An Improved Jackson Inequality for the Best Trigonometric Approximation
The paper presents an improved Jackson inequality and the corresponding inverse inequality for the best trigonometric approximation in terms of the moduli of smoothness equivalent to zero on the trigonometric polynomials whose degree does not exceed a certain number. The deduced inequalities are ana...
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irk-123456789-1656302020-02-15T01:26:33Z An Improved Jackson Inequality for the Best Trigonometric Approximation Draganov, B.R. Статті The paper presents an improved Jackson inequality and the corresponding inverse inequality for the best trigonometric approximation in terms of the moduli of smoothness equivalent to zero on the trigonometric polynomials whose degree does not exceed a certain number. The deduced inequalities are analogous to Timan’s inequalities. The relations between the moduli of different orders are also considered. Отримано покращену оцінку Джексона та відповідну обернену оцінку для найкращого тригонометричного наближення в термінах модулів гладкості, еквівалентних нулю на тригонометричних поліномах степені, що не перевищує певного числа. Отримано нерівності, аналогічні нерівностям Тімана. Розглянуто також співвідношення між модулями різних порядків. 2013 Article An Improved Jackson Inequality for the Best Trigonometric Approximation / B.R. Draganov // Український математичний журнал. — 2013. — Т. 65, № 9. — С. 1219–1226. — Бібліогр.: 18 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165630 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Draganov, B.R. An Improved Jackson Inequality for the Best Trigonometric Approximation Український математичний журнал |
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The paper presents an improved Jackson inequality and the corresponding inverse inequality for the best trigonometric approximation in terms of the moduli of smoothness equivalent to zero on the trigonometric polynomials whose degree does not exceed a certain number. The deduced inequalities are analogous to Timan’s inequalities. The relations between the moduli of different orders are also considered. |
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Draganov, B.R. |
| author_facet |
Draganov, B.R. |
| author_sort |
Draganov, B.R. |
| title |
An Improved Jackson Inequality for the Best Trigonometric Approximation |
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An Improved Jackson Inequality for the Best Trigonometric Approximation |
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An Improved Jackson Inequality for the Best Trigonometric Approximation |
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An Improved Jackson Inequality for the Best Trigonometric Approximation |
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An Improved Jackson Inequality for the Best Trigonometric Approximation |
| title_sort |
improved jackson inequality for the best trigonometric approximation |
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Інститут математики НАН України |
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2013 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165630 |
| citation_txt |
An Improved Jackson Inequality for the Best Trigonometric Approximation / B.R. Draganov // Український математичний журнал. — 2013. — Т. 65, № 9. — С. 1219–1226. — Бібліогр.: 18 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT draganovbr animprovedjacksoninequalityforthebesttrigonometricapproximation AT draganovbr improvedjacksoninequalityforthebesttrigonometricapproximation |
| first_indexed |
2025-07-14T19:16:05Z |
| last_indexed |
2025-07-14T19:16:05Z |
| _version_ |
1837651014962905088 |
| fulltext |
UDC 517.5
B. R. Draganov (Sofia Univ.; Bulgar. Acad. Sci.)
AN IMPROVED JACKSON INEQUALITY
FOR BEST TRIGONOMETRIC APPROXIMATION*
ПОКРАЩЕНА ОЦIНКА ДЖЕКСОНА
ДЛЯ НАЙКРАЩОГО ТРИГОНОМЕТРИЧНОГО НАБЛИЖЕННЯ
The paper presents an improved Jackson inequality and a corresponding inverse one for best trigonometric approximation
in terms of moduli of smoothness that are equivalent to zero on the trigonometric polynomials up to a certain degree. The
inequalities are analogous to M.F. Timan’s. Relations between the moduli of different orders are also considered.
Отримано покращену оцiнку Джексона та вiдповiдну обернену оцiнку для найкращого тригонометричного набли-
ження в термiнах модулiв гладкостi, еквiвалентних нулю на тригонометричних полiномах степенi, що не перевищує
певного числа. Отримано нерiвностi, аналогiчнi нерiвностям Тiмана. Розглянуто також спiввiдношення мiж моду-
лями рiзних порядкiв.
1. Introduction. Let Lp(T), 1 ≤ p ≤ ∞, denote the space of the functions with finite Lp-norm on
the circle T. We can consider C(T) — the space of the continuous functions on T, in the place of
L∞(T). Best trigonometric approximation of a function f ∈ Lp(T) is given by
ETn (f)p = inf
τ∈Tn
‖f − τ‖p,
where Tn denotes the set of the trigonometric polynomials of degree at most n and ‖◦‖p denotes the
usual Lp-norm on T.
The error ETn (f)p is estimated by the so-called classical moduli of smoothness. To recall, the
modulus of smoothness of order r ∈ N of f ∈ Lp(T) is defined by
ωr(f, t)p = sup
0<h≤t
‖∆r
hf‖p,
where the centred finite difference of order r ∈ N of f is given by
∆r
hf(x) =
r∑
k=0
(−1)k
(
r
k
)
f(x+ (r/2− k)h).
The following relation between ETn (f)p and ωr(f, t)p is a classical result in approximation theory
(see, for example, [4] (Ch. 7) or [15] (5.1.32, 6.1.1))
ETn (f)p ≤ c ωr(f, n−1)p, (1.1)
ωr(f, t)p ≤ c tr
∑
0≤k≤1/t
(k + 1)r−1ETk (f)p.
* Supported by grant DDVU 02/30 of the Fund for Scientific Research of the Bulgarian Ministry of Education and
Science.
c© B. R. DRAGANOV, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9 1219
1220 B. R. DRAGANOV
Above and in what follows we denote by c positive constants, which do not depend on the functions
in the relations, nor on n ∈ N or 0 < t ≤ t0. For 1 < p < ∞ these estimates were improved by
Timan [16, 17] (or see [4] (Ch. 7) and [15] (6.1.5)) in the stronger forms
n−r
{
n∑
k=0
(k + 1)sr−1ETk (f)sp
}1/s
≤ c ωr(f, n−1)p, (1.2)
and
ωr(f, t)p ≤ c tr
∑
0≤k≤1/t
(k + 1)σr−1ETk (f)σp
1/σ
, (1.3)
where s = max{p, 2} and σ = min{p, 2}. The improved Jackson estimate (1.2) is also called a sharp
Jackson inequality (see [3]).
Our main goal is to establish analogues of Timan’s inequalities with moduli of smoothness,
which are equivalent to zero if the function f is a trigonometric polynomial of a certain degree,
that is, moduli which are invariant on such trigonometric polynomials. We call them trigonometric
moduli. Such estimates look natural especially with regard to the Jackson-type inequalities (1.1) and
(1.2) and the invariance of best trigonometric approximation. The method we shall use reduces the
new inequalities to the classical. On the one hand, this gives a simpler proof of the Jackson inequality
given in [9] (Theorem 1.1), but on the other hand, we believe, this approach can lead to estimates of
best approximation by other systems as well as of the rate of approximation of linear processes by
moduli possessing a corresponding natural invariance.
A sharp Jackson inequality in a very general setting for multivariate functions was established
by Dai, Ditzian and Tikhonov [3] in terms of K-functionals. Many concrete sharp Jackson estimates
are derived, among which about the univariate best algebraic approximation by the Ditzian – Totik
modulus, about the multivariate best trigonometric approximation by the classical modulus, and about
best approximation by spherical harmonic polynomials by the Ditzian modulus on the sphere [6].
The following two inequalities between the classical moduli of different order are closely related
to (1.2) and (1.3):
tr
t0∫
t
ωr+1(f, u)sp
usr+1
du
1/s
≤ c ωr(f, t)p (1.4)
and
ωr(f, t)p ≤ c tr
t0∫
t
ωr+1(f, u)σp
uσr+1
du
1/σ
, (1.5)
where 0 < t ≤ t0. The former was established in [3] (1.6); the latter is due to Timan [16] (see
also Zygmund [18]) and is referred to as an improved or sharp Marchaud inequality (see [4, p. 49]
and the references cited below). These inequalities were extended to quite general function spaces in
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
AN IMPROVED JACKSON INEQUALITY FOR BEST TRIGONOMETRIC APPROXIMATION 1221
[5, 7, 8]. Also, Dai, Ditzian and Tikhonov [3] (Theorems 5.3 and 5.5) (see also [2]) established more
general forms in terms of K-functionals. There the multivariate form of (1.4) was proved [3] (2.17).
In the present note we verify the analogues of (1.4) and (1.5) for the trigonometric moduli. Let us
mention that their basic properties were established in [9 – 11] — they are just similar to those of the
classical moduli.
The contents of the paper are organised as follows. In Section 2 we define the above-mentioned
trigonometric moduli. Then in Sections 3 and 4 we establish the analogues of (1.2), (1.3) and (1.4),
(1.5), respectively, in their terms.
2. Trigonometric moduli of smoothness. We shall consider two different types of moduli that
are identically zero on the trigonometric polynomials up to a certain degree. The first one is based
on a modification of the finite differences, whereas the second on a modification of the approximated
function.
It was Babenko, Chernykh and Shevaldin [1] who first defined a modulus, which is zero on the
trigonometric polynomials of degree r − 1. It is given by
ω̃Tr (f, t)p = sup
0<h≤t
‖∆̃r,hf‖p,
as the modified finite differences ∆̃r,h were introduced by Shevaldin [13] (see also [12]) and are
defined by
∆̃r,hf(x) = ∆r−1,h · · ·∆1,h∆hf(x), (2.1)
where
∆j,hf(x) = f(x+ h)− 2 cos jh . f(x) + f(x− h), j = 1, 2, . . . .
We have ω̃Tr (f, t)p ≡ 0 iff f ∈ Tr−1.
To define the other modulus, let the 2π-periodic function a be given on [−π, π] by
a(x) =
1
2
|x|(2π − |x|) (2.2)
and let for j ∈ N0 the bounded linear operator Aj : Lp(T)→ Lp(T), 1 ≤ p ≤ ∞, be defined by
Ajf = f + j2a ∗ f.
Above, ∗ denotes the convolution on L1(T)
f ∗ g(x) =
1
2π
∫
T
f(x− y)g(y) dy.
Further, for r ∈ N0 we set
Fr = Ar . . .A0.
Now, we define the trigonometric modulus of smoothness
ωTr (f, t)p = sup
0<h≤t
‖∆2r−1
h Fr−1f‖p.
It has the property that ωTr (f, t)p ≡ 0 iff f ∈ Tr−1 as it was established in [9, 11].
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
1222 B. R. DRAGANOV
In [9 – 11] it was shown that the moduli ωTr (f, t)p and ω̃Tr (f, t)p characterize the rate of best
trigonometric approximation just similarly as the classical modulus in any homogeneous Banach
space of periodic functions and, in particular, in Lp. Babenko, Chernykh and Shevaldin [1] proved
the Jackson estimate (1.1) in the case p = 2 for the modulus ω̃Tr (f, t)2. Shevaldin [14] verified it for
p =∞ and r = 2.
Let us explicitly point out that except for the case r = 1 when both ∆̃r,hf and ∆2r−1
h Fr−1f
coincide with the symmetric finite difference of first order, there does not exist a simple algebraic
connection between them. For r ≥ 2 the finite difference ∆̃r,hf is a linear combination of the values
of the function f at the nodes x+(r−k−1/2)h, k = 0, . . . , 2r−1, whose coefficients depend on h.
Basic properties of these coefficients including their explicit form are given e.g. in [10] (Section 2)
(see also [13]). On the other hand, the finite difference ∆2r−1
h Fr−1f is the classical symmetric finite
difference on the same nodes but of the function Fr−1f.
We can draw another line of comparison between the two finite differences. The relation (2.1)
implies by induction the following representation:
∆̃r,hf = ∆2r−1
h f +
r−1∑
k=1
λk(h) ∆
2(r−k)−1
h f,
where λk(h) are even trigonometric polynomials. Whereas in the other case, since ∆h and the
convolution commute, we have
∆2r−1
h Fr−1f = ∆2r−1
h f + c ∗ (∆2r−1
h f)
with an appropriate even 2π-periodic continuous function c (see [11], (1.10)).
However, the moduli ω̃Tr (f, t)p and ωTr (f, t)p are equivalent in the sense that there exists a
positive constant c, whose value is independent of f and t ≤ t0, such that
c−1ωTr (f, t)p ≤ ω̃Tr (f, t)p ≤ c ωTr (f, t)p,
as it follows from results in [9 – 11] (see the end of the proof of Theorem 3.1 below).
3. An improved Jackson inequality. We shall establish the analogues of (1.2) and (1.3) for the
trigonometric moduli ωTr (f, t)p and ω̃Tr (f, t)p.
Theorem 3.1. Let f ∈ Lp(T), 1 < p <∞, s = max{p, 2}, σ = min{p, 2} and r ∈ N. Then
n1−2r
{
n∑
k=r−1
(k + 1)s(2r−1)−1ETk (f)sp
}1/s
≤ c ωTr (f, n−1)p (3.1)
and
ωTr (f, t)p ≤ c t2r−1
∑
r−1≤k≤1/t
(k + 1)σ(2r−1)−1ETk (f)σp
1/σ
, 0 < t ≤ 1/r. (3.2)
The inequalities remain valid with ω̃Tr (f, t)p in the place of ωTr (f, t)p.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
AN IMPROVED JACKSON INEQUALITY FOR BEST TRIGONOMETRIC APPROXIMATION 1223
Proof. The moduli ωT1 (f, t)p and ω̃T1 (f, t)p coincide with ω1(f, t)p. So it remains to prove the
theorem for r ≥ 2.
Let bj , j ∈ N0, be 2π-periodic and defined on [−π, π] by
bj(x) = j(|x| − π) sin |jx|
and let
BjF = F + bj ∗ F.
In [11] (Proposition 2.4) we showed that for the bounded operator Er=Br . . .B0: Lp(T)→ Lp(T)
we have
ErFrf = f − Srf, f ∈ Lp(T),
where Srf is the rth partial sum of the Fourier series of f. Also, as it follows directly from their
definition, both operators Fr and Er map the set of trigonometric polynomials Tk into itself for any
k. Consequently, we get for k ≥ r − 1 and τk the trigonometric polynomial of degree k of best
Lp-approximation of Fr−1f the relation
ETk (f)p = ETk (Er−1Fr−1f)p ≤ ‖Er−1Fr−1f − Er−1τk‖p ≤
≤ c ‖Fr−1f − τk‖p = cETk (Fr−1f)p.
Now, (1.2) with Fr−1f in the place of f and 2r − 1 in the place of r directly implies (3.1).
Similarly, (1.3) and the estimate ETk (Fr−1f)p ≤ cETk (f)p, k ∈ N0, imply
ωTr (f, t)p ≤ c t2r−1
∑
0≤k≤1/t
(k + 1)σ(2r−1)−1ETk (f)σp
1/σ
.
Next, we split the sum on the right-hand side into two parts for 0 ≤ k ≤ r− 2 and r− 1 ≤ k ≤ 1/t.
We estimate above the summands of the first sum using that k ≤ r − 1 and ETk (f)p ≤ ‖f‖p to get
ωTr (f, t)p ≤ c t2r−1
∑
r−1≤k≤1/t
(k + 1)σ(2r−1)−1ETk (f)σp + rσ(2r−1)−1‖f‖σp
1/σ
.
Now, we replace above f with f − τr−1, where τr−1 is the trigonometric polynomial of best Lp-
approximation of f of degree r − 1, and use the invariance of ωTr (f, t)p and ETk (f)p, k ≥ r − 1,
under addition of trigonometric polynomials of that degree to arrive at (3.2).
The inequalities for ω̃Tr (f, t)p are derived immediately from those for ωTr (f, t)p because both
moduli are equivalent to the same K-functional, namely,
KT
r (f, t)p = inf
g∈W 2r−1
p (T)
{
‖f − g‖p + t2r−1‖D̃rg‖p},
where D̃r is the differential operator whose kernel is Tr−1 (see [10] (Theorem 4.2) and [11] (Theo-
rem 2.1)).
Theorem 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
1224 B. R. DRAGANOV
4. Improved relations between trigonometric moduli of different order. As we mentioned in
the Introduction, the trigonometric moduli possess properties just similar to those of the classical one.
In addition, they satisfy the following sharpened forms of the inequality ωTr+1(f, t)p ≤ c ωTr (f, t)p
and of the Marchaud inequality.
Theorem 4.1. Let f ∈ Lp(T), 1 < p < ∞, s = max{p, 2}, σ = min{p, 2} and r ∈ N. Then
for 0 < t ≤ t0 there hold
t2r−1
t0∫
t
ωTr+1(f, u)sp
us(2r−1)+1
du
1/s
≤ c ωTr (f, t)p (4.1)
and
ωTr (f, t)p ≤ c t2r−1
t0∫
t
ωTr+1(f, u)σp
uσ(2r−1)+1
du
1/σ
+ ‖f‖p
. (4.2)
The inequalities remain valid with ω̃Tr (f, t)p in the place of ωTr (f, t)p.
Proof. Iterating [3] (1.6) (or see (1.4)), we get the inequality
t2r−1
t0∫
t
ω2r+1(f, u)sp
us(2r−1)+1
du
1/s
≤ c ω2r−1(f, t)p. (4.3)
Set F = Fr−1f. Then Frf = F + r2a ∗ F. In [11] (3.2) it was proved that (a ∗ g)′′ = g + const for
any g ∈ Lp(T). Then, using basic properties of the classical modulus, we get
ωTr+1(f, u)sp = ω2r+1(Frf, u)sp ≤ c
[
ω2r+1(F, u)sp + u2sω2r−1(F, u)sp
]
. (4.4)
For the first term on the right above we get by (4.3) with f replaced by F
t2r−1
t0∫
t
ω2r+1(F, u)sp
us(2r−1)+1
du
1/s
≤ c ωTr (f, t)p. (4.5)
To estimate the second term on the right of (4.4), we proceed as follows. Let Ft ∈ W 2r−1
p (T) be
such that
‖F − Ft‖p ≤ c ω2r−1(F, t)p (4.6)
and
t2r−1‖F (2r−1)
t ‖p ≤ c ω2r−1(F, t)p. (4.7)
For Ft one can take the Steklov mean of F (see, e.g., [4, p. 177]). Then we have by basic properties
of the classical modulus
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
AN IMPROVED JACKSON INEQUALITY FOR BEST TRIGONOMETRIC APPROXIMATION 1225
u2sω2r−1(F, u)sp ≤ c ‖F − Ft‖sp + us(2r+1)‖F (2r−1)
t ‖sp,
where 0 < u ≤ t0, and, consequently,
t2r−1
t0∫
t
u2sω2r−1(F, u)sp
us(2r−1)+1
du
1/s
≤ c ‖F − Ft‖p + c t2r−1‖F (2r−1)
t ‖p ≤ c ωTr (f, t)p, (4.8)
as at the last step we have applied (4.6), (4.7) and ωTr (f, t)p = ω2r−1(F, t)p.
Now, (4.4), (4.5) and (4.8) imply (4.1).
We proceed to the proof of (4.2). Iterating the sharp Marchaud inequality (1.5), we arrive at
ω2r−1(f, t)p ≤ c t2r−1
t0∫
t
ω2r+1(f, u)σp
uσ(2r−1)+1
du
1/σ
.
With Frf in the place of f it yields
ω2r−1(Frf, t)p ≤ c t2r−1
t0∫
t
ωTr+1(f, u)σp
uσ(2r−1)+1
du
1/σ
.
Thus it remains to show that
ωTr (f, t)p = ω2r−1(Fr−1f, t)p ≤ c
(
ω2r−1(Frf, t)p + t2r−1‖f‖p
)
. (4.9)
To verify the latter, we take into account that Frf = ArF with F = Fr−1f ; hence BrFrf =
= BrArF = F +ηr ∗F with ηr(x) = −1−2 cos rx as was established in [11] ((2.9)). Set G = Frf
and let Gt ∈W 2r−1
p (T) satisfy (4.6), (4.7) for G in the place of F. Then
ω2r−1(F, t)p ≤ ω2r−1(BrG−BrGt, t)p + ω2r−1(Gt, t)p+
+ω2r−1(br ∗Gt, t)p + ω2r−1(ηr ∗ F, t)p ≤
≤ c ‖BrG−BrGt‖p + t2r−1‖G(2r−1)
t ‖p + t2r−1‖br ∗G(2r−1)
t ‖p + t2r−1‖η(2r−1) ∗ F‖p ≤
≤ c
(
‖G−Gt‖p + t2r−1‖G(2r−1)
t ‖p + t2r−1‖F‖p
)
≤ c
(
ω2r−1(G, t)p + t2r−1‖f‖p
)
.
Thus (4.9) is proved.
Theorem 4.1 is proved.
Remark 4.1. The inequalities (4.1) and (4.2) can be verified by means of (3.1) and (3.2) (see the
proof of [3], Theorem 5.3) and vice versa (the proof of [4], Theorem 3.4, Ch. 7). Moreover, such a
proof maybe considered even simpler and shorter than the one we used here. However, we preferred
to use an approach which is based on the properties of the classical moduli and is independent of
the relation of the new moduli to an approximation process. It demonstrates the advantages of the
connection between ωTr (f, t)p and the classical moduli in transferring properties between them and
can be applied to define moduli appropriate for other approximation operators and establish their
properties.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
1226 B. R. DRAGANOV
Remark 4.2. Replacing in (4.2) f with f − τr−1, where τr−1 is the trigonometric polynomial of
best Lp-approximation of f of degree r − 1, we immediately arrive at its slightly stronger form
ωTr (f, t)p ≤ c t2r−1
t0∫
t
ωTr+1(f, u)σp
uσ(2r−1)+1
du
1/σ
+ ETr−1(f)p
.
Acknowledgments. The research was conducted during my stay in Centre de Recerca Matemàtica,
Bellaterra, Barcelona in February 2012 on the programme Approximation Theory and Fourier Anal-
ysis. I am especially thankful to Prof. Sergey Tikhonov for posing the problem and discussions on
the presented results. I am also thankful to the referee whose remarks improved the exposition.
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Received 02.05.12,
after revision — 08.01.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 9
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