïiââiäíîøåííÿõ (24) i
(25) òåîðåìè 2, ñïiââiäíîøåííÿõ (41) i (42) òåîðåìè 3 òà ñïiââiäíî-
øåííÿõ (43) i (44) òåîðåìè 4 âåëè÷èíè O(1)
∞∑
k=n−p+j
τn,p(k)ψ(k) ìîæ-
íà çàìiíèòè íà O(1)min
{ ∞∑
k=n−p+j
ψ(k), 1
p
∞∑
k=n−p+j
(k − n+ p)ψ(k)
}
,
j = 2, 3.
Çàóâàæåííÿ. Ïðè βk = β, k ∈ N òåîðåìè 1 òà 2 âñòàíîâëåíi â
ðîáîòi [9].
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|
| id | oai:trim.imath.kiev.ua:article-138 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:02:51Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/7f/fd5d9b4a97d8d5de4904bbbc469dab7f.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1382018-01-29T14:44:27Z Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions Нерівності типу Лебега для сум Валле Пуссена та їх інтерполяційних аналогів на класах $(\psi,\bar{\beta})$-диференційовних функцій Voĭtovych, V. A. Musienko, A. P. Войтович, В. А. Мусієнко, А. П. We obtain the estimates of norm of deviations&nbsp;of the de Vall\'{e}e Poussin sums and interpolation analogues of sums of Vall\'{e}e Poussin from the functions that belong to the space $C_{\bar{\beta}}^\psi L_s, \ 1\leq s\leq\infty$ and are&nbsp;represented by the best approximations of $(\psi,\bar{\beta})$-differentiable functions of this sort by trigonometric polynomials in the metric $L_s$ Одержано оцінки норм відхилень сум Валле&nbsp;Пуссена та їх інтерполяційних аналогів від функцій з множин $C_{\bar{\beta}}^\psi L_s, \ 1\leq s\leq\infty$, які виражаються через найкращі наближення $(\psi,\bar{\beta})$-похідних таких функцій тригонометричними поліномами в метриці простору $L_s$ Інститут математики НАН України 2013-07-15 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/138 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 1 (2013): Approximation Theory of Functions and Related Problems; 39-58 Сборник Трудов Института математики НАН Украины; Том 10 № 1 (2013): Tеорiя наближення функцiй та сумiжнi питання; 39-58 Збірник Праць Інституту математики НАН України; Том 10 № 1 (2013): Tеорiя наближення функцiй та сумiжнi питання; 39-58 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/138/113 Авторське право (c) 2013 Інститут математики НАН України |
| spellingShingle | Voĭtovych, V. A. Musienko, A. P. Войтович, В. А. Мусієнко, А. П. Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| title | Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| title_alt | Нерівності типу Лебега для сум Валле Пуссена та їх інтерполяційних аналогів на класах $(\psi,\bar{\beta})$-диференційовних функцій |
| title_full | Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| title_fullStr | Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| title_full_unstemmed | Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| title_short | Lebesgue type inequalities for the de la Vallée Poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| title_sort | lebesgue type inequalities for the de la vallée poussin sums and their interpolation analogues on classes of $(\psi,\bar{\beta})$-differentiable functions |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/138 |
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