No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation
UDC 517.5 We say that a function $f \in C[a,b]$ is $q$-monotone, $q \ge 3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2\pi$-periodic, and change its $q$-monotonicity finitely many times in $[-\pi,\pi]$. We are interested in estimating the degree of appro...
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| author | Leviatan , D. Motorna, O. V. Shevchuk, I. A. Leviatan , D. Motorna, O. V. Shevchuk, I. A. Шевчук, Ігор |
| author_facet | Leviatan , D. Motorna, O. V. Shevchuk, I. A. Leviatan , D. Motorna, O. V. Shevchuk, I. A. Шевчук, Ігор |
| author_sort | Leviatan , D. |
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| datestamp_date | 2022-10-24T09:23:03Z |
| description | UDC 517.5
We say that a function $f \in C[a,b]$ is $q$-monotone, $q \ge 3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2\pi$-periodic, and change its $q$-monotonicity finitely many times in $[-\pi,\pi]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-monotone with it, namely, trigonometric polynomials that change their $q$-monotonicity exactly at the points where $f$ does. Such Jackson type estimates are valid for piecewise monotone ($q = 1$) and piecewise convex ($q = 2$) approximations. However, we prove, that no such estimates are valid, in general, for co-$q$-monotone approximation, when $q \ge 3$. |
| doi_str_mv | 10.37863/umzh.v74i5.7081 |
| first_indexed | 2026-03-24T03:31:21Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i5.7081
UDC 517.5
D. Leviatan (Raymond and Beverly Sackler School Math. Sci., Tel Aviv Univ., Israel),
O. V. Motorna, I. A. Shevchuk (Taras Shevchenko Nat. Univ. Kyiv, Ukraine)
NO JACKSON-TYPE ESTIMATES FOR PIECEWISE \bfitq -MONOTONE \bfitq \geq \bfthree ,
TRIGONOMETRIC APPROXIMATION*
НЕМОЖЛИВI ОЦIНКИ ТИПУ ДЖЕКСОНА ДЛЯ КУСКОВО
\bfitq -МОНОТОННОЇ, \bfitq \geq \bfthree , ТРИГОНОМЕТРИЧНОЇ АПРОКСИМАЦIЇ
We say that a function f \in C[a, b] is q-monotone, q \geq 2, if f \in Cq - 2(a, b), the space of functions possessing a
(q - 2)nd continuous derivative in (a, b), and f (q - 2) is convex there. Let f be continuous and 2\pi -periodic, and change
its q-monotonicity finitely many times in [ - \pi , \pi ]. We are interested in estimating the degree of approximation of f
by trigonometric polynomials which are co-q-monotone with it, namely, trigonometric polynomials that change their q-
monotonicity exactly at the points where f does. Such Jackson-type estimates are valid for piecewise monotone (q = 1)
and piecewise convex (q = 2) approximations. However, we prove, that no such estimates are valid, in general, for
co-q-monotone approximation, when q \geq 3.
Кажуть, що функцiя f \in C[a, b] є q-монотонною, q \geq 2, якщо вона має (q - 2)-ту неперервну похiдну в (a, b)
i f (q - 2) там опукла. Нехай f — неперервна 2\pi -перiодична функцiя, яка змiнює свою q-монотоннiсть скiнченне
число разiв на [ - \pi , \pi ]. Нас цiкавлять оцiнки порядку наближення функцiї f тригонометричними полiномами, якi
змiнюють свою q-монотоннiсть саме в тих точках, де i f. Такi оцiнки типу Джексона справедливi для кусково-
монотонного (q = 1) та кусково-опуклого (q = 2) наближень. Однак ми доводимо, що жодна з таких оцiнок не є
можливою, взагалi кажучи, у ко-q-монотоннiй апроксимацiї, якщо q \geq 3.
1. Introduction and the main results. A function f \in C[a, b] is called q-monotone, q \geq 2, q \in \BbbN ,
if f \in Cq - 2(a, b), the space of functions possessing a (q - 2)nd continuous derivative in (a, b), and
f (q - 2) is convex there. For the sake of uniformity, for q = 1, we say that f \in C[a, b] is 1-monotone,
if it is nondecreasing in [a, b].
Let s \in \BbbN and \BbbY s := \{ Ys\} where Ys = \{ yi\} 2si=1 such that y2s < . . . < y1 < y2s + 2\pi =: y0. We
say that a 2\pi -periodic function f \in C(\BbbR ) is piecewise q-monotone with respect to Ys, if it changes
its q-monotonicity at the points Ys, that is, if ( - 1)i - 1f is q-monotone on [yi, yi - 1], 1 \leq i \leq 2s.
We denote by \Delta (q)(Ys) the collection of all such piecewise q-monotone functions. Note that if, in
addition, f \in Cq(\BbbR ), then f \in \Delta (q)(Ys) if and only if
f (q)(t)
2s\prod
i=1
(t - yi) \geq 0, t \in [y2s, y0].
Remark 1.1. We do not consider the case where Y consists of an odd number of points, since
the only trigonometric polynomials in \Delta (q)(Y ) are constants.
We also need the notation W r, r \in \BbbN , for the Sobolev class of 2\pi -periodic functions f \in
\in AC(r - 1)(\BbbR ), such that \bigm\| \bigm\| \bigm\| f (r)
\bigm\| \bigm\| \bigm\| \leq 2.
* Supported by the National Research Foundation of Ukraine (Project #2020.02/0155).
c\bigcirc D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK, 2022
662 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 663
For a 2\pi -periodic function g, denote
\| g\| := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
| g(x)| .
If, in addition, g is continuous, then, of course,
\| g\| = \mathrm{m}\mathrm{a}\mathrm{x}
x\in \BbbR
| g(x)| .
Similarly, for a function g, defined on the interval [a, b], we denote \| g\| [a,b] := \mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}x\in [a,b]| g(x)| ,
and if g \in C[a, b], then \| g\| [a,b] = \mathrm{m}\mathrm{a}\mathrm{x}x\in [a,b] | g(x)| .
Let \scrT n be the space of trigonometric polynomials
Tn(t) = \alpha 0 +
n\sum
k=1
(\alpha k \mathrm{c}\mathrm{o}\mathrm{s} kt+ \beta k \mathrm{s}\mathrm{i}\mathrm{n} kt), \alpha k, \beta k \in \BbbR ,
of degree \leq n (of order 2n+ 1) and, for 2\pi -periodic function g \in C(\BbbR ), let
En(g) := \mathrm{i}\mathrm{n}\mathrm{f}
Tn\in \scrT n
\| g - Tn\|
denote the error of the best approximation of the function g. If g \in \Delta (q)(Ys), then we would like
to approximate it by trigonometric polynomials that change their q-monotonicity together with g,
namely, are in \Delta (q)(Ys). We call it co-q-monotone approximation. Denote by
E(q)
n (g, Ys) := \mathrm{i}\mathrm{n}\mathrm{f}
Tn\in \scrT n\cap \Delta (q)(Ys)
\| g - Tn\|
the error of the best co-q-monotone approximation of the function g.
It is well-known that for q = 1 and q = 2, if f \in \Delta (q)(Ys) \cap W r, r \geq 1, then
E(q)
n (f, Ys) = O (1/nr) , n \rightarrow \infty (1.1)
(see [2, 4 – 6, 9] for details and references).
It turns out, and proving this is the main purpose of this article, that for q \geq 3, (1.1) is, in general,
invalid for any r, s \in \BbbN and every Ys \in \BbbY s.
Main result of this paper is the following theorem.
Theorem 1.1. For each q \geq 3, r \in \BbbN , s \in \BbbN and any Ys \in \BbbY s, there exists a function
f \in \Delta (q)(Ys) \cap W r such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
nrE(q)
n (f, Ys) = \infty .
We will also prove the following less general but more precise statements. Combining all of
them, in particular yields Theorem 1.1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
664 D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK
Theorem 1.2. For each q \geq 3, s \in \BbbN and any Ys \in \BbbY s, there exists a function
f \in \Delta (q)(Ys) \cap W q - 2
such that
E(q)
n (f, Ys) \geq C(q, Ys), n \in \BbbN , (1.2)
where C(q, Ys) > 0 depends only on q and Ys.
Corollary 1.1. For each q \geq 3, r \leq q - 2, s \in \BbbN and any Ys \in \BbbY s, there exists a function
f \in \Delta (q)(Ys) \cap W r such that
E(q)
n (f, Ys) \geq C(q, Ys), n \in \BbbN ,
where C(q, Ys) > 0 depends only on q and Ys.
Theorem 1.3. For each q \geq 3, s \in \BbbN and any Ys \in \BbbY s, there exists a function
f \in \Delta (q)(Ys) \cap W q - 1
such that
nE(q)
n (f, Ys) \geq C(q, Ys), n \in \BbbN , (1.3)
where C(q, Ys) > 0 depends only on q and Ys.
Final result is the following theorem.
Theorem 1.4. Let q \geq 3, p \geq q, s \in \BbbN and Ys \in \BbbY s. For each sequence \{ \varepsilon n\} \infty n=1 of positive
numbers, tending to infinity, there is a function f \in \Delta (q)(Ys) \cap W p such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\varepsilon nn
p - q+2E(q)
n (f, Ys) = \infty .
We prove Theorem 1.2 in Section 2, Theorem 1.3 in Section 4 and Theorem 1.4 in Section 6.
In the proofs we apply ideas from [3], and we have to overcome the constraints and challenges of
periodicity.
In the sequel, positive constants c and ci either are absolute or may depend only on r, q, p
and m.
2. Eulerian type ideal splines and proof of Theorem 1.2.
Definition 2.1. For each b \in (0, \pi ] and r \in \BbbN denote by \varepsilon r,b the 2\pi -periodic function such that
1) \varepsilon r,b \in Cr - 1,
2)
\int \pi
- \pi
\varepsilon r,b(x)dx = 0,
and
3) \varepsilon
(r)
r,b = \mathrm{s}\mathrm{g}\mathrm{n}x - \gamma b, x \in ( - b, 2\pi - b) \setminus \{ 0\} , where
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 665
\gamma b = 1 - b/\pi , (2.1)
so that
\pi \int
- \pi
\varepsilon
(r)
r,b (x)dx = 0.
Remark 2.1. By its definition, \varepsilon r,b is a spline of minimal defect of degree r, in particular, \varepsilon r,\pi is
called an Eulerian ideal spline.
Put
Fr(x) :=
1
r!
| x| xr - 1.
The following properties of \varepsilon r,b readily follow from its definition
\varepsilon r,b(x) = Fr(x) + pr,b(x), x \in [ - b, 2\pi - b], (2.2)
where pr,b is an algebraic polynomial of degree \leq r;
1 \leq
\bigm\| \bigm\| \bigm\| \varepsilon (r)r,b
\bigm\| \bigm\| \bigm\| < 2, whence \varepsilon r,b \in W r, (2.3)
and, for each collection Ys such that \{ - b, 0\} \in Ys and every q > r, we have
\varepsilon r,b \in \Delta (q)(Ys). (2.4)
We need the following lemma (see [3, Lemma 2.4]).
Lemma 2.1. For each q \geq 3 and any function g \in Cq - 2[ - 1, 1] such that g(q - 2) is convex on
[0, 1] and concave on [ - 1, 0], we have
\| Fq - 2 - g\| [ - 1,1] \geq c.
Proof of Theorem 1.2. Given Ys \in \BbbY s, let
b := \mathrm{m}\mathrm{i}\mathrm{n}
1\leq j\leq 2s
\{ yj - 1 - yj\} ,
and by shifting the periodic function f, we may assume, without loss of generality, that y2s = - b
and y2s - 1 = 0. Obviously, it follows that y2s - 2 \geq b.
We will show that f := \varepsilon q - 2,b is the desired function. Indeed, by (2.3) and (2.4), \varepsilon q - 2,b \in
\in \Delta (q)(Ys) \cap W q - 2. So we have to prove (1.2).
To this end we take an arbitrary polynomial Tn \in \scrT n \cap \Delta (q)(Ys). Then the function gn :=
:= Tn - pq - 2,b satisfies xg
(q)
n (x) \geq 0 for x \in [ - b, b], whence xg
(q)
n (x/b) \geq 0 for x \in [ - 1, 1]. Let
\~Fq - 2(x) := Fq - 2(x/b) and \~gn(x) := gn(x/b). By Lemma 2.1, we obtain
\| f - Tn\| [ - \pi ,\pi ] = \| Fq - 2 - gn\| [ - \pi ,\pi ] \geq \| Fq - 2 - gn\| [ - b,b] =
=
\bigm\| \bigm\| \bigm\| \~Fq - 2 - \~gn
\bigm\| \bigm\| \bigm\|
[ - 1,1]
= b2 - q
\bigm\| \bigm\| Fq - 2 - bq - 2\~gn
\bigm\| \bigm\|
[ - 1,1]
\geq b2 - q c,
which yields (1.2).
Theorem 1.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
666 D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK
3. Approximation of | \bfitx | . Recall that
F1(x) \equiv | x| .
In this section we prove, for trigonometric polynomials, an analog of Bernstein’s estimate
\| F1 - Pn\| [ - b,b] \geq c
b
n
,
which is valid for every algebraic polynomial Pn of degree \leq n (for the exact constant c, see [8]).
To this end, we first extend to an arbitrary interval [ - b, b] the Bernstein – de la Vallée-Poussin
inequality \bigm\| \bigm\| T \prime
n
\bigm\| \bigm\| \leq n\| Tn\| , (3.1)
which is valid for every Tn \in \scrT n.
We begin with the following simple lemma.
Lemma 3.1. If f \in C[ - a, a] is an even function and g \in C[ - a, a] is an odd function, then
\| f\| [ - a,a] \leq \| f + g\| [ - a,a] and \| g\| [ - a,a] \leq \| f + g\| [ - a,a].
Proof. Let M := \| f + g\| [ - a,a] and assume to the contrary, that there is a point x \in [ - a, a] such
that | f(x)| = K > M. Then either | f(x) + g(x)| \geq K, or | f( - x) + g( - x)| = | f(x) - g(x)| \geq K,
a contradiction. The proof for g is similar.
Lemma 3.1 is proved.
The following result is a special case of I. I. Privalov’s theorem (see, e.g., [7, p. 96, 97]).
However, we give another proof that provides sharp estimates.
Lemma 3.2. For each b \in (0, \pi ] and every trigonometric polynomial Tn \in \scrT n, there holds the
inequality \bigm\| \bigm\| T \prime
n
\bigm\| \bigm\|
[ - b/2,b/2]
\leq n
\mathrm{s}\mathrm{i}\mathrm{n}
b
2
\| Tn\| [ - b,b] \leq
\pi n
b
\| Tn\| [ - b,b]. (3.2)
Proof. Let \~b \in (0, \pi /2]. First we prove the inequality\bigm| \bigm| T \prime
n(0)
\bigm| \bigm| \leq n
\mathrm{s}\mathrm{i}\mathrm{n}\~b
\| Tn\| [ - \~b,\~b]. (3.3)
First we show, that (3.3) holds for any odd polynomial Tn \in \scrT n. Indeed, denote by Pn the algebraic
polynomial such that Pn(\mathrm{s}\mathrm{i}\mathrm{n} t) = Tn(t). Then, by Bernstein inequality for the algebraic polynomials,\bigm| \bigm| T \prime
n(0)
\bigm| \bigm| = \bigm| \bigm| P \prime
n(0)
\bigm| \bigm| \leq n
\mathrm{s}\mathrm{i}\mathrm{n}\~b
\| Pn\| [ - sin\~b,sin\~b] =
n
\mathrm{s}\mathrm{i}\mathrm{n}\~b
\| Tn\| [ - \~b,\~b].
Thus, (3.3) is proved for odd polynomials Tn. In order to prove (3.3) for an arbitrary polynomial
Tn \in \scrT n, we represent Tn, in the form Tn := Un + Vn, where Un \in \scrT n is an even polynomial, and
Vn \in \scrT n is an odd polynomial. Then T \prime
n(0) = V \prime
n(0) and by Lemma 3.1 \| Vn\| [ - \~b,\~b] \leq \| Tn\| [ - \~b,\~b].
Hence (3.3) is valid for any Tn \in \scrT n.
Now for the polynomial Tn(x+ t) \in \scrT n, x \in \BbbR , it follows by (3.3) that
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 667\bigm| \bigm| T \prime
n(x)
\bigm| \bigm| \leq n
\mathrm{s}\mathrm{i}\mathrm{n}\~b
\| Tn\| [x - \~b,x+\~b].
Hence, for x \in [ - b/2, b/2], we get
| T \prime
n(x)| \leq
n
\mathrm{s}\mathrm{i}\mathrm{n}
b
2
\| Tn\| [x - b/2,x+b/2] \leq
n
\mathrm{s}\mathrm{i}\mathrm{n}
b
2
\| Tn\| [ - b,b],
which is (3.2).
Lemma 3.2 is proved.
We are ready to prove Lemma 3.3. We follow the arguments in [1, p. 434, 435].
Lemma 3.3. For each b \in (0, \pi ] and polynomial Tn \in \scrT n, we have
\| F1 - Tn\| [ - b,b] \geq
c1b
n
, (3.4)
where c1 \geq (32\pi ) - 1 \approx 0.01.
Proof. Let
c\ast :=
1
16\pi
,
and assume to the contrary, that there is a polynomial \~Tn \in \scrT n such that\bigm\| \bigm\| \bigm\| F1 - \~Tn
\bigm\| \bigm\| \bigm\|
[ - b,b]
<
c\ast b
2n
. (3.5)
Then there is an even polynomial \^Tn \in \scrT n such that\bigm\| \bigm\| \bigm\| F1 - \^Tn
\bigm\| \bigm\| \bigm\|
[ - b,b]
\leq c\ast b
n
and
\^Tn(0) = 0.
Hence \^Tn may be represented in the form
\^Tn(t) = a1(1 - \mathrm{c}\mathrm{o}\mathrm{s} t) + . . .+ an(1 - \mathrm{c}\mathrm{o}\mathrm{s}nt) = 2
n\sum
k=1
ak \mathrm{s}\mathrm{i}\mathrm{n}
2
\biggl(
kt
2
\biggr)
.
Thus, for Tn(t) := \^Tn(2t), we have
\| 2F1 - Tn\| [ - b/2,b/2] \leq
c\ast b
n
. (3.6)
Denote
\tau n(t) :=
Tn(t)
\mathrm{s}\mathrm{i}\mathrm{n} t
\bigl(
\tau n(0) = T \prime
n(0)
\bigr)
.
Then \tau n is an odd trigonometric polynomial of degree < 2n.
First we prove that
\| \tau n\| [ - b/2,b/2] < 4. (3.7)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
668 D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK
Indeed, by virtue of (3.6), one has, for b/8 \leq | t| \leq b/2,
| Tn(t)| \leq 2| t| + c\ast b
n
\leq 2| t| + 8c\ast | t|
n
<
\biggl(
2 +
1
2\pi n
\biggr)
| t| < 2.2| t| .
Hence, if b/8 \leq | t| \leq b/2, then
| \tau n(t)| <
2.2| t|
\mathrm{s}\mathrm{i}\mathrm{n} t
\leq 1.1b
\mathrm{s}\mathrm{i}\mathrm{n} b/2
<
1.1\pi
\mathrm{s}\mathrm{i}\mathrm{n}\pi /2
= 1.1\pi < 4.
Thus, assuming the contrary, that there is a point t0 \in [ - b/2, b/2] such that
\| \tau n\| [ - b/2,b/2] = | \tau n(t0)| = M \geq 4,
we conclude, that t0 \in [ - b/8, b/8]. Since Lemma 3.2 implies that
b
\bigm\| \bigm\| \tau \prime n\bigm\| \bigm\| [ - b/4,b/4]
\leq b
\mathrm{s}\mathrm{i}\mathrm{n} b/4
(2n - 1)M < 2nM
b
\mathrm{s}\mathrm{i}\mathrm{n} b/4
\leq
\leq 2nM
\pi
\mathrm{s}\mathrm{i}\mathrm{n}\pi /4
= 2
\surd
2\pi nM,
we get, for t \in In :=
\biggl[
t0 -
c\ast b
n
, t0 +
c\ast b
n
\biggr]
\subset
\biggl(
- b
4
,
b
4
\biggr)
,
| \tau n(t)| \geq | \tau n(t0)| - | \tau n(t) - \tau n(t0)| \geq
\geq | \tau n(t0)| - | t - t0|
\bigm\| \bigm\| \tau \prime n\bigm\| \bigm\| In \geq M - | t - t0| \| \tau \prime n\| [ - b/4,b/4] \geq
\geq M - c\ast 2
\surd
2\pi M =
\Bigl(
1 -
\surd
2/8
\Bigr)
M > 0.8M.
Hence, for t \in In, we have
| Tn(t)|
| t|
\geq 0.8
| \mathrm{s}\mathrm{i}\mathrm{n} t|
| t|
M \geq 0.8
\mathrm{s}\mathrm{i}\mathrm{n}
\pi
6
\pi
6
M =
2.4
\pi
M >
3
4
M,
which, in turn, implies
\| Tn - 2F1\| In \geq
\biggl(
3M
4
- 2
\biggr)
\| F1\| In \geq \| F1\| In \geq c\ast b
n
,
contradicting (3.6). Therefore, (3.7) is proved.
By virtue of Lemma 3.2 and (3.7),\bigm\| \bigm\| \tau \prime n\bigm\| \bigm\| [ - b/4,b/4]
\leq 2\pi
b
(2n - 1)\| \tau n\| [ - b/2,b/2] <
16\pi
b
n =
n
c\ast b
.
Therefore, for t \in (0, b/4],
| \tau n(t)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\tau \prime n(u) du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < tn
c\ast b
,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 669
whence
| Tn(t)| <
tn
c\ast b
\mathrm{s}\mathrm{i}\mathrm{n} t <
t2n
c\ast b
.
Hence, for t =
c\ast b
n
, we get
2t - Tn(t) > t
\biggl(
2 - tn
c\ast b
\biggr)
= t =
c\ast b
n
,
contradicting (3.6) and, in turn, (3.5).
Lemma 3.3 is proved.
The following lemma is a consequence of Lemma 3.1.
Lemma 3.4. For each b \in (0, \pi ], any linear function l and every trigonometric polynomial
Tn \in \scrT n, we have
\| F1 + l - Tn\| [ - b,b] \geq
c1b
n
. (3.8)
Proof. We represent Tn in the form Tn = Te+To, where Te is an even polynomial, and To is an
odd polynomial. Let l(x) = ax+ k =: lo(x) + le. Denote \~Te := Te - le \in \scrT n, the even polynomial.
By (3.4), \| F1 - \~Te\| \geq c1b/n. Since lo - To is an odd function, it follows by Lemma 3.1 that (3.8)
is valid.
Lemma 3.4 is proved.
4. Proof of Theorem 1.3. The following result readily follows from [3, Lemma 3.1].
Lemma 4.1. Given q \geq 3. If a function f \in Cq - 2[ - 2b, 2b] has a convex (q - 2)nd derivative
f (q - 2) on [0, 2b] and a concave (q - 2)nd derivative f (q - 2) on [ - 2b, 0], then
bq - 2
\bigm\| \bigm\| \bigm\| f (q - 2)
\bigm\| \bigm\| \bigm\|
[ - b,b]
\leq c2\| f\| [ - 2b,2b]. (4.1)
Indeed, let \| f (q - 2)\| [ - b,b] \not = 0 and x\ast \in [ - b, b] be such that
\bigm| \bigm| f (q - 2) (x\ast )
\bigm| \bigm| = \bigm\| \bigm\| f (q - 2)
\bigm\| \bigm\|
[ - b,b]
. If
either x\ast = 0 and f (q - 2)(0) < 0, or x\ast > 0, then [3, (3.1)] yields,
bq - 2
\bigm\| \bigm\| \bigm\| f (q - 2)
\bigm\| \bigm\| \bigm\|
[ - b,b]
= bq - 2
\bigm\| \bigm\| \bigm\| f (q - 2)
\bigm\| \bigm\| \bigm\|
[0,b]
\leq c2\| f\| [0,2b] \leq c2\| f\| [ - 2b,2b].
Otherwise (4.1) follows from [3, (3.2)].
Recall that Fr(x) = | x| xr - 1/r!. We have the following lemma.
Lemma 4.2. For every b \in (0, \pi ], every trigonometric polynomial Tn \in \scrT n, satisfying
tT (r+1)
n (t) \geq 0 for | t| \leq b,
and any algebraic polynomial Pr of degree \leq r, we have
n\| Fr + Pr - Tn\| [ - b,b] \geq c3b
r, n \in \BbbN . (4.2)
Proof. Since F
(r - 1)
r = F1 and P
(r - 1)
r is linear, it follows by Lemma 3.4 that\bigm\| \bigm\| \bigm\| T (r - 1)
n - F (r - 1)
r - P (r - 1)
r
\bigm\| \bigm\| \bigm\|
[ - b/2,b/2]
\geq c1b
2n
.
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670 D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK
Now, T (r - 1)
n - F
(r - 1)
r - P
(r - 1)
r is convex in [0, b] and concave in [ - b, 0], so, by virtue of Lemma 4.1,
\| Tn - Fr - Pr\| [ - b,b] \geq
1
c2
\biggl(
b
2
\biggr) r - 1 \bigm\| \bigm\| \bigm\| T (r - 1)
n - F (r - 1)
r - P (r - 1)
r
\bigm\| \bigm\| \bigm\|
[ - b/2,b/2]
\geq c1
c2n
\biggl(
b
2
\biggr) r
.
Hence, (4.2) follows with c3 \geq 2 - rc1/c2.
Lemma 4.2 is proved.
Proof of Theorem 1.3. Given Ys \in \BbbY s, again, let
b := \mathrm{m}\mathrm{i}\mathrm{n}
j\in \BbbZ
\{ yi+1 - yi\} ,
and by shifting the periodic function f, we may assume, without loss of generality, that y2s = - b
and y2s - 1 = 0. Then f := \varepsilon q - 1,b is the desired function. Indeed, by (2.3) and (2.4), \varepsilon q - 1,b \in
\in \Delta (q)(Ys) \cap W q - 1. So, we have to prove (1.3).
To this end, take an arbitrary polynomial Tn \in \scrT n \cap \Delta (q)(Ys). By (2.2),
\varepsilon q - 1,b(x) = Fq - 1(x) + pq - 1,b(x), x \in [ - b, 2\pi - b],
where pq - 1,b is an algebraic polynomial of degree \leq q - 1. Therefore, Lemma 4.2 implies (1.3) with
C(q, Ys) \geq c3b
q - 1.
Theorem 1.3 is proved.
5. Auxiliary results. Let S \in C\infty (\BbbR ), be a monotone odd function such that S(x) = \mathrm{s}\mathrm{g}\mathrm{n}x,
| x| \geq 1.
Put
sj :=
\bigm\| \bigm\| \bigm\| S(j)
\bigm\| \bigm\| \bigm\| , j \in \BbbN 0.
Fix d \in (0, \pi ], and for \lambda \in (0, d/3], let
\~S\lambda ,d(x) :=
\left\{
S
\biggl(
x - 2\lambda
\lambda
\biggr)
, if x \in [0, 2\pi - d],
- S
\biggl(
x - 2\lambda + d
\lambda
\biggr)
, if x \in [ - d, 0].
Finally, denote
S\lambda ,d(x) := \~S\lambda ,d(x) - \gamma d, x \in [ - d, 2\pi - d],
where \gamma d was defined in (2.1), extended periodically to \BbbR .
Note that \bigm\| \bigm\| \bigm\| S(j)
\lambda ,d
\bigm\| \bigm\| \bigm\| = \lambda - jsj , j \in \BbbN , (5.1)
and
\pi \int
- \pi
S\lambda ,d(x)dx = 0.
Definition 5.1. For each \lambda \in (0, d/3] and r \in \BbbN denote by \varepsilon r,d,\lambda the 2\pi -periodic function
\varepsilon r,d,\lambda \in C\infty (\BbbR ) such that
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NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 671
1)
\int \pi
- \pi
\varepsilon r,d,\lambda (x)dx = 0
and
2) \varepsilon
(r)
r,d,\lambda = S\lambda ,d(x), x \in [ - d, 2\pi - d].
Note that, for each j \in \BbbN , we have
[ - d, 2\pi - d] \cap supp \varepsilon (r+j)
r,d,\lambda = [ - d+ \lambda , - d+ 3\lambda ] \cup [\lambda , 3\lambda ], (5.2)
and that (5.1) implies \bigm\| \bigm\| \bigm\| \varepsilon (r+j)
r,d,\lambda
\bigm\| \bigm\| \bigm\| = \lambda - jsj , j \in \BbbN . (5.3)
Also, \bigm\| \bigm\| \bigm\| \varepsilon (j)r,d,\lambda
\bigm\| \bigm\| \bigm\| < c4, j = 0, . . . , r, in particular,
\bigm\| \bigm\| \bigm\| \varepsilon (r)r,d,\lambda
\bigm\| \bigm\| \bigm\| < 2. (5.4)
Lemma 5.1. We have
\| \varepsilon r,d,\lambda - \varepsilon r,d\| \leq c5\lambda . (5.5)
Proof. Put \varepsilon j := \varepsilon j,d - \varepsilon j,d,\lambda , j = 1, . . . , r. Since
\int \pi
- \pi
\varepsilon j(x)dx = 0, it follows that for any
1 \leq j \leq r there is an xj \in [ - \pi , \pi ] such that \varepsilon j(xj) = 0. Hence, we first conclude that
\| \varepsilon 1\| \leq
2\pi - d\int
- d
\bigm| \bigm| \bigm| \mathrm{s}\mathrm{g}\mathrm{n}x - \~S\lambda ,d(x)
\bigm| \bigm| \bigm| dx = 8\lambda .
Assume by induction that \| \varepsilon j\| \leq c\lambda for some j < r, and note that \varepsilon \prime j+1 = \varepsilon j . Thus, for x \in
\in [xj+1 - \pi , xj+1 + \pi ],
| \varepsilon j+1(x)| = | \varepsilon j+1(x) - \varepsilon j+1(xj+1)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
xj+1
\varepsilon j(t) dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \pi c\lambda .
Lemma 5.1 is proved.
Lemma 5.2. Let 0 < b \leq d and r \in \BbbN be given. Let n \in \BbbN and Tn \in \scrT n be such that
tT
(r+1)
n (t) \geq 0 for | t| \leq b. For any algebraic polynomial Pr of degree \leq r, if
0 < \lambda \leq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
c3b
r
2nc5
,
d
3
\biggr\}
=: \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
c6
br
n
,
d
3
\biggr\}
,
then
2n\| \varepsilon r,d,\lambda + Pr - Tn\| [ - b,b] \geq c3b
r. (5.6)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
672 D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK
Proof. Inequalities (4.2) and (5.5) imply
2n\| \varepsilon r,d,\lambda + Pr - Tn\| [ - b,b] \geq 2n\| \varepsilon r,d + Pr - Tn\| [ - b,b] - 2n\| \varepsilon r,d,\lambda - \varepsilon r,d\| \geq
\geq 2c3b
r - 2nc5\lambda \geq c3b
r.
Fix r \geq 2 and m \in \BbbN , and let q := r + 1 and
c7 := cm6 s - 1
m .
For 0 < b \leq d and each n \geq 3c6b
r, denote
\lambda n,b := c6
br
n
and
fn,b := c7
brm
nm
\varepsilon r,d,\lambda n,b
.
Then we have the following lemma.
Lemma 5.3. We get \bigm\| \bigm\| \bigm\| f (r+m)
n,b
\bigm\| \bigm\| \bigm\| \leq 1, (5.7)
\bigm\| \bigm\| \bigm\| f (r+j)
n,b
\bigm\| \bigm\| \bigm\| \leq c8n
j - m, j = 0, . . . ,m, (5.8)
and \bigm\| \bigm\| \bigm\| f (j)
n,b
\bigm\| \bigm\| \bigm\| \leq c9
nm
, j = 0, . . . , r. (5.9)
For each collection Ys such that y2s = - d, y2s - 1 = 0 and d = \mathrm{m}\mathrm{i}\mathrm{n}1\leq j\leq 2s\{ yj - 1 - yj\} , we have
fn,b \in \Delta (q)(Ys), (5.10)
and, for every polynomial Tn \in \scrT n, satisfying tT
(q)
n (t) \geq 0 for | t| \leq b and any algebraic polynomial
Pr of degree \leq r, we obtain
nm+1\| fn,b + Pr - Tn\| [ - b,b] \geq c10b
r(m+1). (5.11)
Proof. First, (5.9) and (5.10) are clear from the definition of \varepsilon r,d,\lambda n,b
and (5.4), respectively.
We prove (5.7) and (5.8) together. By virtue of (5.3), we have, for j = 0, . . . ,m,\bigm\| \bigm\| \bigm\| f (r+j)
n,b
\bigm\| \bigm\| \bigm\| = c7
brm
nm
\biggl(
c6
br
n
\biggr) - j
sj = cm6 s - 1
m
brm
nm
\biggl(
c6
br
n
\biggr) - j
sj = cm - j
6 nj - mbr(m - j) sj
sm
,
that is, (5.7) and (5.8).
Finally, we prove (5.11). Let \~Pr :=
\biggl(
c7
brm
nm
\biggr) - 1
Pr, \~Tr :=
\biggl(
c7
brm
nm
\biggr) - 1
Tr, apply Lemma 5.2
and get
nm+1\| fn,b + Pr - Tn\| [ - b,b] = nm+1c7
brm
nm
\bigm\| \bigm\| \bigm\| \varepsilon r,b,\lambda + \~Pr - \~Tn
\bigm\| \bigm\| \bigm\|
[ - b,b]
\geq
\geq nm+1c7
brm
nm
c3b
r
2n
=: c10b
r(m+1).
Lemma 5.3 is proved.
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NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 673
6. Proof of Theorem 1.4. Set r := q - 1 and m := p - r. Given Ys \in \BbbY s, let
d := \mathrm{m}\mathrm{i}\mathrm{n}
1\leq j\leq 2s
\{ yj - 1 - yj\} ,
and by shifting the periodic function f, we may assume, without loss of generality, that y2s = - d
and y2s - 1 = 0. Obviously, it follows that y2s - 2 \geq d.
We will prove, that the desired function f may be taken in the form
f(x) :=
\infty \sum
k=1
fnk+1,bk ,
where integers nk and numbers bk are chosen as follows. We put n1 := \lceil 3c6dr\rceil and b1 := d/4.
Then let n2 be such that b2 := \lambda n2,b1 < b1/3. Assume that nk and bk have been chosen. Then we
take nk+1 \geq 2nk, to be such that
3\lambda nk+1,bk < bk, (6.1)
\varepsilon nk+1
c10b
r(m+1)
k \geq k, (6.2)
and
c9
nm
k+1
\leq
c10b
r(m+1)
k - 1
10nm+1
k
. (6.3)
Denote
bk+1 := \lambda nk+1,bk . (6.4)
It follows by (5.2) and (6.4) that, for any j \in \BbbN ,
[ - d, 2\pi - d] \cap supp f (r+j)
nk+1,bk
= [ - d+ bk+1, - d+ 3bk+1] \cup [bk+1, 3bk+1]. (6.5)
Hence by (6.1), for any j \in \BbbN ,
supp f (r+j)
nk+1,bk
\cap supp f (r+j)
nk,bk - 1
= \varnothing . (6.6)
We divide the proof of Theorem 1.4 into two lemmas.
Lemma 6.1. We have
f \in W p \cap \Delta (q)(Ys). (6.7)
Proof. Inequalities (5.8) and (5.9) imply, for all j = 0, . . . , p - 1,\bigm\| \bigm\| \bigm\| f (j)
nk+1,bk
\bigm\| \bigm\| \bigm\| \leq c
nk+1
, k \in \BbbN .
Hence, for each j = 0, . . . , p - 1,
\infty \sum
k=1
\bigm\| \bigm\| \bigm\| f (j)
nk+1,bk
\bigm\| \bigm\| \bigm\| \leq c
\infty \sum
k=1
1
nk+1
\leq c
n2
\infty \sum
k=1
1
2j
= c,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
674 D. LEVIATAN, O. V. MOTORNA, I. A. SHEVCHUK
so that f is well defined on \BbbR , it is periodic, f \in Cp - 1, for each j = 0, . . . , p - 1,
f (j)(x) \equiv
\infty \sum
k=1
f
(j)
nk+1,bk
(x),
which, combined with (5.10), implies that f \in \Delta (q)(Ys).
Then (6.6) means, that for each point x \in ( - d, 0)\cup (0, 2\pi - d) there is neighbourhood, where the
sum in f (r+j) consists of at most one term not identically zero. Hence, f \in C\infty (( - d, 0)\cup (0, 2\pi - d))
and, in particular, f \in Cp(( - d, 0) \cup (0, 2\pi - d)). Combining with (5.7), we have
\bigm\| \bigm\| f (p)
\bigm\| \bigm\| \leq 1.
Lemma 6.1 is proved.
Lemma 6.2. For each k > 2, we have
nm+1
k \varepsilon nk
E(q)
nk
(f, Ys) \geq k/2. (6.8)
Proof. Fix k > 1. Then by (6.1) and (6.4), for every 1 \leq j \leq k - 1,
f
(r+1)
nj+1,bj
(x) = 0, if | x| \leq bk.
Hence,
Pr(x) :=
k - 1\sum
j=1
fnj+1,bj (x), | x| \leq bk, (6.9)
is an algebraic polynomial of degree \leq r.
Now, by (5.9) and (6.3),
\infty \sum
j=k+1
\| fnj+1,bj\| \leq c9
\infty \sum
j=k+1
1
nm
j+1
\leq c9
nm
k+2
\infty \sum
j=0
1
2jm
=
2c9
nm
k+2
\leq
c10b
r(m+1)
k
5nm+1
k+1
. (6.10)
Finally, we take an arbitrary polynomial Tnk+1
\in \scrT nk+1
\cap \Delta (q)(Ys) and note, that tT (q)
nk+1 \geq 0 for
| t| \leq bk \leq d. Therefore (6.9), (6.10) and (5.11), imply
\| f - Tnk+1
\| \geq \| f - Tnk+1
\| [ - bk,bk] =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| Pr +
\infty \sum
j=k
fnj+1,bj - Tnk+1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
[ - bk,bk]
=
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigl( Pr + fnk+1,bk - Tnk+1
\bigr)
+
\infty \sum
j=k+1
fnj+1,bj
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
[ - bk,bk]
\geq
\geq
\bigm\| \bigm\| Pr + fnk+1,bk - Tnk+1
\bigm\| \bigm\|
[ - bk,bk]
-
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
j=k+1
fnj+1,bj
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \geq
\geq
c10b
r(m+1)
k
nm+1
k+1
-
c10b
r(m+1)
k
5nm+1
k+1
=
4c10b
r(m+1)
k
5nm+1
k+1
.
Combining with (6.2), we obtain (6.8).
Lemma 6.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
NO JACKSON-TYPE ESTIMATES FOR PIECEWISE q-MONOTONE . . . 675
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519 – 540 (2009).
3. D. Leviatan, I. A. Shevchuk, Jackson type estimates for piecewise q-monotone approximation, q \geq 3, are not valid,
Pure and Appl. Funct. Anal., 1, 85 – 96 (2016).
4. G. G. Lorentz, K. L. Zeller, Degree of approximation by monotone polynomials I, J. Approx. Theory, 1, 501 – 504
(1968).
5. M. G. Pleshakov, Comonotone Jacksons inequality, J. Approx. Theory, 99, 409 – 421 (1999).
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Received 31.12.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
|
| id | umjimathkievua-article-7081 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:21Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/70/c26425324cc4c51676c0b5cfac910270.pdf |
| spelling | umjimathkievua-article-70812022-10-24T09:23:03Z No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation No Jackson-type estimates for piecewise $q$ -monotone, $q ≥ 3$, trigonometric approximation Leviatan , D. Motorna, O. V. Shevchuk, I. A. Leviatan , D. Motorna, O. V. Shevchuk, I. A. Шевчук, Ігор iecewise $q$-monotone functions, Co-$q$-monotone trigonometric approximation, Degree of approximation UDC 517.5 We say that a function $f \in C[a,b]$ is $q$-monotone, $q \ge 3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2\pi$-periodic, and change its $q$-monotonicity finitely many times in $[-\pi,\pi]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-monotone with it, namely, trigonometric polynomials that change their $q$-monotonicity exactly at the points where $f$ does. Such Jackson type estimates are valid for piecewise monotone ($q = 1$) and piecewise convex ($q = 2$) approximations. However, we prove, that no such estimates are valid, in general, for co-$q$-monotone approximation, when $q \ge 3$. УДК 517.5Неможливi оцiнки типу Джексона для кусково $q$ -монотонної, $q \ge 3$, тригонометричної апроксимацiї Кажуть, що функція $f\in C[a,b]$ є $q$-монотонною,&nbsp;$q\geq 2,$ якщо вона має $(q-2)$-ту неперервну похідну в $(a,b)$ і $f^{(q-2)}$ там опукла.&nbsp;Нехай $f$ ? неперервна $2\pi$-періодична функція, яка змінює свою $q$-монотонність скінченне число разів на $[-\pi,\pi].$&nbsp;Нас цікавлять оцінки порядку наближення функції $f$ тригонометричними поліномами, які змінюють свою&nbsp; $q$-монотонність саме в тих точках, де і $f.$&nbsp;Такі оцінки типу Джексона справедливі для кусково-монотонного $(q=1)$ та кусково-опуклого&nbsp;$(q=2)$ наближень.&nbsp;Однак ми доводимо, що жодна з таких оцінок не є можливою, взагалі кажучи, у ко-$q$-монотонній апроксимації, якщо $q\geq 3.$ Institute of Mathematics, NAS of Ukraine 2022-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7081 10.37863/umzh.v74i5.7081 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 5 (2022); 662 - 675 Український математичний журнал; Том 74 № 5 (2022); 662 - 675 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7081/9239 Copyright (c) 2022 Ігор Олександрович Шевчук |
| spellingShingle | Leviatan , D. Motorna, O. V. Shevchuk, I. A. Leviatan , D. Motorna, O. V. Shevchuk, I. A. Шевчук, Ігор No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| title | No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| title_alt | No Jackson-type estimates for piecewise $q$ -monotone, $q ≥ 3$, trigonometric approximation |
| title_full | No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| title_fullStr | No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| title_full_unstemmed | No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| title_short | No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| title_sort | no jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation |
| topic_facet | iecewise $q$-monotone functions Co-$q$-monotone trigonometric approximation Degree of approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7081 |
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