PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 96 (110) (2014), 169–180
DOI: 10.2298/PIM1410169K
NEW MODULI OF SMOOTHNESS
K. A. Kopotun, D. Leviatan, and I. A. Shevchuk
Abstract. We discuss various properties of the new modulus of smoothness
ϕ
r
ωk,r
(f (r) , t)p := sup kWkh
(·)∆khϕ(·) (f (r) , ·)kLp [−1,1] ,
0<h6t
1/2
√
where ϕ(x) := 1 − x2 and Wδ (x) = (1 − x − δϕ(x)/2)(1 + x − δϕ(x)/2)
.
Related moduli with more general weights are also considered.
1. Introduction
e p , 1 6 p 6 ∞, denote the space
1.1. Trigonometric approximation. Let L
1/p
Rπ
of 2π-periodic measurable functions for which the norm kf ke
:= −π |f (x)|p dx
Lp
e ∞ we mean the space of continuous 2π-periodic functions C
e
is finite. Here, by L
equipped with the uniform norm, i.e., kf ke
:= maxx∈[−π,π] |f (x)|.
C
Let Tn , n ∈ N, be the space of (n − 1)st degree trigonometric polynomials
Tn (x) =
n−1
X
(aj cos jx + bj sin jx).
j=0
e p , denote by
For f ∈ L
(1.1)
∆kh (f, x) =
k X
k
(−1)k−i f (x + (i − k/2)h)
i
i=0
the kth symmetric difference of the function f , and by
ωk (f, t)p := inf k∆kh (f, ·)ke
L
h∈[0,t]
p
en (f )p := inf Tn ∈Tn kf − Tn k denote
its kth modulus of smoothness. Finally, let E
e
Lp
the degree of approximation of f by trigonometric polynomials from Tn .
In 1908, de la Vallée Poussin (see [20, Section 7], for example) posed a problem
on a connection between the rate of polynomial approximation of functions and
2010 Mathematics Subject Classification: Primary 41A17; Secondary 41A10, 42A10, 41A25,
41A27.
The first author acknowledges support of NSERC of Canada.
169
170
KOPOTUN, LEVIATAN, AND SHEVCHUK
their differential properties. To quote de la Vallée Poussin [20, p. 119], “It is the
memoir by D. Jackson [9] which answers most completely the direct question, and
that of S. Bernstein [3] which answers most completely the inverse problem”. These
results were generalized by de la Vallée Poussin in [21], though as he writes in [20,
p. 119], “I combined the results obtained by the two authors above named, and filled
them out in many points; I changed or simplified the proofs; but I contributed little
in the way of new materials to the construction”.
In 1911, Jackson [9, Theorem VIII] (see also [8, p. 428]) proved the following
inequality (which is now commonly known as one of “Jackson’s inequalities”):
en (f )∞ 6 cω1 (f, n−1 )∞ , n > 1.
E
This result was later extended by Zygmund [22, Theorems 8 and 8′ ], Bernstein [2],
Akhiezer [1, Section 89], and Stechkin [15, Theorem 1] as follows.
e 0 (Direct theorem, r = 0). Let k ∈ N. If f ∈ L
e p , 1 6 p 6 ∞,
Theorem D
en (f )p 6 c(k)ωk (f, n−1 )p , n > 1.
then E
e 0 ” will become clear once one
We note that “r = 0” and the subscript “0” in “D
e r below.
compares this result with Theorem D
Matching inverse theorems are due to Bernstein [3], de la Vallée Poussin [21,
Section 39], Quade [19, Theorem 1], Salem [14, Chapter V], Zygmund [22, Theorems 8, 8′ , 9, and 9′ ], the Timan brothers [18], and Stechkin [15, Theorem 8].
e p , 1 6 p 6 ∞. Then
Theorem 1.1. Let k ∈ N and f ∈ L
n
ωk (f, n−1 )p 6
c(k) X k−1 e
ν
Eν (f )p ,
nk ν=1
n > 1.
This theorem can be restated in the following form.
Theorem eI0 (Inverse theorem, r = 0). Let k ∈ N and let φ : [0, 1] → [0, ∞) be
e p , 1 6 p 6 ∞,
a nondecreasing function such that φ(0+) = 0. If a function f ∈ L
−1
en (f )p 6 φ n
is such that E
, n > 1, then
Z 1
φ(u)
ωk (f, t)p 6 c(k)tk
du, 0 < t 6 1/2.
k+1
u
t
These direct and inverse theorems yield aconstructive characterization of the
e p | ω⌊α⌋+1 (f, t)p 6 ctα .
class g
Lip(α, p) = f ∈ L
e 0 (Constructive characterization, r = 0). Let f ∈ L
e p , 1 6 p 6 ∞,
Theorem C
α
−α
en (f )p 6 c(k)n , n > 1.
and α > 0. If ωk (f, t)p 6 t , then E
e
Conversely, if 0 < α < k and En (f )p 6 n−α , n > 1, then ωk (f, t)p 6 c(k, α)tα .
Jackson’s inequalities of the second type involve differentiable functions.
Let f
Wrp , r ∈ N, be the space of 2π-periodic functions f such that f (r−1) is
e p , where by f
er.
absolutely continuous and f (r) ∈ L
Wr∞ we mean C
e 0 and the well
The following result is an immediate consequence of Theorem D
r
(r)
known property ωk+r (f, t)p 6 t ωk (f , t)p , t > 0.
NEW MODULI OF SMOOTHNESS
171
e r (Direct theorem, r ∈ N). Let k ∈ N and r ∈ N. If f ∈ f
Theorem D
Wrp , then
en (f )p 6 c(k, r)n−r ωk (f (r) , n−1 )p ,
E
n > 1.
The following inverse theorems are due to Bernstein [3], de la Vallée Poussin
[21, Section 39], Quade [19, Theorem 1], Zygmund [22, Theorems 8, 8′ , 9 and 9′ ],
Stechkin [15, Theorem 11], and A. Timan [17], [16, Theorem 6.1.3].
e p , 1 6 p 6 ∞. If P∞ ν r−1 E
eν (f )p < ∞,
Theorem 1.2. Let r ∈ N and f ∈ L
ν=1
then f is a.e. identical with a function from f
Wrp . In addition, for any k ∈ N,
ωk (f (r) , n−1 )p 6
n
∞
X
c(k, r) X k+r−1 e
eν (f )p ,
ν
E
(f
)
+
c(k,
r)
ν r−1 E
ν
p
nk ν=1
ν=n+1
n > 1.
This theorem can be restated as follows.
Theorem eIr 1 (Inverse theorem, r ∈ N). Let k ∈ N, r ∈ N and φ : [0, 1] →
[0, ∞) be a nondecreasing function such that φ(0+) = 0 and
Z 1
φ(t)
dt < ∞.
r+1
t
0
e p be such that E
en (f )p 6 φ n−1 , n > 1, then f is a.e. identical with a
If f ∈ L
function from f
Wrp , and
Z t
Z 1
φ(u)
φ(u)
k
(r)
ωk (f , t)p 6 c(k, r)
du + t
du , 0 < t 6 1/2.
r+1
k+r+1
0 u
t u
f
(r)
Finally, we have a constructive characterization of functions f ∈ f
Wr such that
g
∈ Lip(α − r, p).
e r (Constructive characterization, r ∈ N). Let r ∈ N, α > r,
Theorem C
r
en (f )p 6 c(k, r)n−α , n > 1.
f ∈f
Wp , 1 6 p 6 ∞, and ωk (f (r) , t)p 6 tα−r . Then E
e p , 1 6 p 6 ∞, r < α < k + r, and E
en (f )p 6 n−α , n > 1,
Conversely, if f ∈ L
then f is a.e. identical with a function from f
Wrp and ωk (f (r) , t)p 6 c(k, r, α)tα−r .
1.2. Algebraic approximation. Let Lp [−1, 1], 1 6 p 6 ∞ denote the usual
1/p
R1
Lp space equipped with the norm kf kp := −1 |f (x)|p dx
, where by L∞ [−1, 1]
we mean C[−1, 1] equipped with the uniform norm.
Let Pn denote the space of algebraic polynomials of degree < n and set
En (f )p := inf kf − Pn kp
Pn ∈Pn
the degree of best approximation of f by algebraic polynomials in Lp .
Define
k
∆h (f, x), x ± kh/2 ∈ [−1, 1],
,
∆kh (f, x; [−1, 1]) :=
0,
otherwise
where ∆kh (f, x) was defined in (1.1).
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KOPOTUN, LEVIATAN, AND SHEVCHUK
Finally, define the Ditzian–Totik (DT) moduli of smoothness [6], by
ωkϕ (f, t)p := sup k∆khϕ(·) (f, ·; [−1, 1])kp
(1.2)
0<h6t
where ϕ(x) := (1 − x2 )1/2 .
It is well known that the DT moduli of smoothness yield results which are
e 0 , eI0 and C
e 0 . Namely, we have the following
completely analogous to Theorems D
results (see [6]).
Theorem D0 . Let k ∈ N. If f ∈ Lp [−1, 1], 1 6 p 6 ∞, then
En (f )p 6 c(k)ωkϕ (f, n−1 )p ,
n > k.
Theorem I0 . Let k ∈ N and φ : [0, 1] → [0, ∞) be a nondecreasing function
such that φ(0+) = 0. If a function f ∈ Lp [−1, 1], 1 6 p 6 ∞, is such that
En (f )p 6 φ(n−1 , n > k, then
Z 1
φ(u)
ϕ
k
du, t ∈ [0, 1/2].
ωk (f, t)p 6 c(k)t
k+1
t u
Theorem C0 . Let α > 0 and f ∈ Lp [−1, 1], 1 6 p 6 ∞. If ωkϕ (f, t)p 6 tα ,
then En (f )p 6 c(k)n−α , n > k.
Conversely, if 0 < α < k and En (f )p 6 n−α , n > k, then ωkϕ (f, t)p 6 c(k, α)tα .
The purpose of this paper is to discuss our new moduli of smoothness (introe r , eIr and C
e r.
duced in [10]) that allow to obtain the analogs of Theorems D
2. New moduli of smoothness
2.1. Definitions. For 1 6 p < ∞ and r ∈ N, denote
Brp := {f : f (r−1) ∈ ACloc (−1, 1) and kf (r) ϕr kp < +∞}.
If p = ∞, then
Br∞ := {f : f ∈ C r (−1, 1) and
lim f (r) (x)ϕr (x) = 0}.
x→±1
Finally, if r = 0, then B0p := Lp [−1, 1], 1 6 p < ∞ and B0∞ := C[−1, 1].
For f ∈ Brp , define
ϕ
r
ωk,r
(f (r) , t)p := sup kWkh
(·)∆khϕ(·) (f (r) , ·)kp ,
0<h6t
where

1/2

,
 (1 − x − δϕ(x)/2)(1 + x − δϕ(x)/2)
Wδ (x) :=
1 ± x − δϕ(x)/2 ∈ [−1, 1],


0,
otherwise.
ϕ
Note that, if r = 0, then ωk,0
(f, t)p = ωkϕ (f, t)p are the usual DT moduli defined in
(1.2).
ϕ
It turns out (see [10, Lemma 3.2]) that if f ∈ Brp , then limt→0+ ωk,r
(f (r), t)p = 0.
NEW MODULI OF SMOOTHNESS
173
2.2. Weighted DT moduli of smoothness. Let
Pk
k
k−i
−
→k
f (x + ih), if x, x + kh ∈ [−1, 1],
i=0 i (−1)
∆ h f (x) :=
0,
otherwise,
Pk
k
i
←
−
i=0 i (−1) f (x − ih), if x − kh, x ∈ [−1, 1],
∆ kh f (x) :=
0,
otherwise,
be the forward and backward kth differences, respectively. Note that
−
→k
←
−
∆ h f (x) := ∆kh (f, x + kh/2) and ∆ kh f (x) := ∆kh (f, x − kh/2).
Let
(2.1)
w(x) := wα,β (x) := (1 − x)α (1 + x)β ,
α, β > 0,
and denote Lp (w) := Lp (wα,β ) := {f : [−1, 1] → R | kwα,β f kp < ∞}. For f ∈
Lp (w), the weighted DT moduli of smoothness were defined (see [6, (8.2.10) and
Appendix B]) by
(2.2)
ωkϕ (f, t)w,p := sup kw∆khϕ f kLp [−1+2k2 h2 ,1−2k2 h2 ]
0<h6t
+
+
sup
−
→
kw ∆ kh f kLp [−1,−1+2k2 t2 ]
sup
←
−
kw ∆ kh f kLp [1−2k2 t2 ,1] .
0<h62k2 t2
0<h62k2 t2
The first term on the right-hand side of (2.2) is the main part modulus which is
denoted by Ωϕ
k (f, t)w,p (see [6, (8.1.2)]) and is further discussed in Section 5.
It was shown in [6, Theorem 6.1.1] that ωkϕ (f, t)w,p is equivalent to the following
weighted K-functional Kk,ϕ (f, tk )w,p (with 0 < t 6 t0 ):
Kk,ϕ (f, tk )w,p :=
inf
k(f − g)wkp + tk kwϕk g (k) kp .
g(k−1) ∈ACloc
2.3. Properties of the new moduli. For r > 0 and f ∈ Brp , we denote
ϕ
Kk,r
(f (r) , tk )p :=
inf (k(f (r) − g (r) )ϕr kp + tk kg (k+r) ϕk+r kp ).
g∈Bk+r
p
Then, we have the following equivalence results (see [10, Theorem 2.7]).
Theorem 2.1. If k ∈ N, r ∈ N0 , 1 6 p 6 ∞ and f ∈ Brp , then, for all
0 < t 6 2/k,
ϕ
ϕ
ϕ
cKk,r
(f (r) , tk )p 6 ωk,r
(f (r) , t)p 6 cKk,r
(f (r) , tk )p ,
where constants c > 0 and may depend only on k, r and p.
Corollary 2.1. If k ∈ N, r ∈ N0 , 1 6 p 6 ∞ and f ∈ Brp , then, for all
0 < t 6 2/k,
ϕ
cKk,ϕ (f (r) , tk )ϕr ,p 6 ωk,r
(f (r) , t)p 6 cKk,ϕ (f (r) , tk )ϕr ,p .
Also, the following was proved in [10, Theorem 7.1].
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KOPOTUN, LEVIATAN, AND SHEVCHUK
Theorem 2.2. If f ∈ Br+1
p , 1 6 p 6 ∞, r ∈ N0 and k > 2, then
ϕ
ϕ
ωk,r
(f (r) , t)p 6 ctωk−1,r+1
(f (r+1) , t)p .
The following sharp Marchaud inequality was proved in [4].
Theorem 2.3. [4, Theorem 7.5] For α > −1/p, β > −1/p, 1 < p < ∞, m ∈ N
and a weight w defined in (2.1), we have
1/q
Z 1
Km+1,ϕ (f, um+1 )qw,p
q
du
+
E
(f
)
Km,ϕ (f, tm )w,p 6 Ctm
m
w,p
umq+1
t
X
1/q
Km,ϕ (f, tm )w,p 6 Ctm
nqm−1 En (f )qw,p
,
n<1/t
where q = min(2, p) and En (f )w,p is the degree of best weighted approximation of
f by polynomials from Pn , i.e., En (f )w,p := inf {k(f − Pn )wkp | Pn ∈ Pn }.
Corollary 2.2. For 1 < p < ∞, r ∈ N0 , m ∈ N and f ∈ Brp , we have
1/q
Z 1 ϕ
ωm+1,r (f (r) , u)qp
(r) q
ϕ
(r)
m
ωm,r (f , t)p 6 Ct
du + Em (f )ϕr ,p
,
umq+1
t
X
1/q
ϕ
(r)
m
qm−1
(r) q
ωm,r (f , t)p 6 Ct
n
En (f )ϕr ,p
,
n<1/t
where q = min(2, p).
The following sharp Jackson inequality was proved in [5].
Theorem 2.4. [5, Theorem 6.2] For α > −1/p, β > −1/p, 1 < p < ∞, m ∈ N
and a weight w defined in (2.1), we have
X
1/s
n
−nm
mjs
s
2
2
E2j (f )w,p
6 CKm,ϕ (f, 2−nm )w,p ,
2
−nm
X
n
j=j0
2
mjs
Km+1,ϕ (f, 2−j(m+1) )sw,p
j=j0
where 2
j0
1/s
6 CKm,ϕ (f, 2−nm )w,p ,
> m and s = max(p, 2).
Corollary 2.3. For 1 < p < ∞, r ∈ N0 , m ∈ N and f ∈ Brp , we have
X
1/s
n
ϕ
2−nm
2mjs E2j (f (r) )sϕr ,p
6 Cωm,r
(f (r) , 2−n )p
, 2−nm
X
n
j=j0
ϕ
2mjs ωm+1,r
(f (r) , 2−j )sp
j=j0
where 2j0 > m and s = max(p, 2).
1/s
ϕ
6 Cωm,r
(f (r) , 2−n )p ,
NEW MODULI OF SMOOTHNESS
175
Corollary 2.4. For 1 < p < ∞, r ∈ N0 , m ∈ N and f ∈ Brp , we have
Z 1/m ϕ
1/s
ωm+1,r (f (r) , u)sp
m
ϕ
t
du
6 Cωm,r
(f (r) , t)p , 0 < t 6 1/m,
ums+1
t
where s = max(p, 2).
3. Algebraic polynomial approximation in Lp
e r , eIr and C
er
In [10], we proved the following results analogous to Theorems D
(see also [11, Theorem 3.2] for the inverse result for p = ∞).
Theorem Dr . If f ∈ Brp , 1 6 p 6 ∞, then
(3.1)
ϕ
En (f )p 6 c(k, r)n−r ωk,r
(f (r) , n−1 )p ,
n > k + r.
Note that it follows from the DT estimates that if f ∈ Brp , then
En (f )p 6 c(r)n−r kf (r) ϕr kp ,
n > r,
which is asymptotically weaker than (3.1).
It is also known that if, for some r > 1, f (r) ∈ Lp [−1, 1], 1 6 p 6 ∞, then
En (f )p 6 c(k, r)n−r ωkϕ (f (r) , n−1 )p ,
n > k + r.
But we should emphasize that here we have to assume that f (r) ∈ Lp [−1, 1], as
the DT-moduli are not well defined if the function is not in Lp [−1, 1] and, clearly,
ϕ
ωk,r
(f (r) , n−1 )p is smaller than ωkϕ (f (r) , n−1 )p .
Theorem Ir . Let r ∈ N0 , k > 1, and N ∈ N, and let φ : [0, 1] → [0, ∞) be a
nondecreasing function such that φ(0+) = 0 and
Z 1
φ(u)
r r+1 du < ∞.
u
0
If f ∈ Lp [−1, 1], 1 6 p 6 ∞, and En (f )p 6 φ(n−1 ), for all n > N , then f is
a.e. identical with a function from Brp , and
Z t
Z 1
φ(u)
φ(u)
ϕ
(r)
k
ωk,r (f , t)p 6 c(k, r)
r r+1 du + c(k, r)t
du
k+r+1
u
u
0
t
+ c(N, k, r)tk Ek+r (f )p ,
t ∈ [0, 1/2].
If, in addition, N 6 k + r, then
Z 1
Z t
φ(u)
φ(u)
ϕ
du, t ∈ [0, 1/2].
ωk,r
(f (r) , t)p 6 c(k, r)
r r+1 du + c(k, r)tk
k+r+1
u
0
t u
Taking N = 1 and appropriately choosing the function φ, we get the following
corollary of Theorem Ir in terms of the degrees of approximation.
P∞
r−1
Corollary 3.1. Given 1 6 p < ∞, k ∈ N, r ∈ N0 . If
En (f )p <+∞,
n=1 rn
then f is a.e. identical with a function from Brp , and
X
X
ϕ
ωk,r
(f (r) , t)p 6 c
rnr−1 En (f )p + ctk
nk+r−1 En (f )p , t ∈ [0, 1/2].
n>1/t
16n61/t
176
KOPOTUN, LEVIATAN, AND SHEVCHUK
Theorem Cr . Let r ∈ N0 , α > r, k > 1 and f ∈ Brp , 1 6 p 6 ∞. If
ϕ
ωk,r
(f (r) , t)p 6 tα−r , then En (f )p 6 cn−α , n > k + r.
Conversely, if r < α < r + k and f ∈ Lp [−1, 1] and En (f )p 6 n−α , n > N ,
then f is a.e. identical with a function from Brp , and
ϕ
ωk,r
(f (r) , t)p 6 c(α, k, r)tα−r + c(N, k, r)tk Ek+r (f )p ,
t ∈ [0, 1/2].
ϕ
If, in addition, N 6 k + r, then ωk,r
(f (r) , t)p 6 c(α, k, r)tα−r .
4. Further characterizations
In addition to characterizations in the previous section, we can also characterize
certain smoothness classes of functions via the growth of certain weighted norms
of their polynomials of best approximation.
Theorem 4.1. Let f ∈ Lp [−1, 1], 1 6 p 6 ∞, k ∈ N, r ∈ N0 , r < α < r + k,
and suppose that Pn denotes the (n − 1)st degree polynomial of best approximation
of f in Lp [−1, 1]. Then
(4.1)
kϕr+k Pn(r+k) kp 6 cnr+k−α ,
n > r + k,
if and only if f is a.e. identical with a function from Brp , and
(4.2)
ϕ
ωk,r
(f (r) , t)p 6 ctα−r ,
t > 0.
Proof. By virtue of [6, Theorem 7.3.1] we conclude that, for every k ∈ N and
r ∈ N0 ,
ϕ
kϕr+k Pn(r+k) kp 6 cnk+r ωk+r
(f, n−1 )p .
Hence, if f ∈ Brp and (4.2) is valid, then (4.1) follows immediately from the inequality
(4.3)
ϕ
ϕ
ωk+r
(f, t)p 6 ctr ωk,r
(f (r) , t)p
which is an immediate consequence of Theorem 2.2.
Conversely, if (4.1) holds, then it follows by [6, Theorem 7.3.2] that En (f )p 6
cn−α , n > r + k. Hence (4.2) follows from Theorem Cr .
We note that while inequality (4.3) cannot be reversed for a general function
f , the following is an immediate consequence of Theorem Cr .
Corollary 4.1. Let r ∈ N0 , k > 1, f ∈ Lp [−1, 1], 1 6 p 6 ∞, r < α < r + k.
If
ϕ
ωr+k
(f, t)p 6 ctα ,
t > 0,
then f is a.e. identical with a function from Brp , and
ϕ
ωk,r
(f (r) , t)p 6 ctα−r ,
t > 0.
NEW MODULI OF SMOOTHNESS
177
5. Further results for Weighted DT moduli
The proofs (and therefore the results) of [10] may be extended to the weighted
DT moduli with weight w which satisfies the conditions of [6, Section 6.1]. So,
in particular, we have the hierarchy relations between the weighted moduli of the
function (of course, provided its derivative exists), extending Theorem 2.2.
Theorem 5.1. Let 0 < r < k, and assume that f is such that f (r−1) is locally
absolutely continuous in (−1, 1) and wϕr f (r) ∈ Lp [−1, 1], 1 6 p 6 ∞. Then
(5.1)
ϕ
ωkϕ (f, t)w,p 6 ctr ωk−r
(f (r) , t)wϕr ,p ,
t > 0.
Remark 5.1. The inequality (5.1) extends [6, Corollary 6.3.3(b)], as we do not
require the condition of β(c) > 1, for c = ±1, that appears there.
ϕ
Proof. Recall thatthe main
part modulus Ωk is defined in [6, (8.1.2)] by
k
:= sup0<h6t w∆hϕ f Lp [−1+2k2 h2 ,−1+2k2 h2 ] . Then, [6, (6.2.9)] implies
that
Z
Ωϕ
k (f, t)w,p
ωkϕ (f, t)w,p 6 c
t
0
(Ωϕ
k (f, τ )w,p /τ ) dτ.
ϕ
′
Also, by [6, (6.3.2)], we have Ωϕ
k (f, t)w,p 6 ctΩk−1 (f , t)wϕ,p . Hence,
Z t
′
ωkϕ (f, t)w,p 6 c
Ωϕ
k−1 (f , τ )wϕ,p dτ
0
ϕ
′
′
6 ctΩϕ
k−1 (f , t)wϕ,p 6 ctωk−1 (f , t)wϕ,p ,
′
where for the second inequality we used the monotonicity of Ωϕ
k−1 (f , t)wϕ,p , and
for the third one we applied [6, (6.2.9)]. Applying this inequality r times we get
the desired estimate.
For the Jacobi weights w = wα,β defined in (2.1), it was proved by Ky [12,
Theorem 4] (see also Luther and Russo [13, Corollary 2.2]) that there is an n0 ∈ N
such that
(5.2)
En (f )w,p 6 cωkϕ (f, n−1 )w,p ,
n > n0 .
Thus, by (5.1), we have the following Jackson-type result.
Theorem 5.2. Let 0 < r < k and assume that f (r−1) is locally absolutely
continuous in (−1, 1) and wϕr f (r) ∈ Lp [−1, 1], 1 6 p 6 ∞. Then
ϕ
En (f )w,p 6 cn−r ωk−r
(f (r) , n−1 )wϕr ,p ,
It was proved in [6, Theorem 8.2.4] that
X
ωkϕ (f, t)w,p 6 ctk
nk−1 En (f )w,p ,
n > n0 .
t 6 t0 .
0<n61/t
This readily implies that, if 0 < α < k and En (f )w,p 6 n−α , for n > 1, then
ωkϕ (f, t)w,p 6 ctα ,
t 6 t0 .
In fact, it is possible to prove the following more general result.
178
KOPOTUN, LEVIATAN, AND SHEVCHUK
Theorem 5.3. Let 0 6 r < α < k, and let f be such that wf ∈ Lp [−1, 1],
1 6 p 6 ∞. If, for an N ∈ N,
En (f )w,p 6 n−α ,
(5.3)
n > N,
then f is a.e. identical with a function that has a locally absolutely continuous
derivative f (r−1) in (−1, 1), and
ϕ
ωk−r
(f (r) , t)wϕr ,p 6 c(w, α, k, r)tα−r + c(w, N, k, r)tk−r Ek (f )w,p ,
ϕ
ωk−r
(f (r) , t)wϕr ,p
In particular, if N 6 k, then
6 c(w, α, k, r)t
α−r
t > 0.
, t > 0.
Proof. Let Pk ∈ Pk be a polynomial of best approximation to f in the
weighted norm kw · kp , and set F := f − Pk . Then En (F )w,p = kwF kp = Ek (f )w,p ,
n < k, and En (F )w,p = En (f )w,p , n > k. Hence, in particular, En (F )w,p 6
Ek (f )w,p , for all n ∈ N.
Combining [6, Theorem 8.2.1] and (5.3), we obtain
X
k
Ωϕ
nk−1 En (F )w,p
k (F, t)w,p 6 ct
0<n61/t
k
6 ct N k Ek (f )w,p + ctk
X
nk−1 En (f )w,p
N 6n61/t
k
α
6 c(N )t Ek (f )w,p + ct ,
Hence,
Z 1
0
(Ωϕ
k (F, τ )w,p
/τ
r+1
) dτ 6
Z
1
0
t > 0.
cτ α−r−1 + c(N )tk−r−1 Ek (f )w,p dτ < ∞,
which, by [6, Theorem 6.3.1(a)], implies that F (r−1) is locally absolutely continuous
in (−1, 1) and
Z t
ϕ
(r)
r+1
Ωk−r (F , t)wϕr ,p 6 c
(Ωϕ
) dτ
k (F, τ )w,p /τ
0
Z t
6c
cτ α−r−1 + c(N )τ k−r−1 Ek (f )w,p dτ
6 ct
0
α−r
+ c(N )tk−r Ek (f )w,p ,
t > 0.
Finally, taking into account that
ϕ
ϕ
ωk−r
(F (r) , t)wϕr ,p = ωk−r
(f (r) , t)wϕr ,p ,
t > 0,
we apply [6, (6.2.9)] to get
ϕ
ϕ
ωk−r
(f (r) , t)wϕr ,p = ωk−r
(F (r) , t)wϕr ,p
Z t
(r)
6c
(Ωϕ
, τ )wϕr ,p /τ ) dτ
k−r (F
6 ct
This completes the proof.
0
α−r
+ c(N )tk−r Ek (f )w,p .
NEW MODULI OF SMOOTHNESS
179
Finally, we have the following result analogous to Corollary 4.1 which immediately follows from (5.2) and Theorem 5.3.
Theorem 5.4. Let wf ∈ Lp [−1, 1], 1 6 p 6 ∞ and 0 6 r < α < k. If
ωkϕ (f, t)w,p 6 ctα ,
t > 0,
then f is a.e. identical with a function that has a locally absolutely continuous
ϕ
derivative f (r−1) in (−1, 1), and ωk−r
(f (r) , t)wϕr ,p 6 ctα−r , t > 0.
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Department of Mathematics
University of Manitoba
Winnipeg, Manitoba R3T 2N2
Canada
kopotunk@cc.umanitoba.ca
Raymond and Beverly Sackler School of Mathematical Sciences
Tel Aviv University
Tel Aviv 69978
Israel
leviatan@post.tau.ac.il
Faculty of Mechanics and Mathematics
Taras Shevchenko National University of Kyiv
Kyiv 01601
Ukraine
shevchuk@univ.kiev.ua