Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics
In the paper, exact order estimates of nonlinear approximative characteristics (such as the best $m$-member of the trigonometric approximation, better than the $m$-th term of the orthogonal trigonometric approximation, approximation of the $m$-member of the grid by polynomials) are found in the clas...
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2017
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552590029357056 |
|---|---|
| author | Shidlich, A. L. Шыдлич, А. Л. Шидліч, А. Л. |
| author_facet | Shidlich, A. L. Шыдлич, А. Л. Шидліч, А. Л. |
| author_institution_txt_mv | [
{
"author": "A. L. Shidlich",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Shidlich, A. L. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-13T11:57:10Z |
| description | In the paper, exact order estimates of nonlinear approximative characteristics (such as the best $m$-member of the trigonometric approximation, better than the $m$-th term of the orthogonal trigonometric approximation, approximation of the $m$-member of the grid by polynomials) are found in the class $ℱ_{q,r}^{\psi}$ of functions of several variables in the integral metric. |
| first_indexed | 2026-08-04T01:01:51Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2016, т. 13, № 3, 293–312
УДК 517.983.27
Nonlinear approximation of the
classes ℱ𝜓
𝑞,𝑟 of functions of several
variables in the integral metrics
A. L. Shidlich1*
1 Institute of Mathematics of NAS of Ukraine, Kyiv;
shidlich@gmail.com, andy709@list.ru
У роботi знайдено точнi порядковi оцiнки нелiнiйних апроксимативних
характеристик (таких як найкраще 𝑚-членне тригонометричне набли-
ження, найкраще 𝑚-членне ортогональне тригонометричне наближе-
ння, наближення 𝑚-членними грiдi полiномами) класiв ℱ𝜓
𝑞,𝑟 функцiй
багатьох змiнних у iнтегральнiй метрицi.
В работе найдены точные порядковые оценки нелинейных аппрокси-
мативных характеристик (таких как лучше 𝑚–членне тригонометри-
ческое приближения, лучше 𝑚–членне ортогональное тригонометри-
ческое приближения, приближение 𝑚 –членнимы гриди полиномами)
классов ℱ𝜓
𝑞,𝑟 функций многих переменных в интегральной метрике.
Introduction
Let 𝑑 be a fixed natural number, let ℝ𝑑 and ℤ𝑑 be the sets of all ordered
collections 𝑘 := (𝑘1, . . . , 𝑘𝑑) of 𝑑 real and integer numbers correspondi-
ngly. Let also 𝕋𝑑 := [0, 2𝜋]𝑑 denote 𝑑-dimensional torus.
Further, let 𝐿𝑝 := 𝐿𝑝(𝕋𝑑), 1 ≤ 𝑝 ≤ ∞, be the space of all Lebesgue-
measurable on ℝ𝑑 2𝜋-periodic in each variable functions 𝑓 with finite
norm
||𝑓 ||
𝐿𝑝
:=
⎧⎨⎩
(︂
(2𝜋)−𝑑
∫︁
𝕋𝑑
|𝑓(𝑥)|𝑝𝑑𝑥
)︂ 1
𝑝
, 1 ≤ 𝑝 <∞,
ess sup𝑥∈𝕋𝑑 |𝑓(𝑥)|, 𝑝 = ∞.
*The present investigation was supported, in part, by the FP7-People-2011-IRSES
project number 295164 (EUMLS: EUUkrainian Mathematicians for Life Sciences).
c○ Shidlich A. L., 2016
294 Shidlich A. L.
Set (𝑘, 𝑥) := 𝑘1𝑥1+𝑘2𝑥2+ . . .+𝑘𝑑𝑥𝑑, 𝑒𝑘(𝑥) := 𝑒𝑖(𝑘,𝑥) and for any 𝑓 ∈ 𝐿1,
we denote the Fourier coefficients of 𝑓 bŷ︀𝑓(𝑘) := (2𝜋)−𝑑
∫︁
𝕋𝑑
𝑓(𝑥)𝑒𝑘(𝑥)𝑑𝑥, 𝑘 ∈ ℤ𝑑,
where 𝑧 is the complex conjugate of 𝑧.
The space 𝑆𝑝 := 𝑆𝑝(𝕋𝑑), 0 < 𝑝 < ∞, (see, for example, [5] (Ch. XI))
is the space of all functions 𝑓 ∈ 𝐿1 such that
||𝑓 ||
𝑆𝑝 := ||{| ̂︀𝑓(𝑘)|}𝑘∈ℤ𝑑 ||𝑙𝑝(ℤ𝑑) =
(︂ ∑︁
𝑘∈ℤ𝑑
| ̂︀𝑓(𝑘)|𝑝)︂ 1
𝑝
<∞. (1)
The functions 𝑓 ∈ 𝐿1 and 𝑔 ∈ 𝐿1 are equivalent in the space 𝑆𝑝, when
‖𝑓 − 𝑔‖
𝑆𝑝=0.
We denote by 𝑙𝑑𝑝, 0 < 𝑝 ≤ ∞, the space ℝ𝑑 equipped with 𝑙𝑝-(quasi-
)norm that is defined for 𝑥 = {𝑥𝑖}𝑑𝑖=1 ∈ ℝ𝑑 by
|𝑥|𝑝 := ||𝑥||𝑙𝑝 =
⎧⎨⎩
(︂∑︀𝑑
𝑖=1 |𝑥𝑖|𝑝
)︂ 1
𝑝
, 0 < 𝑝 <∞,
sup1≤𝑖≤𝑑 |𝑥𝑖|, 𝑝 = ∞.
Let also 𝜓 = 𝜓(𝑡), 𝑡 ≥ 1, be a positive decreasing function, 𝜓(0) := 𝜓(1)
and 0 < 𝑞, 𝑟 ≤ ∞.
We investigate asymptotical behavior of some important approximati-
ve characteristics (in the sense of order estimates) of the classes of functi-
ons of several variables ℱ𝜓
𝑞,𝑟, defined by the following equality:
ℱ𝜓
𝑞,𝑟 :=
{︂
𝑓 ∈ 𝐿1 : ||{| ̂︀𝑓(𝑘)|/𝜓(|𝑘|𝑟)}𝑘∈ℤ𝑑 ||𝑙𝑞(ℤ𝑑) ≤ 1
}︂
.
If 𝜓(𝑡) = 𝑡−𝑠, 𝑠 ∈ ℕ and 𝑟 = ∞, then ℱ𝜓
𝑞,∞ =: ℱ𝑠
𝑞,∞ is a set of functions
whose 𝑠th partial derivatives have absolutely convergent Fourier series.
When 𝑞=2, ℱ𝑠
𝑞,∞ is equivalent (modulo constants) to the unit ball of the
Sobolev class 𝑊 𝑠
2 .
Approximative characteristics of the classes ℱ𝜓
𝑞,𝑟 for different 𝑟 ∈
(0,∞] and for the various functions 𝜓 were investigated by many authors
(see, for example, [1]– [5]). In particular, in [1], the authors found the
exact order estimates of the quantities of the best 𝑚-term trigonometric
approximations of the classes ℱ𝑠
𝑞,∞, 𝑠 > 0, in the spaces 𝐿𝑝. Temlyakov [2]
obtained the exact order estimates of approximations of these classes by
Nonlinear approximation of the classes ... 295
𝑚-term greedy polynomials in 𝐿𝑝. In the case where 𝜓(𝑡) is a positive
function that decreases to zero no faster than some power function, the
quantities of the best 𝑚-term one-sided trigonometric approximations
and the quantities of approximations by 𝑚-term one-sided Greedy-liked
polynomials of the classes ℱ𝜓
𝑞,∞ were studied in [3].
It should be noted that in [4], [5] (Ch. XI) Stepanets got the exact
values the best 𝑚-term trigonometric approximations of the classes ℱ𝜓
𝑞,𝑟
in the spaces 𝑆𝑝. These results are used in the proof and presented in
section 4.
1 Approximative characteristics
In this section, we give the definition of the approximation quantities for
the functions of the classes ℱ𝜓
𝑞,𝑟, which are considered in this paper.
Further, for 𝑓 ∈ 𝐿1, let {𝑘𝑙}∞𝑙=1 = {𝑘𝑙(𝑓)}∞𝑙=1 denote the rearrangement
of vectors of ℤ𝑑 such that
| ̂︀𝑓(𝑘1)| ≥ | ̂︀𝑓(𝑘2)| ≥ . . . . (2)
In general case, this rearrangement is not unique. In such case, we take
any rearrangement satisfying (2).
We define Σ𝑚 to be the class of all complex trigonometric polynomials
of the form 𝑇 =
∑︀
𝑘∈𝛾𝑚 𝑐𝑘𝑒𝑘, where 𝛾𝑚 is any collection of 𝑚 different
vectors from the set ℤ𝑑.
For 𝑓 ∈ ℱ𝜓
𝑞,𝑟, we consider the following quantities:
||𝑓 −𝐺𝑚(𝑓)||
𝑋
:= ||𝑓(·)−
𝑚∑︁
𝑙=1
̂︀𝑓(𝑘𝑙)𝑒𝑘𝑙 ||𝑋 , (3)
𝜎⊥
𝑚(𝑓)
𝑋
:= inf
𝛾𝑚
||𝑓 −
∑︁
𝑘∈𝛾𝑚
̂︀𝑓(𝑘)𝑒𝑘||𝑋 , (4)
and
𝜎𝑚(𝑓)
𝑋
:= inf
𝑇∈Σ𝑚
||𝑓 − 𝑇 ||
𝑋
= inf
𝛾𝑚,𝑐𝑘
||𝑓 −
∑︁
𝑘∈𝛾𝑚
𝑐𝑘𝑒𝑘||𝑋 , (5)
where 𝑋 is one of the spaces 𝐿𝑝, 1 ≤ 𝑝 ≤ ∞, or 𝑆𝑝, 0 < 𝑝 < ∞, 𝑐𝑘 are
any complex numbers. Here, it is assumed that the embedding ℱ𝜓
𝑞,𝑟 ⊂ 𝑋
is true.
296 Shidlich A. L.
The quantities (5) and (4) are respectively called the best 𝑚-term
trigonometric and the best 𝑚-term orthogonal trigonometric approxi-
mations of the function 𝑓 in the space 𝑋. The quantity (3) is called the
approximation of the function 𝑓 by 𝑚-term greedy polynomials in the
space 𝑋.
For a set N ⊂ 𝑋, we put
𝜎⊥
𝑚(N)
𝑋
:= sup
𝑓∈N
𝜎⊥
𝑚(𝑓)
𝑋
and 𝜎𝑚(N)
𝑋
:= sup
𝑓∈N
𝜎𝑚(𝑓)
𝑋
.
In general case, the quantities (3) depend on the choice of the rearrangement
satisfying (2). So, for the unique definition, we put
𝐺𝑚(N)
𝑋
:= sup
𝑓∈N
inf
{𝑘𝑙(𝑓)}∞
𝑙=1
||𝑓(·)−
𝑚∑︁
𝑙=1
̂︀𝑓(𝑘𝑙(𝑓))𝑒𝑘𝑙(𝑓)||𝑋 . (6)
In (6), for any function 𝑓 ∈ N, we consider the infimum on all rearrange-
ments, satisfying (2), but it should be noted that results, formulated in
this paper, are also true for any other rearrangements, satisfying (2).
Research of the quantities of the form (3)–(5) goes back to the paper of
S.B. Stechkin [6]. Order estimates of these quantities on different classes
of functions of one and several variables were obtained by many authors.
In particular, the bibliography of papers with the similar results can be
found in [7], [8], [9].
Note that for 𝑓 ∈ 𝐿𝑝,
𝜎𝑚(𝑓)
𝐿𝑝
≤ 𝜎⊥
𝑚(𝑓)
𝐿𝑝
≤ ||𝑓 −𝐺𝑚(𝑓)||
𝐿𝑝
. (7)
and by virtue of (1), for 𝑓 ∈ 𝑆𝑝,
𝜎𝑚(𝑓)
𝑆𝑝 = 𝜎⊥
𝑚(𝑓)
𝑆𝑝 = ||𝑓 −𝐺𝑚(𝑓)||
𝑆𝑝 . (8)
2 Main results
The main purpose of this work is to find the dependence of the choice of
the parameters 𝑟, 𝜓 and 𝑞 on the rate of convergence to zero, as 𝑚→ ∞,
of the approximative characteristics of the classes ℱ𝜓
𝑞,𝑟.
For a real number 𝑎, we denote (𝑎)+ = max{0, 𝑎}. As mentioned
above, in the case where 𝜓(𝑡) is a power function, i.e., 𝜓(𝑡) = 𝑡−𝑠,
𝑠 > 0, for all 1 ≤ 𝑝 ≤ ∞, the exact order estimates of the quantities
Nonlinear approximation of the classes ... 297
𝜎𝑚(ℱ𝜓
𝑞,∞)
𝐿𝑝
and 𝐺𝑚(ℱ𝜓
𝑞,∞)
𝐿𝑝
were obtained in [1] and [2], correspondi-
ngly. In particular, from Theorems 6.1 [1] and 3.1 [2], it follows that for
all 𝑠 > 𝑑(1− 1
𝑞 )+,
𝜎𝑚(ℱ𝑠
𝑞,∞)
𝐿𝑝
≍ 𝑚− 𝑠
𝑑−
1
𝑞+
1
2 , 1 ≤ 𝑝 ≤ ∞, (9)
and
𝐺𝑚(ℱ𝑠
𝑞,∞)
𝐿𝑝
≍
{︃
𝑚− 𝑠
𝑑−
1
𝑞+
1
2 , 1 ≤ 𝑝 < 2,
𝑚− 𝑠
𝑑−
1
𝑞+1− 1
𝑝 , 2 ≤ 𝑝 <∞.
(10)
For positive sequences 𝛼(𝑚) and 𝛽(𝑚), the expression ’𝑎(𝑚) ≍ 𝑏(𝑚)’
means that there are constants 0 < 𝐾1 < 𝐾2 such that for any 𝑚 ∈ ℕ,
𝛼(𝑚) ≤ 𝐾2𝛽(𝑚) (in this case, we write ’𝛼(𝑚) ≪ 𝛽(𝑚)’) and 𝛼(𝑚) ≥
𝐾1𝛽(𝑚) (in this case, we write ’𝛼(𝑚) ≫ 𝛽(𝑚)’).
From the following Theorem 2.6, in particular, it follows that for the
quantities 𝜎𝑚(ℱ𝜓
𝑞,∞)
𝐿𝑝
and 𝐺𝑚(ℱ𝜓
𝑞,∞)
𝐿𝑝
, the estimates of forms (9) and
(10) are satisfied for a wider set of the functions 𝜓. To formulate this
statement, we use the following notation: let 𝐵 denote the set of all
positive decreasing functions such that
lim
𝑡→∞
𝜓(𝑡) = 0, (11)
and for all 𝑡 ≥ 1, the following relation is true:
1 < 𝜓(𝑡)/𝜓(2𝑡) ≤ 𝐾. (12)
Here and in what follows, 𝐾, 𝐾0, . . . are positive constants which are
independent of the variable 𝑡.
Теорема 2.6. Assume that 1 ≤ 𝑟 ≤ ∞, 1 ≤ 𝑝 <∞, 0 < 𝑞 <∞, 𝜓 ∈ 𝐵
and in the case 𝑝/(𝑝− 1) < 𝑞, moreover, for all 𝑡, larger than a certain
number 𝑡0, 𝜓(𝑡) is convex and satisfies the condition
𝑡|𝜓′(𝑡)|/𝜓(𝑡) ≥ 𝐾0 > 𝛽, 𝜓′(𝑡) := 𝜓′(𝑡+), (13)
where 𝛽 := 𝑑( 12 − 1
𝑞 ), when 1 < 𝑝 ≤ 2 and 𝛽 := 𝑑(1 − 1
𝑝 − 1
𝑞 ), when
2 ≤ 𝑝 <∞. Then
𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≍ 𝜎⊥
𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≍
{︃
𝜓(𝑚
1
𝑑 )𝑚
1
2−
1
𝑞 , 1 ≤ 𝑝 ≤ 2,
𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑝−
1
𝑞 , 2≤ 𝑝<∞
,
298 Shidlich A. L.
for all 1 ≤ 𝑝 ≤ 2,
𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≍ 𝜓(𝑚
1
𝑑 )𝑚
1
2−
1
𝑞 , (14)
and for all 2 < 𝑝 <∞,
𝜓(𝑚
1
𝑑 )𝑚
1
2−
1
𝑞 ≪ 𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≪ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑝−
1
𝑞 .
In the case 2 < 𝑝 ≤ ∞, the following theorem is true.
Теорема 2.7. Assume that 1 ≤ 𝑟 ≤ ∞, 2 < 𝑝 ≤ ∞, 0 < 𝑞 < ∞, the
function 𝜓 belongs to the set 𝐵 and for all 𝑡, larger than a certain number
𝑡0, 𝜓(𝑡) is convex and satisfies condition (13) with 𝛽 = 𝑑(1− 1
𝑞 )+. Then
relation (14) holds.
Note that conditions in Theorems 2.6 and 2.7 guarantee the embeddi-
ng ℱ𝜓
𝑞,𝑟⊂𝐿𝑝.
Putting 𝑟 = ∞ and 𝜓(𝑡) = 𝑡−𝑠, 𝑠 > 0, from Theorems 2.6 and 2.7 we
obtain the following corollary:
Наслiдок 2.2. Assume that 1 ≤ 𝑝 < ∞, 0 < 𝑞 < ∞, 𝑠 is a positive
number, which in the case 𝑝/(𝑝 − 1) < 𝑞, satisfies the inequality 𝑠 > 𝛽,
where 𝛽 is defined in Theorem 2.6. Then for all 1 ≤ 𝑝 < ∞, relation
relation (10) holds and for all 1 ≤ 𝑝 ≤ 2, relation (9) holds. If 𝑠 >
𝑑(1− 1
𝑞 )+, then relation (9) holds for all 1 ≤ 𝑝 ≤ ∞.
This statement complements the results mentioned above of [1] and [2]
in the following sense:
∙ from Corollary 2.2, in particular, it follows that in the case 1 <
𝑞 ≤ 𝑝/(𝑝− 1), relation (9) (for 1 ≤ 𝑝 ≤ 2) and relation (10) (for
1 ≤ 𝑝 <∞) also hold for all 𝑠 > 0,
∙ if 1 < 𝑝 ≤ 2 and 𝑞 > 𝑝/(𝑝−1), then relations (9) and (10) also hold
for all 𝑠 such that 𝑑( 12 − 1
𝑞 ) < 𝑠 ≤ 𝑑(1− 1
𝑞 ),
∙ if 2 < 𝑝 < ∞ and 𝑞 > 𝑝/(𝑝 − 1), then relation (10) also holds for
all 𝑠 such that 𝑑(1− 1
𝑝 −
1
𝑞 ) < 𝑠 ≤ 𝑑(1− 1
𝑞 ),
∙ in the case 2 < 𝑝 ≤ ∞, conditions on 𝑠 in Corollary 2.2 (for validity
of relation (9)) are the same as in Theorem 6.1 [1].
Nonlinear approximation of the classes ... 299
Note also that if 0 < 𝑞 ≤ 𝑝/(𝑝 − 1), then the conditions of Theorem
2.6 are satisfied, for example, for the function 𝜓(𝑡) = 𝑡−𝑠 ln𝜀(𝑡+𝑒), where
𝑠 > 0, 𝜀 ∈ ℝ, as well as for the function 𝜓(𝑡) = ln𝜀(𝑡 + 𝑒), 𝜀 < 0. If
1 < 𝑝/(𝑝 − 1) < 𝑞 and 1 < 𝑝 ≤ 2, then the conditions of Theorem 2.6
are satisfied for the function 𝜓(𝑡) = 𝑡−𝑠 ln𝜀(𝑡 + 𝑒), where 𝜀 ∈ ℝ and
𝑠 > 𝑑( 12 − 1
𝑞 ). If 1 < 𝑝/(𝑝 − 1) < 𝑞 and 2 < 𝑝 < ∞, then the conditions
of Theorem 2.6 are satisfied for the function 𝜓(𝑡) = 𝑡−𝑠 ln𝜀(𝑡+ 𝑒), where
𝜀 ∈ ℝ and 𝑠 > 𝑑(1− 1
𝑝 −
1
𝑞 ). The conditions of Theorem 2.7 are satisfied
for the function 𝜓(𝑡) = 𝑡−𝑠 ln𝜀(𝑡+ 𝑒), where 𝜀 ∈ ℝ and 𝑠 > 𝑑(1− 1
𝑞 )+.
The proof of Theorems 2.6 and 2.7 will be given in Section 5.
3 Order estimates for some functionals and their
applications
4.1. Let Ψ = {Ψ(𝑗)}∞𝑗=1 be a nonincreasing positive sequence such that
lim
𝑗→+∞
Ψ(𝑗) = 0. (15)
The following Lemma 3.2 is essentially used for proving upper estimates
in Theorem 2.6. This lemma gives exact order estimates for the following
functionals 𝐻𝑚(Ψ, 𝑠), which in the case 𝑠 ∈ (0, 1], are defined by the
equality
𝐻𝑚(Ψ, 𝑠) := sup
𝑙>𝑚
(𝑙 −𝑚)
(︂ 𝑙∑︁
𝑗=1
Ψ−𝑠(𝑗)
)︂− 1
𝑠
, (16)
and for 𝑠 ∈ (1,∞), they are defined by the equality
𝐻𝑚(Ψ, 𝑠) :=
(︂
(𝑙𝑚 −𝑚)𝑠
′
(︂ 𝑙∑︁
𝑗=1
Ψ−𝑠(𝑗)
)︂− 𝑠′
𝑠
+
∞∑︁
𝑗=𝑙𝑚+1
Ψ𝑠
′
(𝑗)
)︂ 1
𝑠′
, (17)
where 1/𝑠+ 1/𝑠′ = 1,
∞∑︁
𝑗=1
Ψ𝑠
′
(𝑗) <∞, (18)
and the number 𝑙𝑚 is given by relation
Ψ−𝑠(𝑙𝑚) ≤ 1
𝑙𝑚 −𝑚
𝑙𝑚∑︁
𝑗=1
Ψ−𝑠(𝑗) < Ψ−𝑠(𝑙𝑚 + 1). (19)
300 Shidlich A. L.
Note that in the terms of similar functionals, solutions of many problems
of approximation theory are formulated (see, eg, [4], [5] (Ch. XI), [10]
(Ch.VI), [11], [12], [13]). Therefore, the problem of finding such estimates
is interesting.
Let 𝑑 ∈ ℕ, 𝑀0, 𝑐1 and 𝑐2 be fixed positive numbers. Let also 𝜈 =
{𝜈𝑖}∞𝑖=0 be an increasing sequence of natural numbers such that 𝜈0 := 1
and for all 𝑛, greater than a certain number 𝑛0,
𝑀0(𝑛− 𝑐1)
𝑑 < 𝑉𝑛 :=
𝑛∑︁
𝑘=0
𝜈𝑘 ≤𝑀0(𝑛+ 𝑐2)
𝑑. (20)
Further, let 𝒮𝑑(𝑀0) = 𝒮𝑑(𝑀0, 𝑐1, 𝑐2) denote the set of all positive noni-
ncreasing step sequences Ψ, satisfying condition (15), which are represented
as
Ψ(𝑡) = 𝜓(𝑛), 𝑡 ∈ (𝑉𝑛−1, 𝑉𝑛], 𝑛 = 1, 2, . . . , (21)
where 𝜓 is the decreasing sequence of different values of the sequence Ψ.
Without loss of generality, we assume that the sequences 𝜓 are restri-
ctions of certain positive continuous functions 𝜓(𝑡) of continuous argument
𝑡 ≥ 1 on the set of natural numbers ℕ.
Лема 3.2. Let 𝑠 ∈ (0,∞), 𝑑 ∈ ℕ, the sequence Ψ belongs to the set
𝒮𝑑(𝑀0) and the sequence of its different values is a restriction of a certain
function 𝜓∈𝐵 on the set ℕ. Furthermore, in case 𝑠 > 1, we also assume
that for all 𝑡, greater than a certain number 𝑡0, the function 𝜓(𝑡) is convex
and satisfies condition (13) with 𝛽 = 𝑑/𝑠′. Then the following relation is
true:
𝐻𝑚(Ψ; 𝑠) ≍ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑠 . (22)
Let us note that for any Ψ ∈ 𝒮𝑑(𝑀0), condition (13) (with 𝛽 = 𝑑/𝑠′)
guarantees convergence of the series in (18), when 𝑠 > 1. Indeed, in this
case, for all 𝜏 ≥ 𝑡0,
|𝜓′(𝜏)|/𝜓(𝜏) ≥ 𝐾0/𝜏. (23)
Integrating each part of this relation in the range from 𝑡0 to 𝑡, 𝑡>𝑡0,
we obtain 𝜓(𝑡)≪ 𝑡−𝐾0 ≪ 𝑡−𝑑/𝑠
′
. Therefore, in view of (21) and (20), we
conclude
∞∑︁
𝑗=1
Ψ𝑠
′
(𝑗)=
∞∑︁
𝑛=1
𝜈𝑛𝜓
𝑠′(𝑛)≪
∞∑︁
𝑛=1
𝑛𝑑−1𝜓𝑠
′
(𝑛)≪
∞∑︁
𝑛=1
𝑛𝑑−1𝑛−𝑠
′𝐾0<∞.
Nonlinear approximation of the classes ... 301
Also note that in the case where the sequences Ψ are restrictions of
certain positive convex functions 𝜓(𝑡) of continuous argument 𝑡 ≥ 1 on
the set ℕ, the functionals 𝐻𝑛(Ψ, 𝑠) were considered in [13].
4.2. Proof of Lemma 3.2. First, consider the case 𝑠 ∈ (0, 1]. In view
of (16) and (21), the functionals 𝐻𝑚(Ψ; 𝑠) can be represented as
𝐻𝑚(Ψ, 𝑠) = sup
𝑙>𝑚
(𝑙 −𝑚)
(︂ 𝑛𝑙−1∑︁
𝑛=1
𝜈𝑛
𝜓𝑠(𝑛)
+
𝑙 − 𝑉𝑛𝑙−1
𝜓𝑠(𝑛𝑙)
)︂− 1
𝑠
=: ̃︀𝐻𝑚(𝜓, 𝑠),
where 𝑛𝑙 denote a number such that
𝑉𝑛𝑙−1 < 𝑙 ≤ 𝑉𝑛𝑙
. (24)
By virtue of (20), we see that
𝜈𝑛 ≍ 𝑛𝑑−1. (25)
and for all 𝑙 > 𝑚 ≥ 𝑛0,
(𝑙/𝑀0)
1
𝑑 − 𝑐2 ≤ 𝑛𝑙 < (𝑙/𝑀0)
1
𝑑 + 𝑐1 + 1. (26)
If 𝜓 ∈ 𝐵, then for any 𝑠 > 0 and 𝑙 = 2, 3, . . .,
𝑙𝑑
𝜓𝑠(𝑙)
≪ (𝑙/2)𝑑
𝜓𝑠(𝑙/2)
≪
∑︁
𝑙/2≤𝑛≤𝑙
𝑛𝑑−1
𝜓𝑠(𝑛)
≪
𝑙∑︁
𝑘=1
𝑛𝑑−1
𝜓𝑠(𝑛)
≪ 𝑙𝑑
𝜓𝑠(𝑙)
.
Therefore,
𝑙∑︁
𝑛=1
𝜈𝑛
𝜓𝑠(𝑛)
≍
𝑙∑︁
𝑛=1
𝑛𝑑−1
𝜓𝑠(𝑛)
≍ 𝑙𝑑
𝜓𝑠(𝑙)
. (27)
Further, by virtue of (26) and the definition of the set 𝐵, we see that
𝜓(𝑛𝑙)≍𝜓((𝑙/𝑀0)
1
𝑑 )≍𝜓(𝑙
1
𝑑 ). In view of (27), we conclude that
̃︀𝐻𝑚(𝜓, 𝑠) ≍ sup
𝑙>𝑚
(𝑙 −𝑚)
(︂
𝑛𝑑𝑙
𝜓𝑠(𝑛𝑙)
)︂− 1
𝑠
≍
≍ sup
𝑙>𝑚
𝜓(𝑙
1
𝑑 )(𝑙 −𝑚)/𝑙
1
𝑠 ≪ 𝜓(𝑚
1
𝑑 ) sup
𝑙>𝑚
(𝑙 −𝑚)/𝑙
1
𝑠 . (28)
For 𝑡 > 0, 𝑚 ∈ ℕ and 𝑠 ∈ (0, 1), the function (𝑡) = (𝑡, 𝑠) =
(𝑡−𝑚)/𝑡
1
𝑠 attains its maximal value at the point 𝑡* = 𝑚/(1− 𝑠), and
(𝑡*, 𝑠) = 𝑠(𝑚/(1− 𝑠))1−
1
𝑠 . (29)
302 Shidlich A. L.
If 𝑠 = 1, then the function (𝑡) = (𝑡; 1) is non-decreasing and tends to
1 as 𝑡 increases. Therefore,
sup
𝑡>0
(𝑡; 1) = sup
𝑡>0
(𝑡−𝑚)/𝑡 = lim
𝑡→+∞
(𝑡−𝑚)/𝑡 = 1. (30)
Combining (3)–(30), we obtain necessary upper estimates:
𝐻𝑚(Ψ, 𝑠) = ̃︀𝐻𝑚(𝜓, 𝑠) ≍ sup
𝑙>𝑚
𝜓(𝑙
1
𝑑 )(𝑙 −𝑚)/𝑙
1
𝑠 ≪ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑠 .
Taking into account (3) and the inclusion 𝜓 ∈ 𝐵, we also obtain the
lower estimates
𝐻𝑚(Ψ, 𝑠) = ̃︀𝐻𝑚(𝜓, 𝑠) ≫ 𝜓((2𝑚)
1
𝑑 )(2𝑚−𝑚)/(2𝑚)
1
𝑠 ≍ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑠 .
Now, we consider the case 𝑠 > 1. To simplify the notes, we set
𝑄𝑚(Ψ, 𝑙) := (𝑙 −𝑚)
(︂ 𝑙∑︁
𝑗=1
Ψ−𝑠(𝑗)
)︂−1
, 𝑙 ≥ 𝑚, 𝑙 ∈ ℕ.
For any 𝑙 > 𝑚, we have
𝑄𝑚(Ψ, 𝑙 + 1) = 𝑄𝑚(Ψ, 𝑙)+
+
(︂
Ψ𝑠(𝑙 + 1)−𝑄𝑚(Ψ, 𝑙)
)︂
Ψ−𝑠(𝑙 + 1)
(︂ 𝑙+1∑︁
𝑖=1
Ψ−𝑠(𝑖)
)︂−1
and
Ψ𝑠(𝑙 + 1) = 𝑄𝑚(Ψ, 𝑙 + 1)+
+
(︂
Ψ𝑠(𝑙 + 1)−𝑄𝑚(Ψ, 𝑙)
)︂ 𝑙∑︁
𝑗=1
Ψ−𝑠(𝑗)
(︂ 𝑙+1∑︁
𝑖=1
Ψ−𝑠(𝑖)
)︂−1
,
Therefore, in view of monotonicity of the function Ψ and relation (19),
we conclude that for all 𝑙 ≥ 𝑙𝑚, 𝑄𝑚(Ψ, 𝑙) > 𝑄𝑚(Ψ, 𝑙+1) > Ψ𝑠(𝑙+1) and
for all 𝑙 ∈ [𝑚, 𝑙𝑚), 𝑄𝑚(Ψ, 𝑙) ≤ 𝑄𝑚(Ψ, 𝑙+1) ≤ Ψ𝑠(𝑙+1). This yields that
sup
𝑙>𝑚
𝑄𝑛(Ψ, 𝑙) = 𝑄𝑚(Ψ, 𝑙𝑚). (31)
According to (19), we get Ψ(𝑙𝑚+1) > Ψ(𝑙𝑚). Hence, if the function Ψ(𝑡)
is represented in the form (21), then
𝑙𝑚 = 𝑉𝑘𝑙𝑚 =
𝑛𝑙𝑚∑︁
𝑖=0
𝜈𝑖 (32)
Nonlinear approximation of the classes ... 303
where 𝑛𝑙𝑚 is defined in (24) for 𝑙 = 𝑙𝑚. In this case, the functionals
𝐻𝑚(Ψ, 𝑠), 𝑠 ∈ (1,∞) can be represented as
𝐻𝑚(Ψ, 𝑠)=
(︂
(𝑙𝑚 − 𝑛)𝑠
′
(︂ 𝑛𝑙𝑚∑︁
𝑛=1
𝜈𝑛
𝜓𝑠(𝑛)
)︂− 𝑠′
𝑠
+
+
∞∑︁
𝑛=𝑛𝑙𝑚+1
𝜈𝑛𝜓
𝑠′(𝑛)
)︂ 1
𝑠′
:= ̃︀𝐻𝑚(𝜓, 𝑠), (33)
where
𝜓−𝑠(𝑛𝑙𝑚) ≤ 1
𝑙𝑚 −𝑚
𝑛𝑙𝑚∑︁
𝑗=1
𝜈𝑛
𝜓𝑠(𝑛)
< 𝜓−𝑠(𝑛𝑙𝑚 + 1). (34)
By virtue of (31), for the function
̃︀𝑄𝑚(𝜓, 𝑙) := (𝑙 −𝑚)
(︂ 𝑛𝑙−1∑︁
𝑛=1
𝜈𝑛
𝜓𝑠(𝑛)
+
𝑙 − 𝑉𝑛𝑙−1
𝜓𝑠(𝑛𝑙)
)︂−1
,
where 𝑛𝑙 is defined in (24), the following relation is satisfied:
sup
𝑙>𝑚
̃︀𝑄𝑚(𝜓, 𝑙) = ̃︀𝑄𝑚(𝜓, 𝑙𝑚) = (𝑙𝑚 −𝑚)
(︂ 𝑛𝑙𝑚∑︁
𝑛=1
𝜈𝑛
𝜓𝑠(𝑛)
)︂−1
. (35)
Then similarly to the case 𝑠 ∈ (0, 1], we show that̃︀𝑄𝑚(𝜓, 𝑙𝑚) = sup
𝑙>𝑚
̃︀𝑄𝑚(𝜓, 𝑙) ≍ 𝜓𝑠(𝑚
1
𝑑 ). (36)
Taking into account (34)–(36) and the definition of the set 𝐵, we see that
𝜓(𝑛𝑙𝑚) ≍ 𝜓(𝑚
1
𝑑 ). (37)
Since 𝜓 ∈ 𝐵, then in view of (25), we conclude that for any 𝑙 ∈ ℕ,
∞∑︁
𝑛=𝑙+1
𝜈𝑛𝜓
𝑠′(𝑛) ≫
2𝑙∑︁
𝑛=𝑙+1
𝑛𝑑−1𝜓𝑠
′
(𝑛) ≫ 𝑙𝑑𝜓𝑠
′
(𝑙). (38)
By virtue of (13), the function 𝑡𝑑𝜓𝑠
′
(𝑡) decreases to zero at 𝑡 > 𝑡0.
Therefore,
∞∑︁
𝑛=𝑙+1
𝜈𝑛𝜓
𝑠′(𝑛) ≪
∞∑︁
𝑛=𝑙+1
𝑛𝑑−1𝜓𝑠
′
(𝑛) ≪
∞∫︁
𝑙
𝑡𝑑−1𝜓𝑠
′
(𝑡)𝑑𝑡 =: 𝒥𝑙.
304 Shidlich A. L.
Integrating by parts, we obtain
𝒥𝑙 ≤
1
𝐾0
∞∫︁
𝑙
𝑡𝑑𝜓𝑠
′−1(𝑡)|𝜓′(𝑡)|𝑑𝑡 = 𝑙𝑑𝜓𝑠
′
(𝑙)
𝐾0𝑠′
+
𝑑
𝐾0𝑠′
𝒥𝑙.
Then in view of (13), we see that 𝒥𝑙 ≪ 𝑙𝑑𝜓𝑠
′
(𝑙). Hence, we get the
estimate
∞∑︁
𝑛=𝑙+1
𝜈𝑛𝜓
𝑠′(𝑛) ≪ 𝑙𝑑𝜓𝑠
′
(𝑙), (39)
that together with (38) proves the relation
∞∑︁
𝑛=𝑙+1
𝜈𝑛𝜓
𝑠′(𝑛) ≍ 𝑙𝑑𝜓𝑠
′
(𝑙). (40)
In the end of the proof, let us show that
𝑛𝑙𝑚 ≍ 𝑚
1
𝑑 . (41)
Indeed, by virtue of (32) and (26), we see that ̃︀𝑚 := (𝑚/𝑀0)
1
𝑑 − 𝑐2 ≤
(𝑙𝑚/𝑀0)
1
𝑑 − 𝑐2 ≤ 𝑛𝑙𝑚 . On the other hand, integrating each part of
(23) in the range from ̃︀𝑚 to 𝑛𝑙𝑚 , ̃︀𝑚 > 𝑡0, we obtain 𝜓(̃︀𝑚)/𝜓(𝑛𝑙𝑚) ≥
(𝑛𝑙𝑚/̃︀𝑚)𝐾0 .Therefore, in view of (37) and (12), we see that ̃︀𝑚≫ 𝑛𝑙𝑚 and
relation (41) is true.
Thus, combining the relations (3), (27), (36), (37), (40) and (41) we
obtain the estimate (22), i.e.,
𝐻𝑚(Ψ, 𝑠) = ̃︀𝐻𝑚(𝜓, 𝑠) ≍
(︂
𝜓𝑠𝑠
′
(𝑚
1
𝑑 ) ·
(︂
𝑚/𝜓𝑠(𝑚
1
𝑑 )
)︂(1− 1
𝑠 )𝑠
′
+
+𝑚𝜓𝑠
′
(𝑚
1
𝑑 )
)︂1/𝑠′
≍ 𝜓(𝑚
1
𝑑 )𝑚
1
𝑠′ ≍ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑠 .
4.3. In this section, we apply Lemma 3.2 for estimation of the exact
upper bounds of the quantities (3)–(5) on the classes ℱ𝜓
𝑞,𝑟 in the spaces
𝑆𝑝(𝕋𝑑). For all 0<𝑝, 𝑞 <∞, the exact values of the quantities 𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝 ,
as well as the exact values of the quantities 𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝 and 𝜎⊥
𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝
(due to (8)) were obtained by A.I. Stepanets ( [4], [5] (Ch. XI)). In parti-
cular, from Theorem 9.1 of [5] (Ch. XI), it follows that for all 𝑚 ∈ ℕ,
Nonlinear approximation of the classes ... 305
0 < 𝑞 ≤ 𝑝 < ∞ and for any positive function 𝜓 = 𝜓(𝑡), 𝑡 ≥ 0, satisfying
condition (11),
𝜎𝑝𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝 = sup
𝑙>𝑚
(𝑙 −𝑚)
(︁ 𝑙∑︁
𝑗=1
𝜓−𝑞(𝑗)
)︁− 𝑝
𝑞
, (42)
where 𝜓 = 𝜓(𝑗), 𝑗 = 1, 2, . . ., is the decreasing rearrangement of the
system of numbers 𝜓(|𝑘|𝑟), 𝑘∈ℤ𝑑. If 0 < 𝑝 < 𝑞 < ∞ and the positive
function 𝜓 = 𝜓(𝑡), 𝑡 ≥ 0, satisfies the condition∑︁
𝑘∈ℤ𝑑
𝜓
𝑝𝑞
𝑞−𝑝 (|𝑘|𝑟) <∞, (43)
then from Theorem 9.4 of [5] (Ch. XI) it follows that
𝜎𝑝𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝=
(︂
(𝑙𝑚 −𝑚)
𝑞
𝑞−𝑝
(︂ 𝑙𝑚∑︁
𝑗=1
𝜓−𝑞(𝑗)
)︂ 𝑝
𝑞−𝑝
+
∞∑︁
𝑗=𝑙𝑚+1
𝜓
𝑝𝑞
𝑞−𝑝 (𝑗)
)︂ 𝑞−𝑝
𝑞
, (44)
where 𝜓 = 𝜓(𝑗), 𝑗 = 1, 2, . . ., is the decreasing rearrangement of the
system of numbers 𝜓(|𝑘|𝑟), 𝑘∈ℤ𝑑, and the number 𝑙𝑚 is defined by
𝜓−𝑞(𝑙𝑚) ≤ 1
𝑙𝑚 −𝑚
𝑙𝑚∑︁
𝑗=1
𝜓−𝑞(𝑗) < 𝜓−𝑞(𝑙𝑚 + 1).
Taking into account notation (16) and (17), we can write relations
(42) and (44) as
𝜎𝑝𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝 = 𝐻𝑚(𝜓𝑝, 𝑞/𝑝), 0 < 𝑝, 𝑞 <∞.
Furthermore, if the number 𝑉𝑛 := |Δ𝑑
𝑛,𝑟| of elements of the set
Δ𝑑
𝑛,𝑟 := {𝑘 ∈ ℤ𝑑 : |𝑘|𝑟 ≤ 𝑛, 𝑛 = 0, 1, . . .}.
for all sufficiently large 𝑛 ∈ ℕ (𝑛 is greater than some positive number
𝑛0) satisfies the following condition:
𝑀𝑟(𝑛− 𝑐1)
𝑑 < 𝑉𝑛 = |Δ𝑑
𝑛,𝑟| ≤𝑀𝑟(𝑛+ 𝑐2)
𝑑, (45)
where 𝑀𝑟, 𝑐1 and 𝑐2 are certain positive constants, then the sequence
𝜓 = 𝜓(𝑗), 𝑗 = 1, 2, . . ., belongs to the set 𝒮𝑑(𝑀𝑟) = 𝒮𝑑(𝑀𝑟, 𝑐1, 𝑐2). Thus,
by virtue of Lemma 3.2, we can formulate the following statement:
306 Shidlich A. L.
Assertion 4.1. Assume that 0 < 𝑟 ≤ ∞, 0 < 𝑝 < ∞, 0 < 𝑞 < ∞,
condition (45) holds, 𝜓 ∈ 𝐵 and in the case 𝑝 < 𝑞, moreover, for all 𝑡,
larger than a certain number 𝑡0, 𝜓𝑝 is convex and condition (13) holds
with 𝛽 = 𝑑( 1𝑝 −
1
𝑞 ). Then
𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝 ≍ 𝜓(𝑚
1
𝑑 )𝑚
1
𝑝−
1
𝑞 . (46)
It is clear that in the case 𝑟 = ∞, condition (45) is satisfied and 𝑀∞=
𝑣𝑜𝑙{𝑘∈ℝ𝑑 : |𝑘|∞ ≤ 1}=2𝑑. If 𝑟=1, then𝑀1 = 𝑣𝑜𝑙{𝑘∈ℝ𝑑 : |𝑘|1 ≤ 1}=2𝑑/𝑑!.
Unfortunately, we do not know whether a similar relation for other 𝑟 is
valid. However, one can formulate the following corollary:
Наслiдок 3.3. Assume that 0 < 𝑝 < ∞, 0 < 𝑞 < ∞, condition (45)
holds, 𝜓 ∈ 𝐵 and in the case 𝑝 < 𝑞, moreover, for all 𝑡, larger than
a certain number 𝑡0, 𝜓𝑝 is convex and condition (13) holds with 𝛽 =
𝑑( 1𝑝 −
1
𝑞 ). Then for all 1 ≤ 𝑟 ≤ ∞, relation (46) is true.
Indeed, for any numbers 𝑟 ∈ [1,∞], 0 < 𝑞 < ∞ and for any positive
decreasing function 𝜓
ℱ𝜓
𝑞,1 ⊂ ℱ𝜓
𝑞,𝑟 ⊂ ℱ𝜓
𝑞,∞. (47)
Therefore, if conditions of Corollary 3.3 are satisfied, then for all 𝑟 ∈
[1,∞],
𝜓(𝑚
1
𝑑 )
𝑚
1
𝑞−
1
𝑝
≪ 𝜎𝑚(ℱ𝜓
𝑞,1)𝑆𝑝 ≪ 𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝑆𝑝 ≪ 𝜎𝑚(ℱ𝜓
𝑞,∞)
𝑆𝑝 ≪ 𝜓(𝑚
1
𝑑 )
𝑚
1
𝑞−
1
𝑝
.
4 Proof of Theorems 2.6 and 2.7.
5.1. Proof of Theorem 2.6.
Upper estimates. In the case 1 ≤ 𝑝 ≤ 2, by virtue of (7) and (8), we
have
𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≪ 𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≪ 𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿2
≪ 𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝑆2 .
Thus, to obtain the required upper estimates, it is sufficient to use Corollary
3.3 for 𝑆𝑝 = 𝑆2.
If 2 ≤ 𝑝 < ∞, then using the Hausdorff–Young inequality (see, for
example, [14] (p. 16)), relation (8) and Corollary 3.3, we get
𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≪ 𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≪
Nonlinear approximation of the classes ... 307
≪ 𝐺𝑚(ℱ𝜓
𝑞,𝑟)
𝑆𝑝′
≪ 𝜎𝑚(ℱ𝜓
𝑞,𝑟)
𝑆𝑝′
≪ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑝−
1
𝑞 .
Lower estimate. Let 𝒯𝑛, 𝑛 ∈ ℕ, denote the set of all polynomials of
the form
𝑇𝑛 =
∑︁
|𝑘|∞≤𝑛
̂︀𝑇𝑛(𝑘)𝑒𝑘,
and let 𝒜𝑞(𝒯𝑛), 0 < 𝑞 <∞, denote the subset of all polynomials 𝑇𝑚 ∈ 𝒯𝑚
such that ||𝑇 ||
𝑆𝑞 ≤ 1. From Theorem 5.2 of [1], it follows that for any
0 < 𝑞 <∞, 1 ≤ 𝑝 <∞, 𝑛 = 1, 2, . . . and 𝑚 = ((2𝑛+ 1)𝑑 − 1)/2,
𝜎𝑚(𝒜𝑞(𝒯𝑛))𝐿𝑝
≥ 𝐾𝑚1/2−1/𝑞.
For a fixed 𝑛 ∈ ℕ, consider the set
𝜓(𝑑𝑛)𝒜𝑞(𝒯𝑛)={𝑇∈𝒯𝑛 : ||𝑇 ||𝑆𝑞≤𝜓(𝑑𝑛)}.
Due to monotonicity 𝜓, for any polynomial 𝑇 ∈ 𝜓(𝑑𝑛)𝒜𝑞(𝒯𝑛), we have∑︁
𝑘∈ℤ𝑑
| ̂︀𝑇 (𝑘)/𝜓(|𝑘|1)|𝑞 ≤ ∑︁
|𝑘|∞≤𝑛
| ̂︀𝑇 (𝑘)/𝜓(𝑑|𝑘|∞)|𝑞 ≤
≤
∑︁
|𝑘|∞≤𝑛
| ̂︀𝑇 (𝑘)/𝜓(𝑑𝑛)|𝑞 ≤ 1
Therefore, 𝜓(𝑑𝑛)𝒜𝑞(𝒯𝑛) is contained in the set ℱ𝜓
𝑞,1. In view of definition
of the set 𝐵, for all 𝑛 = 1, 2, . . . and 𝑚 = ((2𝑛+ 1)𝑑 − 1)/2, we obtain
𝜎𝑚(ℱ𝜓
𝑞,1)𝐿𝑝
≥𝜎𝑚(𝜓(𝑑𝑛)𝒜𝑞(𝒯𝑛))𝐿𝑝
≫𝜓(𝑑𝑛)𝑚
1
2−
1
𝑞 ≫𝜓(𝑚
1
𝑑 )𝑚
1
2−
1
𝑞 .
Taking into account the relations (7) and (47), monotonicity of the quanti-
ty 𝜎𝑚 and inclusion 𝜓 ∈ 𝐵, we see that for all 1 ≤ 𝑝 ≤ ∞ and all
1 ≤ 𝑟 ≤ ∞,
𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≫ 𝜎⊥
𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≫ 𝜎𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≫
≫ 𝜎𝑚(ℱ𝜓
𝑞,1)𝐿𝑝
≫ 𝜓(𝑚
1
𝑑 )𝑚
1
2−
1
𝑞 .
In the case 2 < 𝑝 <∞, for the quantities 𝜎⊥
𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
and 𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
,
this estimate can be improved. For this purpose, consider the function
𝑓1 =
∑︁
|𝑘|1≤𝑛𝑚
̂︀𝑓1(𝑘)𝑒𝑘 = 𝐶𝑚
∑︁
|𝑘|1≤𝑛𝑚
𝑒𝑘,
308 Shidlich A. L.
where 𝐶−𝑞
𝑚 :=
∑︀
|𝑗|1≤𝑛𝑚
𝜓−𝑞(|𝑗|1), 𝑛𝑚 := [(2𝑚/𝑀1)
1/𝑑] and 𝑀1 = 2𝑑/𝑑!.
It is obviously that 𝑓1 ∈ ℱ𝜓
𝑞,1. Due to (45), for all sufficiently large 𝑛, the
number |̃︀Δ𝑑
𝑛,1| of elements of the set ̃︀Δ𝑑
𝑛,1 := {𝑘 ∈ ℤ𝑑 : |𝑘|1 = 𝑛, 𝑛 ∈ ℕ}
satisfies the condition
𝑀1(𝑛− 𝑐3)
𝑑−1 < |̃︀Δ𝑑
𝑛,1| = |Δ𝑑
𝑛,1| − |Δ𝑑
𝑛−1,1| ≤𝑀1(𝑛− 𝑐4)
𝑑−1,
where 𝑐3 and 𝑐4 are some positive numbers. Therefore, by virtue of (27),
𝐶−𝑞
𝑚 ≍
𝑛𝑚∑︁
𝑛=1
𝑛𝑑−1
𝜓𝑞(𝑛)
≍ 𝑛𝑑𝑚
𝜓𝑞(𝑛𝑚)
≍ 𝑚
𝜓𝑞(𝑚
1
𝑑 )
.
For any collection 𝛾𝑛 ⊂ ℤ𝑑, using Nikol’skii’s inequality [15] and (45), we
obtain⃒⃒⃒⃒⃒⃒⃒⃒
𝑓1 −
∑︁
𝑘∈𝛾𝑛
̂︀𝑓1(𝑘)𝑒𝑘 ⃒⃒⃒⃒⃒⃒⃒⃒
𝐿𝑝
≫ 𝐶𝑚𝑚
− 1
𝑝
⃒⃒⃒⃒⃒⃒⃒⃒ ∑︁
|𝑘|1≤𝑛𝑚: 𝑘/∈𝛾𝑛
𝑒𝑘
⃒⃒⃒⃒⃒⃒⃒⃒
𝐿∞
≍ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑝−
1
𝑞 .
Therefore, for all 2 ≤ 𝑝 <∞, the following estimates are true:
𝐺𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≫ 𝜎⊥
𝑚(ℱ𝜓
𝑞,𝑟)𝐿𝑝
≫ 𝜎⊥
𝑚(ℱ𝜓
𝑞,1)𝐿𝑝
≫
≫ 𝜎⊥
𝑚(𝑓1)𝐿𝑝
≫ 𝜓(𝑚
1
𝑑 )𝑚1− 1
𝑝−
1
𝑞 .
Theorem 2.6 is proved.
5.2. Proof of Theorem 2.7 is similar to the proof of the upper
estimates in Theorem 6.1 [1].
Лема 4.3. [1] For each 0 < 𝑞 ≤ ∞, each 𝑛 = 1, 2, . . ., and 1 ≤ 𝑚 ≤
(2𝑛+ 1)𝑑, we have
𝜎𝑚(𝒜𝑞(𝒯𝑛))𝐿∞
≤ 𝐶𝑚
1
2−
1
𝑞𝐿(𝑛𝑑/𝑚), 0 < 𝑞 ≤ 1, (48)
where 𝐿(𝑥) = (1 + (ln𝑥)+)
1/2 and
𝜎𝑚(𝒜𝑞(𝒯𝑛))𝐿∞
≤ 𝐶𝑛𝑑−
𝑑
𝑞𝑚− 1
2𝐿(𝑛𝑑/𝑚), 1 < 𝑞 ≤ ∞, (49)
with 𝐶 depending only on 𝑞 and 𝑑.
Nonlinear approximation of the classes ... 309
For any 𝑓 ∈ ℱ𝜓
𝑞,∞, we use the decomposition
𝑓 =
∞∑︁
𝑗=0
𝑓𝑗 ,
where 𝑓𝑗 :=
∑︀
2𝑗−1≤|𝑘|∞<2𝑗
̂︀𝑓(𝑘)𝑒𝑘, 𝑗 ≥ 1, and 𝑓0 = ̂︀𝑓(0). We note that
𝑓𝑗/𝜓(2
𝑗−1) ∈ 𝒜𝑞(𝒯2𝑗 ), 𝑗 = 1, 2, . . . (50)
For any 𝑁 = 1, 2, . . ., we approximate 𝑓 as follows. Let 𝑁0 be the
largest integer 𝑗 such that 𝑚𝑗 := [(𝑗 −𝑁)−22𝑁𝑑] ≥ 1 (with [𝑥] denoting
the greatest integer in 𝑥), i.e. 𝑁0 = [2
𝑁𝑑
2 +𝑁 ]. If 𝑗 ≤ 𝑁 , we set 𝑃𝑗 := 𝑓𝑗 .
If 𝑁 < 𝑗 ≤ 𝑁0, then by virtue of (48), (49) and (50), there is polynomial
𝑃𝑗 ∈ Σ𝑚𝑗 such that
||𝑓𝑗 − 𝑃𝑗 ||𝐿𝑝
≪ 𝑚
1
2−
1
𝑞
𝑗 𝐿(2𝑗𝑑/𝑚𝑗)𝜓(2
𝑗−1), 0 < 𝑞 ≤ 1. (51)
and
||𝑓𝑗 − 𝑃𝑗 ||𝐿𝑝
≪ 2𝑗(𝑑−
𝑑
𝑞 )𝑚
− 1
2
𝑗 𝐿(2𝑗𝑑/𝑚𝑗)𝜓(2
𝑗−1), 1 < 𝑞 <∞. (52)
Set 𝑃 =
𝑁0∑︀
𝑗=0
𝑃𝑘. Since
(2 · 2𝑁 + 1)𝑑 +
𝑁0∑︁
𝑗=𝑁+1
(𝑗 −𝑁)−22𝑁𝑑 ≤ 𝑎2𝑁𝑑,
where 𝑎 depends only on 𝑑, then 𝑃 is a linear combination of at most
𝑎2𝑁𝑑 exponentials 𝑒𝑘. Hence, 𝑃 is in Σ𝑎2𝑁𝑑 . We also have
||𝑓 − 𝑃 ||
𝐿𝑝
≤
𝑁0∑︁
𝑗=𝑁+1
||𝑓𝑗 − 𝑃𝑗 ||𝐿𝑝
+
∞∑︁
𝑗=𝑁0+1
||𝑓𝑗 ||𝐿𝑝
=: 𝑆1 + 𝑆2. (53)
For all 𝑥 ≥ 1, we have [𝑥] ≥ 𝑥/2. Therefore, for sufficiently large 𝑁 and
𝑁 < 𝑗 ≤ 𝑁0, from the definition of 𝐿(𝑥), we have
𝐿(2𝑗𝑑/𝑚𝑗) ≤ (1 + ln(2𝑑(𝑗−𝑁)+1(𝑗 −𝑁)2))
1
2 ≪ (𝑗 −𝑁)
1
2 . (54)
First, consider the case 0 < 𝑞 ≤ 1. Reasoning similar to the proof of
(39), it is easy to show that if the function 𝜓 belongs to the set 𝐵 and
310 Shidlich A. L.
for all 𝑡, larger than a certain number 𝑡0, 𝜓 is convex and it satisfies
condition (13) with a fixed 𝛽 ≥ 0, then for any 𝛼 ∈ ℝ and sufficiently
large 𝑡 > 𝑁 , the function 𝛼,𝛽(𝑡) := 2𝛽𝑡(𝑡 − 𝑁)𝛼𝜓(2𝑡−1) decreases to
zero, as well as
∞∑︁
𝑗=𝑁+1
2𝛽𝑗(𝑗 −𝑁)𝛼𝜓(2𝑗−1) ≪ 2𝛽𝑁𝜓(2𝑁 ). (55)
In this case, 𝛽 = 0. By virtue of (51), (54) and (55), we obtain the
estimate of the first sum 𝑆1 in (53):
𝑆1 ≪
∞∑︁
𝑗=𝑁+1
(𝑗 −𝑁)2(
1
𝑞−
1
2 )2−𝑁𝑑(
1
𝑞−
1
2 )(𝑗 −𝑁)
1
2𝜓(2𝑗−1) ≪
≪ 2−𝑁( 𝑑
𝑞−
𝑑
2 )
∞∑︁
𝑗=𝑁+1
(𝑗 −𝑁)
2
𝑞−
1
2𝜓(2𝑗−1) ≪ 2−𝑁( 𝑑
𝑞−
𝑑
2 )𝜓(2𝑁 ). (56)
To estimate 𝑆2, we note that from (50)
𝑆2 ≤
∞∑︁
𝑗=𝑁0+1
||𝑓𝑗 ||𝐿∞
≤
∞∑︁
𝑗=𝑁0+1
(︁ ∑︁
2𝑗−1≤|𝑘|∞<2𝑗
| ̂︀𝑓(𝑘)|)︁ ≤
≤
∞∑︁
𝑗=𝑁0+1
||𝑓𝑗 ||𝑆𝑞 ≪
∞∑︁
𝑗=𝑁0+1
𝜓(2𝑗−1) ≪ 𝜓(2𝑁0).
Further, let us note that if for all 𝑡, larger than a certain number 𝑡0,
𝜓 is convex and satisfies the condition (13), then for any 𝛼 > 0, we have
𝜓(2𝑁(𝛼+1)) ≪ 𝜓(2𝑁 )2−𝑁𝛼.
From the definition of 𝑁0, we have 𝑁0 ≥ 𝑁 + 2
𝑁𝑑
2 − 1. It follows
that if 𝑁 is sufficiently large (depending only on 𝑑 and 𝑞), then 𝑁0 ≥
𝑁(1 + 𝑑/𝑞 − 𝑑/2). Hence,
𝑆2 ≪ 𝜓(2𝑁(1+ 𝑑
𝑞−
𝑑
2 )) ≪ 2−𝑁( 𝑑
𝑞−
𝑑
2 )𝜓(2𝑁 ).
Using this and (56) in (53), we find that
𝜎𝑎2𝑁𝑑(𝑓)
𝐿𝑝
≤ ||𝑓 − 𝑃 ||
𝐿𝑝
≪ 2−𝑁( 𝑑
𝑞−
𝑑
2 )𝜓(2𝑁 ). (57)
Nonlinear approximation of the classes ... 311
In the case 1 < 𝑞 < ∞, condition (13) is satisfied with 𝛽 = 𝑑 − 𝑑/𝑞.
By virtue of (52), (54) and (55), we have
𝑆1≪ 2−
𝑁
2
∞∑︁
𝑗=𝑁+1
2𝑗(𝑑−
𝑑
𝑞 )(𝑗 −𝑁)
3
2𝜓(2𝑗−1) ≪ 𝜓(2𝑁 )2−𝑁( 𝑑
𝑞−
𝑑
2 ). (58)
To estimate 𝑆2, we use Hölder’s inequality, (50), (55) and the inequalities
𝑁0 ≥ 𝑁+2
𝑁𝑑
2 −1 and 𝛼,𝛽(𝑁0) ≤ 𝛼,𝛽(𝑁+1) with 𝛼 = 1 and 𝛽 = 𝑑−𝑑/𝑞,
𝑆2 ≤
∞∑︁
𝑗=𝑁0+1
||𝑓𝑗 ||𝐿∞
≤
∞∑︁
𝑗=𝑁0+1
𝜓(2𝑗−1)
(︁ ∑︁
2𝑗−1≤|𝑘|∞<2𝑗
⃒⃒⃒⃒ ̂︀𝑓(𝑘)
𝜓(2𝑗−1)
⃒⃒⃒⃒)︁
≤
≤
∞∑︁
𝑗=𝑁0+1
𝜓(2𝑗−1)2(𝑗−1)(𝑑− 𝑑
𝑞 ) ≪ 𝜓(2𝑁0)2𝑁0(𝑑− 𝑑
𝑞 ) ≪ 𝜓(2𝑁 )2𝑁( 𝑑
2−
𝑑
𝑞 ).
Using this and (58) in (53), we see that in this case, relation (57) is also
true. Therefore, the upper estimate in (13) follows from the monotonicity
of 𝜎𝑚 and inclusion 𝜓 ∈ 𝐵.
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1. Луковський І.О., Гаврилюк І.О., Василик В.Б., Ситник Д.О.
2. Біленко В. І., Божонок К. В., Дзядик С. Ю., Стеля О. Б.
Інтегро–апроксимаційний алгоритм
Вступ
Постановка задачі
Алгоритм
Похибка алгоритму
Застосування a–методу для алгебраїчно–нелінійних рівнянь гіперболічного типу
Задача Дирихле для алгебраїчно–нелінійних рівнянь еліптичного типу на прямокутнику
Наближений розв'язок початкової задачі для алгебраїчно–нелінійних рівнянь параболічного типу на прямокутнику
Сплайн–алгоритм
Монотонна схема для рівняння конвекції–дифузії
Висновки
3. Василик В.Б., Макаров В.Л., Ситник Д.О.
Вступ
Регуляризація та явне зображення розв'язку
Вибір контуру інтегрування
Чисельний метод
4. Веселовська Г.М.
5. Грушковская В.В.
Введение
Построение модельной системы
Условия устойчивости
Оценка скорости убывания решений
Пример: оценка скорости затухания колебаний маятниковой системы с частичной диссипацией
Выводы
6. Дзюбенко Г.А.
Вступ
Допоміжні факти
Доведення Теореми ??
7. Діденко Ю.Ф., Денисенко В.І.
8. Елишевич М.А.
Постановка задачи
Полученный результат
Пример
9. Константинов А.В., Лимарченко О.С., Кинебас К.В., Паранькина О.Ю.
Введение
Объект исследования и математическая модель
Результаты вычислительных экспериментов
Выводы
10. Мазко О.Г., Кусій С.М.
Вступ
Допоміжні твердження
Лінійні системи з керованими і спостережуваними виходами
Статичний регулятор по вимірюваному виходу
Динамічний регулятор
Алгоритм побудови динамічного регулятора
Приклад. Гасіння коливань лінійного осцилятора.
Висновок
11. Працьовитий М. В., Маслова Ю. П.
Вступ
Функція Радемахера і ряди Уолша
Узагальнення функцій Радемахера
Узагальнення функцій Уолша
12. Працьовитий М.В., Чуйков А.С.
Вступ
Оператори лівостороннього та правостороннього зсуву елементів ланцюгового дробу
Інші функції, пов'язані з оператором T(x)
13. Новицький В.В., Зінчук М.О., Коломійчук О.П., Тетерятник О.В.
Вступ
Оптимальне керування лінійними неперервними майже консервативними системами
Оптимальне керування лінійними дискретними майже консервативними системами
14. Осауленко Р. Ю.
Вступ
Перетворення, які зберігають хвости Qs–зображення чисел
Група перетворень, які зберігають частоти цифр Qs–зображення числа
Приклад функції, яка зберігає частоти, але не зберігає хвости зображення Qs-ірраціональних чисел
15. Слинько В.І., Кравчук С.В.
Постановка задачі.
Основний результат.
Умови стійкості
16. Солодун А. В.
Постановка задачи
Численные результаты
17. Ситник Д.О.
Вступ
Sinc–апроксимація
Sinc-апроксимація функції за її значеннями поза інтерполяційною сіткою
18. Сосницький С.П.
Вступ
Про рівняння збуреного руху в околі стаціонарних лагранжевих трикутників
Теорема про орбітальну нестійкість лагранжевих стаціонарних рухів у задачі трьох тіл
Висновок
19. Сосницький С.П.
Вступ
Про достатні умови відсутності осцилюючих симетричних рухів
20. Чернецька Л.О.
21. Timokha A.N.
Statement
Asymptotic steady-state solutions of (??)–(??)
The reciprocating excitation type
The axisymmetric elliptic excitation type
The oblique elliptic excitation type
Conclusions
22. Shlepakov L.N.
Main relationships for a non-inflated system
Construction of enlarged systems
Defining the task mathematical programming
Case of multiple channels with same probability characteristics in the same system of channels.
23. Shidlich A.L.
Approximative characteristics
Main results
Order estimates for some functionals and their applications
Proof of Theorems ?? and ??.
24. Луковський І.О., Стороженко В.О.
25. Луковський І.О., Пустовойтов М.О.
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| id | oai:trim.imath.kiev.ua:article-62 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English |
| last_indexed | 2026-08-04T01:01:51Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
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| spelling | oai:trim.imath.kiev.ua:article-622018-02-13T11:57:10Z Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics Нелинейная аппроксимация классов $F_{q,r}^{\psi}$ функций нескольких переменных в интегральной метрике Нелінійна апроксимація класів $F_{q,r}^{\psi}$ функций нескольких переменніх в интегральній метриці Shidlich, A. L. Шыдлич, А. Л. Шидліч, А. Л. In the paper, exact order estimates of nonlinear approximative characteristics (such as the best $m$-member of the trigonometric approximation, better than the $m$-th term of the orthogonal trigonometric approximation, approximation of the $m$-member of the grid by polynomials) are found in the class $ℱ_{q,r}^{\psi}$ of functions of several variables in the integral metric. В работе найдены точные порядковые оценки нелинейных аппроксимативных характеристик (таких как лучше $m$–членне тригонометрическое приближения, лучше $m$–членне ортогональное тригонометрическое приближения, приближение $m$–членнимы гриди полиномами) классов&nbsp;$ℱ_{q,r}^{\psi}$ функций многих переменных в интегральной метрике. У роботi знайдено точнi порядковi оцiнки нелiнiйних апроксимативних характеристик (таких як найкраще $m$-членне тригонометричне наближення, найкраще $m$-членне ортогональне тригонометричне наближення, наближення $m$-членними грiдi полiномами) класiв $ℱ_{q,r}^{\psi}$ функцiй багатьох змiнних у iнтегральнiй метрицi. Інститут математики НАН України 2017-12-22 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/62 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 13 No. 3 (2016): Mathematical problems of mechanics and computational mathematics; 293-312 Сборник Трудов Института математики НАН Украины; Том 13 № 3 (2016): Математичні проблеми механіки та обчислювальної математики; 293-312 Збірник Праць Інституту математики НАН України; Том 13 № 3 (2016): Математичні проблеми механіки та обчислювальної математики; 293-312 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/62/57 Авторське право (c) 2016 Праці Інституту математики НАН України |
| spellingShingle | Shidlich, A. L. Шыдлич, А. Л. Шидліч, А. Л. Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| title | Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| title_alt | Нелинейная аппроксимация классов $F_{q,r}^{\psi}$ функций нескольких переменных в интегральной метрике Нелінійна апроксимація класів $F_{q,r}^{\psi}$ функций нескольких переменніх в интегральній метриці |
| title_full | Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| title_fullStr | Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| title_full_unstemmed | Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| title_short | Nonlinear approximation of the classes $F_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| title_sort | nonlinear approximation of the classes $f_{q,r}^{\psi}$ of functions of several variables in the integral metrics |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/62 |
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