Abstract
In this paper, we consider the generalized Lorentz space of periodic functions of several variables and the Nikol’skii–Besov space of functions. The article establishes a sufficient condition for a function to belong from one generalized Lorentz space to another space in terms of the difference of the partial sums of the Fourier series of a given function. Exact in order estimates of the best approximation by trigonometric polynomials of functions of the Nikol’skii–Besov class are obtained.
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1 Introduction
Let \({\mathbf {R}}^{m}\) be a m – dimensional Euclidean space of points \({\overline{x}} = (x_{1}, \dots , x_{m})\) with real coordinates; \({\mathbf {I}}^{m} = \{{\overline{x}}\in {\mathbf {R}}^{m}; 0 \le x_{j}\le 1;\ j=1,\ldots ,m\}\) — m – dimensional cube.
Definition 1.1
(see [21, Chapter 2, Sect. 2]). Two nonnegative Lebesgue measurable functions f, g are called equimeasurable if
where \(\mu e\) — Lebesgue measure of the set \(e \subset {\mathbf {I}}^{m}\).
Let X be a Banach space of Lebesgue measurable functions on \({\mathbf {I}}^{m}\) of functions f with the norm \(\Vert f\Vert _{X}\). The space X is called symmetric
-
(1)
if \(|f({\overline{x}})| \le |g({\overline{x}})|\) almost everywhere on \({\mathbf {I}}^{m}\) and \(g \in X\), then \(f \in X\) and \(\Vert f\Vert _{X}\le \Vert g\Vert _{X}\);
-
(2)
if \(f \in X\) and |f| , |g| are equimeasurable, then \(g \in X\) and \(\Vert f\Vert _{X} = \Vert g\Vert _{X} \) (see [21, Chapter 2, Sect. 4]).
The norm \(\Vert \chi _{e}\Vert _{X}\) of the characteristic function \(\chi _{e}(t)\) of the measurable set \( e \subset {\mathbf {I}}^{m}\) is called is the fundamental function of space X and is denoted by \(\varphi (\mu e) = \Vert \chi _{e}\Vert _{X}\). Further, the symmetric space X with the fundamental function \(\varphi \) will be denoted by \(X(\varphi )\).
It is known that the fundamental function of the symmetric space X is the function \(\varphi (t) = \Vert \chi _{[0, t]}\Vert _{X}\) defined on the interval [0, 1]. She is a concave, non-decreasing, continuous function on [0, 1], and \(\varphi (0) = 0 \) (see [21, p. 70, 137, 164]). Such functions are called \(\varPhi \) - functions.
For this function \(\varphi (t),\) \(t \in [0,1]\), put \( \alpha _{\varphi }={{{\underline{\lim }}}}_{t\rightarrow 0}\frac{\varphi (2t)}{\varphi (t)},\quad \beta _{\varphi }={\overline{\lim }}_{t\rightarrow 0}\frac{\varphi (2t)}{\varphi (t)}. \) It is known that for any symmetric space \(X(\varphi )\) we have inequalities \(1 \le \alpha _{\varphi } \le \beta _{\varphi } \le 2\) (see [26]).
One example of a symmetric space is \(L_{q}({\mathbf {T}}^{m})\) — Lebesgue space \(2\pi \) periodic for each variable of the function f with norm (see [24, Chapter 1, Sect. 1.1])
Here and in after, \({\mathbf {T}}^{m} = [0, 2 \pi ]^{m}\).
The space \(C({\mathbf {T}}^{m})\) consists of continuous functions f with the norm \(\Vert f\Vert _{\infty }=\max \limits _{{\overline{x}} \in {\mathbf {I}}^{m}}|f(2\pi {\overline{x}})|\).
Let the function \(\psi \) be continuous, non-decreasing, concave by [0, 1], \(\psi (0) = 0\) and \(0< \tau <\infty \). A generalized Lorentz space \(L_{\psi , \tau }({\mathbf {T}}^{m})\) is the set of measurable on \({\mathbf {T}}^{m} = [0,2\pi ]^{m}\) having \(2\pi \)-period for each variable \( x_j, j = 1, \dots , m, \) of functions \(f({\overline{x}}) = f (x_1, \dots , x_m)\), for which (see [27])
where \(f^{*}\) denotes the nonincreasing rearrangement of the function \(|f(2\pi {\overline{x}})|\), \({\overline{x}}\in {\mathbf {I}}^{m}\) (see e.g. [21, 27]). It is known that under the conditions \( 1<\alpha _{\psi }, \beta _{\psi } <2\), the space \(L_{\psi , \tau }({\mathbf {T}}^{m})\) will be a symmetric space with the fundamental function \(\psi \) and the functional \(\Vert f\Vert _{\psi , \tau }\) will be equivalent to the norm
spaces \(L_{\psi , \tau }({\mathbf {T}}^{m})\) [27, Lemma 3.1].
Note that for \(\psi (t) = t^{1/q} \) the space \(L_{\psi , \tau }({\mathbf {T}}^{m})\) coincides with the Lorentz space denoted by \( L_{q, \tau }({\mathbf {T}}^{m}) \), \( 1<q, \tau <\infty \) (see [32, p. 228]).
For a given positive integer M, consider the set \( \varDelta _{M} = \{{\overline{k}} = (k_{1},\dots ,k_{m}) \in {\mathbf {Z}}^{m} \,\, : |k_{j}| < M, \,\, j=1,\dots , m\}. \) We will consider the multiple Dirichlet kernel
and the convolution of a function \(f \in L_{\psi,\tau }({\mathbf {T}}^{m})\)
where \(s\in {\mathbf {N}}_{0} = {\mathbf {N}} \cup \{0\}, {\mathbf {N}}\) is the set of natural numbers.
Let \(E_{M}(f)_{\psi ,\tau } \equiv E_{M,\dots , M}(f)_{\psi ,\tau } = \inf \limits _{T\in \varGamma _{\varDelta _{M}}}||f-T||_{\psi,\tau }\) is the best approximation of the function \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\) by the set \(\varGamma _{\varDelta _{M}}\) of trigonometric polynomials of order at most \(M-1\) in each variable. For a given class \(F \subset L_{\psi ,\tau }({\mathbf {T}}^{m})\) we put \(E_{M}(F)_{\psi ,\tau } = \sup \limits _{f\in F}E_{M}(f)_{\psi ,\tau }\).
Let \(0 < \theta \le \infty \) and a number \( r> 0. \) We consider the space of all functions \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\) for which
for \(0< \theta <\infty \) and
for \(\theta = \infty \).
This space is denoted by the symbol \(B_{\psi , \tau , \theta }^{r}\) and is called the Nikol’skii–Besov space. In this space, we consider a unit ball
where
for \(0< \theta <\infty \) and
for \(\theta = \infty \).
In the case \(\psi (t)=t^{1/p}\) and \(\tau = p\), the space \(B_{\psi , \tau , \theta }^{r}\) is defined in [7, 24] and is denoted by \(B_{p, \theta }^{r}\).
Note that the generalized Nikol’skii–Besov space in the Lebesgue space is defined and investigated in [8, 15, 16].
One of the generalizations of the Nikol’skii–Besov space is the Nikol’skii–Besov space with logarithmic smoothness, defined as a subset of \(L_{p}({\mathbf {T}}^{m})\) (see [9,10,11, 13]). Dominguez O., Tikhonov S. [13] established characterizations and embeddings of Besov functional spaces with logarithmic smoothness.
In the space of continuous functions \(C({\mathbf {T}}^{1})\) S.B.Kashin and V.N.Temlyakov [18] defined the following class:
Now we define a similar Nikol’skii–Besov space with logarithmic smoothness in the generalized Lorentz space.
Let \(0< \theta \le \infty \) and a number \(\alpha > 0.\) Consider the space of all functions \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\) for which
for \(0< \theta < \infty \) and
for \(\theta = \infty \).
This space is denoted by \(B_{\psi , \tau , \theta }^{0, \alpha }\) and is called the Nikol’skii–Besov space of logarithmic smoothness in the generalized Lorentz space.
In this space, we consider a unit ball
where
for \(0< \theta < \infty \) and
In the case \(\psi (t)=t^{1/p}\) and \(\tau = p\), the space \(B_{\psi , \tau , \theta }^{0, \alpha }=B_{p, \theta }^{0, \alpha }\) is defined in [30, 31]. Other generalizations ofthe Nikol’skii–Besov space are given in [8, 15, 16].
In the case of \(\tau _{1} = p\), \(\tau _{2} = q\) for the Nikol’skii–Besov class, \({\mathbf {B}}_{p, \theta }^{r}\) in order exact estimates of the best approximation in the space \(L_{q}(T^{m})\) received A.S. Romanyuk [25]. In the case \(\tau = p\), estimates of the approximative characteristics of the class \({\mathbf {B}}_{p, \tau , \theta }^{0, \alpha }\) got S.A. Stasyuk [30, 31]. In [4], estimates of the best approximations of functions of the class \({\mathbf {B}}_{\psi , \tau _{1}, \theta }^{0, \alpha }\) in the Lorentz space \(L_{\psi , \tau _{ 2}}({\mathbf {T}}^{m})\) in the case of \(\psi (t) = t^{1/p}\). A survey of results on the theory of approximation of functions of many classes of Sobolev, Nikol’skii and Besov is given in [14], also see the bibliography in [34, 35].
It is known that \(L_{\psi , \tau _{2}}({\mathbf {T}}^{m}) \subset L_{\psi , \tau _{1}}({\mathbf {T}}^{m})\) for \( 0<\tau _{2}<\tau _{1} <\infty \) and the fundamental functions of these spaces are equivalent to the function \(\psi \).
In [5], the following statement was proved.
Lemma 1.1
Let \(1< \tau _{2}< \tau _{1} < \infty \) and the functions \(\psi _{1}, \psi _{2}\) satisfy the conditions \(\alpha _{\psi _{1}}=\alpha _{\psi _{2}}\), \(\beta _{\psi _{1}}=\beta _{\psi _{2}}\) and
Then \(L_{\psi _{2}, \tau _{2}}({\mathbf {T}}^{m}) \subset L_{\psi _{1}, \tau _{1}}({\mathbf {T}}^{m})\) and \(\Vert f\Vert _{\psi _{1}, \tau _{1}} \le C\Vert f\Vert _{\psi _{2}, \tau _{2}}\).
Therefore, the main goal of this article is to find the exact order
in various relations between the parameters \(p, \tau _{1}, \tau _{2}, \theta \).
The article consists of three sections. In the Sect. 2, several statements are proved necessary to prove the main results. In the Sect. 3, estimates for the value \(E_{M}({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r})_{\psi _{2},\tau _{2}}\).
In the Sect. 4, we establish estimates of \(E_{M}({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha })_{\psi _{2},\tau _{2}}\) . The main result of this section is Theorems 4.1, 4.2.
For theorems, lemmas, formulas, double numbering is used. Further, \(a_{+} = \max \{a, 0 \}\) and the record \(A(y) \asymp B(y)\) means that there are positive numbers \(C_{1} \, \, C_{2}\) independent of y such that \(C_{1} A(y) \le B(y)\le C_{2} A(y)\). For brevity, in the case of the inequalities \( B \ge C_{1} A\) or \(B \le C_{2}A\), we often write \(B>> A\) or \(B<< A \), respectively.
For a function g defined on [0, 1], the notation \(g \uparrow \) (respectively \( g \downarrow \)) means that the function g is non-decreasing (respectively non-increasing) by [0, 1].
2 Auxiliary results
Theorem 2.1
(see [23]). Let \( 1<p <\infty \). Then for any function \( f \in L_{p}({\mathbf {T}}^{m})\) the following relation holds
Theorem 2.2
Let \( 1<\tau <\infty \) and give \(\varPhi - \) function \(\psi \), \( 1<\alpha _{\psi }, \beta _{\psi } <2\). Then for any function \( f \in L_{\psi , \tau }({\mathbf {T}}^{m})\) the relation
Proof
Let \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\). We consider the operator P:
P is known to be a sublinear operator. By Theorem 2.1, this operator acts boundedly in the space \(L_{p}({\mathbf {T}}^{m})\), \( 1<p <\infty \). Therefore, by the Janson interpolation theorem [17], this operator is bounded in the space \(L_{\psi , \tau }({\mathbf {T}}^{m})\) i.e. \(\Vert P(f)\Vert _{\psi , \tau } \le C_{2}(p, \tau )\Vert f\Vert _{\psi , \tau }\) for any function \(f \in L_{\psi , \tau }({\mathbf {T}}^{m})\).
The converse inequality follows from the duality principle. Let \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\), \(g \in L_{\bar{\psi }, \tau ^{'}}({\mathbf {T}}^{m})\), \(1< \tau <\infty \), \(\frac{1}{\tau } + \frac{1}{\tau ^{'}} = 1\). Here and in the sequel \(\bar{\psi }(t)=t/\psi (t)\) for \(t\in (0, 1]\) and \(\bar{\psi }(0)=0\). Then, due to the orthogonality of the functions \(\sigma _{s}(f, 2\pi \bar{x}) \), we have
Further, applying the Hölder inequalities for the sum and the integral, we obtain
for any function \(g \in L_{{\bar{\psi }}, \tau ^{'}}({\mathbf {T}}^{m})\). Therefore, taking into account the well-known relation (see [27])
and the boundedness of the operator P, we have
\(\square \)
Lemma 2.1
Let \(\varPhi - \) function \(\psi \) satisfy the condition \(1< \alpha _{\psi }, \beta _{\psi } < 2^{1/\tau }\) and \(1 < \tau \le 2\). Then for an arbitrary system of functions \(\{\varphi _{j}\}_{j=1}^{n} \subset L_{\psi ,\tau }({\mathbf {T}}^{m})\) the inequality hold
where the constant C is independent of \(\varphi _{j}\) and n.
Proof
It is known that \((f^{*})^{\theta } = (|f|^{\theta })^{*}\) for the number \(\theta > 0 \). Therefore,
Now, using Jensen’s inequality (see [24, Lemma 3.3.3]) and taking into account that the function \(f^{*}\) is non-increasing from (2.2), we obtain
Applying the formula (see [21, p.89])
where \(\mu E\) is the Lebesgue measure of the set E and the properties of the integral we have
Now it follows from inequalities (2.3) and (2.5) that
Changing the order of integration, we have
We will consider the function \(\varphi (t) = t^{1/\tau }\). By the assumption of the lemma, \(\beta _{\psi } <2^{1/\tau } \) i.e. \(\alpha _{\varphi } = 2^{1/\tau } > \beta _{\psi }\). Therefore, by [22, Lemma 4] there exists a \(\varPhi - \) function g(t) such that \(\varphi (t)/\psi (t) \asymp g(t)\) and \(\alpha _{g}> 1\). Therefore, the [28, Lemma 3] holds the estimate
Now, according to this estimate, from equality (2.7) we obtain
It follows from inequalities (2.6) and (2.8) that
\(\square \)
Lemma 2.2
Let \(2<\tau <\infty \) and give a \(\varPhi - \) function \(\psi \) and \(1< \alpha _{\psi }, \beta _{\psi } < 2^{1/2}\). Then for an arbitrary system of functions \(\{\varphi _{j} \}_{j = 1}^{n} \subset L_{\psi , \tau }({\mathbf {T}}^{m})\) the inequality hold
where the constant C is independent of \(\varphi _{j}\) and n.
Proof
By the property of nonincreasing rearrangment of the function, we have (see (2.2))
We will consider the function \(\varphi (t) = t^{1/2}\), \(t\in (0, 1]\) and \(\varphi (0)=0\). By the assumption of the lemma, \(\beta _{\psi } <2^{1/2} \) i.e. \(\alpha _{\varphi } = 2^{1/2} > \beta _{\psi }\). Therefore, by [22, Lemma 4] there exists a \(\varPhi - \) function g(t) such that \(\varphi (t)/\psi (t) \asymp g(t) \) and \(\alpha _{g }> 1\). Therefore,
The functions \(\psi (t)\) and \(\varphi (t) = t^{1/2}\) are concave, so their product is a concave function. Therefore, \(L_{\psi \varphi , \tau /2}({\mathbf {T}}^{m})\) is a generalized Lorentz space and, moreover, \(\tau /2> 1 \). Now, taking into account that \(\frac{1}{g(t)}\) decreasing on (0, 1] and applying the triangle inequality, we obtain
Since, by the assumption of the lemma, \(\beta _{\psi } <2 ^ {1/2} \), then
Therefore, according to Hardy’s inequality in the generalized Lorentz space , we have
Now, inequalities (2.9)–(2.11) imply the assertion of Lemma 2.2. \(\square \)
Remark 2.1
These lemmas in the one-dimensional case in the Lorentz weighted space were proved by Kokilashvili and Yildirir [20].
Lemma 2.3
Let \(\psi \) a given \(\varPhi \) be a function. If \( 1<\alpha _{\psi }, \beta _{\psi } < 2^{1/\tau }\) and \(1<\tau \le 2\) or \(1< \alpha _{\psi }, \beta _{\psi } < 2^{1/2}\) and \(2 \le \tau < \infty \), then for any function \( f \in L_{\psi , \tau }({\mathbf {T}}^{m})\) the inequality hold
where \(\tau _{0} = \min \{\tau , 2\}\).
Proof
Let \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\). Then by Theorem 2.2 we have
From this inequality, according to the Lemma 2.1 and the Lemma 2.2, we obtain
It is known that the Fourier series of the function \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\) converges to it in the norm of the space \(L_{\psi ,\tau }({\mathbf {T}}^{m})\). Therefore, in inequality (2.12), passing to the limit for \(n \rightarrow \infty \), we obtain the assertion of Lemma. \(\square \)
Lemma 2.4
Let \(\varPhi \) - the function \(\psi \) satisfy the condition \(1< \alpha _{\psi }, \beta _{\psi } < 2^{1/2}\) and \(2 \le \tau < \infty \). Then for an arbitrary system of functions \(\{\varphi _{j}\}_{j=1}^{n} \subset L_{\psi ,\tau }({\mathbf {T}}^{m})\) the inequality hold
Proof
It is known that \((f^{*})^{\theta } = (|f|^{\theta })^{*}\) for the number \(\theta > 0 \). Therefore,
We consider the function \(\varphi (t) = t^{1/2}\), \(t\in [0, 1]\). This function is increasing, continuous, concave, and \(\alpha _{\varphi } = \beta _{\varphi } = 2^{1/2}\).
By the assumption of Lemma 2.4, \(\beta _{\psi } < 2^{1/2}\) i.e. \(\alpha _{\varphi } = 2^{1/2} > \beta _{\psi }\). Therefore, by [22, Lemma 4] there exists a \(\varPhi \)–function g(t) equivalent to the function \(\varphi /\psi \) and \(\alpha _{g}> 1\) (also see the proof of Lemma 2.2). Then \(\psi (t) = \frac{\psi (t)}{t^{1/2}}t^{1/2} \asymp \frac{1}{g(t)}t^{1/2}\). Therefore,
The functions \(\psi (t)\) and \(\varphi (t) = t^{1/2}\) are concave, so their product is a concave function. Therefore, \(L_{\psi \varphi , \tau /2}\) is a generalized Lorentz space and, moreover, \(\tau /2> 1\). Take into account these considerations, we have
Now in the space \(L_{\varphi \psi , \frac{\tau }{2}}({\mathbf {T}}^{m})\) applying Hardy’s inequality (see [27]) from (2.14) we get
Now, taking into account that the function \(\frac{1}{g(u)}\) decreasing from the inequalities (2.13) and (2.15) we get
Further, using equality (2.5), Jensen’s inequality (since \(\frac{2}{\tau } \le 1 \)) (see [24, Lemma 3.3.3]) from (2.16) we get
\(\square \)
Lemma 2.5
Let \(\varPhi \)-the function \(\psi \) satisfy the condition \(2^{1/2}< \alpha _{\psi }, \beta _{\psi } < 2\) and \(1< \tau \le 2\). Then for the function \(f \in L_{\psi , \tau }({\mathbf {T}}^{m})\) the inequality hold
Proof
Let \(f \in L_{\psi ,\tau }({\mathbf {T}}^{m})\), \(g \in L_{{\bar{\psi }}, \tau ^{'}}({\mathbf {T}}^{m})\), \(1< \tau <\infty \), \(\frac{1}{\tau } + \frac{1}{\tau ^{'}} = 1\). Then, taking into account the orthogonality of the function \(\sigma _{s}(f, {\bar{x}})\), we have
Here and in the sequel \({\bar{\psi }}(t)=t/\psi (t)\) for \(t\in (0, 1]\) and \({\bar{\psi }}(0)=0\). Further, applying the Hölder inequalities for the sum and the integral, we obtain
for any function \(g \in L_{{\bar{\psi }}, \tau ^{'}}({\mathbf {T}}^{m})\). Now, taking into account the relation (2.1) from the inequalities (2.17), (2.18) we have
Since \(2^{1/2}< \alpha _{\psi }, \beta _{\psi } < 2\) and \(1< \tau \le 2\), then \(\beta _{{{\bar{\psi }}}} < 2^{1/2}\) and \(2< \tau ^{'} < \infty \). Therefore, by applying Lemma 2.4, from the inequality (2.19) we obtain the assertion of Lemma 2.5.\(\square \)
Lemma 2.6
Let the \(\varPhi \) - function \(\psi \) satisfy the condition \(2^{1/2}< \alpha _{\psi }, \beta _{\psi } < 2\) and \(1< \tau \le 2\). Then for the function \(f\in L_{\psi ,\tau }({\mathbf {T}}^{m})\) the inequality hold
Proof
To prove this lemma, we use the method applied by V.N. Temlyakov (see [34, p.28-29] and [35, p.98]) . From the formulas (2.17), (2.1) we get
Consider the set
where \(\frac{1}{\tau }+\frac{1}{\tau ^{'}}=1\) and the number sequence \(\{\varepsilon _{s}\}\) satisfies the condition
The set of such sequences \(\{\varepsilon _{s}\}\) is denoted by \(\varLambda _{2}\).
Since \(2^{1/2}< \alpha _{\psi }, \beta _{\psi } < 2\) and \(1< \tau \le 2\), then \(1< \alpha _{{{\bar{\psi }}}}, \beta _{{{\bar{\psi }}}} < 2^{1/2}\) and \(2 \le \tau ^{'} < \infty \). Therefore, according to Lemma 2.2, we have
Therefore, from the inequality (2.20) we obtain
As in the article by V.N. Temlyakov (see [34, p.28-29] and [35, p.98]) we can prove that
Therefore, from the inequality (2.21) and taking into account the properties of the norm in the space \(l_{2}\), we obtain
\(\square \)
Theorem 2.3
Let \(\varPhi \) - functions \(\psi _{1}, \psi _{2}\) be given such that
\(1< \alpha _{\psi _{1}} = \alpha _{\psi _{2}} \le \beta _{\psi _{1}} = \beta _{\psi _{2}} < 2^{1/\tau _{2}}\) and \(1<\tau _{2} \le 2\). If \(\tau _{2} <\tau _{1}\) and the function \(f \in L_{\psi _{1},\tau _{1}}({\mathbf {T}}^{m})\) satisfies the condition
then \(f \in L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\) and the inequality hold
Proof
Let the function \(f \in L_{\psi _{1},\tau _{1}}({\mathbf {T}}^{m})\) and the conditions of the theorem be satisfied. Then using the inequality different for trigonometric polynomials (see [5, Theorem 1]) we have
Therefore, taking into account that \(1<\tau _{2} \le 2\), by Lemma 2.3 we obtain the assertions of the theorem. \(\square \)
Definition 2.1
(see [29, 33]). We denote by SVL the set of all non-negative functions on [0, 1] of \(\psi (t)\) for which \((\log \, 2/t)^{\varepsilon }\psi (t) \uparrow + \infty \) and \((\log \,2/t)^{- \varepsilon } \psi (t) \downarrow 0 \) for \(t\downarrow 0\).
Here and below, the notation \(\log x\) means the logarithm with base 2 of the number \(x>0\).
Corollary 2.1
Let \(\varPhi \) - functions \(\psi _{1}, \psi _{2}\) satisfy the conditions of Theorem 2.3and \(\frac{\psi _{2}}{\psi _{1}} \in SVL\). If \(\tau _{2} <\tau _{1}\) and the function \(f \in L_{\psi _{1},\tau _{1}}({\mathbf {T}}^{m})\) satisfies the condition
then \(f \in L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\) and the inequality hold
Proof
By condition, the function \(\frac{\psi _{2}}{\psi _{1}} \in SVL\). Let \(0<t_{1} < t_{2} \le 1\) i.e. \(1/t_{1}> 1/t_{2} >0\). Consequently
for the number \(\varepsilon > 0\). Put \(t_{1} = \prod \limits _{j=1}^{m}(n_{j} + 1)^{-1}\) and \(t_{2} = t\). Then
Put \(\varepsilon = \frac{1}{2}(\frac{1}{\tau _{2}} - \frac{1}{\tau _{1}})\). Then
Therefore, for \(\varepsilon = \frac{1}{2}(\frac{1}{\tau _{2}} - \frac{1}{\tau _{1}})\) from inequality (2.22) we obtain
Now, using this inequality and Theorem 2.3 we obtain the corollary. \(\square \)
Theorem 2.4
Let \(\varPhi \)–functions \(\psi _{1}, \psi _{2}\) satisfy the conditions \(1< \alpha _{\psi _{2}} \le \beta _{\psi _{2}}< \alpha _{\psi _{1}} \le \beta _{\psi _{1}} < 2\) and \(1< \tau _{1}, \tau _{2} < \infty \). If function \(f \in L_{\psi _{1},\tau _{1}}({\mathbf {T}}^{m})\) and
then \(f \in L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\) and the inequality
Proof
Using the formula (2.4) and [3, Lemma 5], we can prove that
Further, using this inequality due to the boundedness of the Hardy operator in the generalized Lorentz space \(L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\) we can verify that the statement of the theorem is true. \(\square \)
Corollary 2.2
Let the functions \(\psi _{1}, \psi _{2}\) satisfies the conditions of the Theorem 2.4. If function \(f \in L_{\psi _{1},\tau _{1}}({\mathbf {T}}^{m})\) and
then \(f \in L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\).
Proof
We consider the Fourier sum
where \(a_{{\overline{k}}}(f)\) as usual denote the Fourier coefficients of the function f with respect to the system \(\{e^{i\langle {\overline{k}}, 2\pi {\overline{x}}\rangle }\}\). Then, by the property of the norm and the best approximation of the function, the following inequalities hold:
Further, from the properties of the functions \(\psi _{1}, \psi _{2}\), the best approximation of the function, and [2, Lemma 2] it follows that
Now it follows from inequalities (2.25) and (2.26) that conditions (2.23) and (2.24) are equivalent. Therefore, the assertion of Corollary 2.2 follows from Theorem 2.4. \(\square \)
Remark 2.2
In case \(\psi _{1}(t)=t^{1/p},\) \(\psi _{2}(t)=t^{1/q}\) for \(1< \tau _{1}=p < \tau _{2}=q\) Corollary 2.2 is proved in [12, Theorem 2.3], and for \(1< \tau _{1}=p< q<\infty \) and \(0< \tau _{2} < \infty \) in [1, Theorem 1].
3 On orders of approximation of functions of Nikol’skii–Besov classes
In this section, we prove estimates of the best approximations of a function from the class \({\mathbf {B}}_{\psi , \tau , \theta }^{r}\).
Theorem 3.1
Let \(\varPhi \) - the function \(\psi \) satisfy the conditions \(1< \alpha _{\psi } \le \beta _{\psi } < 2\) and \(1\le \tau < \infty \), \(0< \theta < \infty \). Then for the number \(r> 0\) the relation hold
Proof
Let \(f \in {\mathbf {B}}_{\psi , \tau , \theta }^{r}\) and a positive integer l such that \(2^{l-1} \le n < 2^{l}\). Then by the property of best approximation and norm we have
If \(1<\theta <\infty \), then applying the Hölder‘s inequality (\(\frac{1}{\theta } + \frac{1}{\theta ^{'}} = 1\)) from (3.1) we obtain
If \( 0 <\theta \le 1\), then applying Jensen’s inequality (see [24, Lemma 3.3.3]) from (3.1) we obtain
Thus, \(E_{n}(f)_{\psi , \tau } \le Cn^{-r}\) for any function \(f \in {\mathbf {B}}_{\psi , \tau , \theta }^{r}\), \(0< \theta < \infty \). The upper bound is proved.
Let us prove the lower bound for \(E_{n}({\mathbf {B}}_{\psi , \tau , \theta }^{r})_{\psi ,\tau }\). Let a natural number l be such that \(2^{l-1} \le n < 2^{l}\). We will consider the function
According to the estimate of the norm of the Dirichlet kernel in the generalized Lorentz space, we have (see [3, p.67])
for \(1< \tau< \infty ,\,\, 1<\alpha _{\psi }\le \beta _{\psi }< 2\). Therefore,
Therefore, the function \(F_{0} = C_{0}^{-1}f_{0}\in {\mathbf {B}}_{\psi , \tau , \theta }^{r}\). Now, by the best approximation property and relation (3.2), we have
Consequently \( E_{n}({\mathbf {B}}_{\psi , \tau , \theta }^{r})_{\psi ,\tau } \ge Cn^{-r}, n\in {\mathbf {N}}. \) \(\square \)
Remark 3.1
In the case \(\psi (t) = t^{1/p}\) and \(\tau =p, \,\, 1\le \theta < \infty \) , Theorem 3.1 was proved in [25, Theorem 1].
Theorem 3.2
Let \(\varPhi \) - functions \(\psi _{1}, \psi _{2}\) satisfy the conditions of Theorem 2.3and \(\frac{\psi _{2}}{\psi _{1}} \in SVL\), \(0 < \theta \le \infty \). If \(1<\tau _{2}<\tau _{1}< \infty \), \( r> 0 \), then the relation hold
Proof
Let \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\). If \(\tau _{2} <\theta \), then put \(q = \frac{\theta }{\tau _{2}} > 1, \frac{1}{q} + \frac{1}{q^{'}} = 1\). Applying the Hölder’s inequality, we obtain
Since the function \(\frac{\psi _{2}}{\psi _{1}} \in SVL\), then
for \(\varepsilon > 0\), \(s=0,1,2,...\). Therefore,
Therefore, it follows from (3.4) that the series
converges for any function \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\) for \(\tau _{2} < \theta \). Therefore, according to Corollary 2.1, the inclusion \({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r} \subset L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\) for \(\tau _{2} < \theta \). If \(\theta \le \tau _{2}\), then applying Jensen’s inequality (see [24, Lemma 3.3.3]) and taking into account (3.5) we obtain
Therefore, it follows from 3.4 that
Therefore, again according to Corollary 2.1, we can state that
for \(\theta \le \tau _{2}\).
Now we prove relation (3.3). Let us prove an upper bound for the quantity \(E_{n}({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r})_{\psi _{2},\tau _{2}}\). Let a natural number l be such that \(2^{l-1} \le n <2^{l}\). For the function \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\) by Corollary 2.1 we have
If \(\tau _{2}< \theta < \infty \), then put \(q = \frac{\theta }{\tau _{2}} > 1, \frac{1}{q} + \frac{1}{q^{'}} = 1\). Applying the Hölder’s inequality and taking into account the condition \(\frac{\psi _{2}}{\psi _{1}} \in SVL\) (see the proof of (3.6) from (3.7) we obtain
for any function \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\) in the case of \(\tau _{2}<\theta <\infty \).
If \(\theta = \infty \), then from inequalities (3.5) and (3.7) we obtain
for any function \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \infty }^{r}\).
If \(\theta \le \tau _{2}\), then applying Jensen’s inequality (see [24, Lemma 3.3.3]) and taking into account the condition \(\frac{\psi _{2}}{\psi _{1}} \in SVL\) from (3.7) we get
for any function \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\) in the case of \(\theta \le \tau _{2}\). Now it follows from inequalities (3.8) and (3.9) that
The upper bound is proved.
Let us prove the lower bound for the quantity \(E_{n}({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r})_{\psi _{2},\tau _{2}}\). Let a natural number l be such that \(2^{l-1} \le n < 2^{l}\). We will consider the function
for \({\bar{x}} \in {\mathbf {I}}^{m}, \,\, l \in {\mathbf {N}}_{0}\).
Since \(|\prod _{j=2}^{m} e^{i2^{l}2\pi x_{j}}| = 1\), \(x_{j}\in [0, 1], \,\, j=2,...,m\), then
Therefore, non-increasing rearrangement of these functions are equal. Hence,
By the norm property and taking into account the boundedness of the conjugate function operator in the space \(L_{\psi _{1}, \tau _{1}}({\mathbf {T}}^{m})\) we have
In the article [6] it was proved that
Therefore, from the estimate (3.11) it follows that
Therefore, from equality (3.10) we obtain \( \Bigl \Vert f_{1}\Bigr \Vert _{\psi _{1}, \tau _{1}} \le C2^{-lr}, \,\, l=1,2,... \) Then
Therefore, the function \(F_{1} = C_{1}^{-1}f_{1} \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\). In the article [6] it was proved that
on condition \(\frac{\psi _{2}}{\psi _{1}} \in SVL\). Using this inequality, we can verify that
Now, by the property of the best approximation of the function and inequality (3.13), we obtain
Hence,
\(\square \)
4 Estimates of the best approximations of logarithmic smoothness functions in a generalized Lorentz space
In this section, we prove estimates of the best approximations of functions from the class \({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha }\).
Theorem 4.1
Let \(\varPhi \) - functions \(\psi _{1}, \psi _{2}\) satisfy the conditions of Theorem 2.3and \(\frac{\psi _{2}}{\psi _{1}} \in SVL\), \(0 < \theta \le \infty \). If \(1 < \tau _{2} \le 2, \tau _{2} < \tau _{1} < \infty\) and \(\alpha > (\frac{1}{\tau _{2}} - \frac{1}{\tau _{1}}) + (\frac{1}{\tau _{2}} - \frac{1}{\theta })_{+}\), then the relation hold
where \(a_{+} = \min \{0, a\}\). In the case of \(\theta \leq \tau _{2}\), this estimate is sharp in order.
In case \(1 < \tau _{2}< \theta \leq \infty\), if \(2^{1/2} < {\alpha}_{\psi{_2}}, {\beta}_{\psi{_2}} < 2 \; and \; 1 < \tau _{2} \leq 2\, or \,1 < {\alpha}_{\psi{_2}}, {\beta}_{\psi{_2}} < 2^{1/2} \; and \; 2 < \tau _{2} < \infty\), then
where \({\gamma} = \max \{2, \tau _2\}\).
Proof
Let \(f \in {\mathbf {B}}_{\psi_{1}, \tau _{1}, \theta }^{0, \alpha }\). If \(\tau _{2} <\theta \), then for \(q = \frac{\theta }{\tau _{2}}, \frac{1}{q} + \frac{1}{q^{'}} = 1)\) applying the Hölder’s inequality we get
Since the function \(\frac{\psi _{2}}{\psi _{1}} \in SVL\), then
for \(\varepsilon > 0\), \(s=l, l+1, l+2,...\). Therefore,
for \(l=0, 1, 2,...\).
Since \(\tau _{2} < \theta \) , then \(\alpha > (\frac{1}{\tau _{2}} - \frac{1}{\tau _{1}}) + (\frac{1}{\tau _{2}} - \frac{1}{\theta })\). Therefore, you can choose a number \(\varepsilon \) such that \(0< \varepsilon < \alpha + (\frac{1}{\tau _{1}} - \frac{1}{\tau _{2}}) - (\frac{1}{\tau _{2}} - \frac{1}{\theta })\). Then the series
converges and
Now it follows from inequalities (4.3) and (4.4) that
for \(l=0, 1, 2,...\). From inequalities (4.1) and (4.5) we obtain
in the case of \(\tau _{2}< \theta <\infty \), for \(l=0, 1, 2,...\).
If \(\theta =\infty \), then
for any function \(f \in {\mathbf {B}}_{\psi , \tau _{1}, \infty }^{0, \alpha }\).
If \(\theta \le \tau _{2}\), then applying Jensen’s inequality (see [24, Lemma 3.3.3]) we have
Since \(\alpha + \frac{1}{\tau _{1}} - \frac{1}{\tau _{2}} > 0\) and \(\frac{\psi _{2}}{\psi _{1}} \in SVL\), then using inequality (4.2) for \(\varepsilon = \alpha + \frac{1}{\tau _{1}} - \frac{1}{\tau _{2}}\) from the formula (4.7) we get
in the case \(\theta \le \tau _{2}\) for \(l=0,1,2,...\).
In particular, for \( l = 0 \) it follows from estimates (4.6), (4.8) that
for any function \(f \in {\mathbf {B}}_{\psi_{1} , \tau _{1}, \theta }^{0, \alpha }\). Therefore, according to Corollary 2.1, the inclusion \({\mathbf {B}}_{\psi_{1} , \tau _{1}, \theta }^{0, \alpha } \subset L_{\psi _{2},\tau _{2}}({\mathbf {T}}^{m})\) is true.
Now we estimate the value \(E_{n}({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha })_{\psi _{2},\tau _{2}}\). Let a natural number l be such that \(2^{l-1} \le n < 2^{l}\). Using the properties of the best approximation function and Corollary 2.1, we have
Further, using inequalities (4.6), (4.8) and the properties of the functions \(\psi _{1}, \psi _{2}\), we obtain
for any function \(f \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha }\). Thus,
This proves the upper bound.
Now we prove the lower bounds. We will consider the function
for \( {\bar{x}} \in {\mathbf {I}}^{m}, \,\, l \in {\mathbf {N}}_{0}\). By continuity, the function \(f_{2} \in L_{\psi _{1}, \tau _{1}}({\mathbf {T}}^{m})\). Using estimate (3.12) we have
If \(s \ne l+1\), then \( \Bigl \Vert \sigma _{s}(f_{2})\Bigr \Vert _{\psi _{1}, \tau _{1}} = 0\). Therefore,
Therefore, the function \(F_{2}=C_{2}^{-1}f_{2} \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha }\).
Further, taking into account the definition of the best approximation and using inequality (3.12), we have
in the case \(\theta \le \tau _{2}\). Hence,
in the case \(\theta \le \tau _{2}\).
Let \(\tau _{2} < \theta < \infty\). We will consider the function
where \({\bar{x}} \in {\mathbf {I}}^{m}, \,\, n \in {\mathbf {N}}_{0}\). Then
for \(1<p, \tau _{1}<\infty \), \(s\in {\mathbf {N}}_{0}\). By continuity, the function \(f_{3} \in L_{\psi _{1},\tau _{1}}({\mathbf {T}}^{m})\) and using relations (3.12) and (4.9) we obtain
Hence, the function \(F_{3}=C_{3}^{-1}f_{3} \in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha }\). Further, by definition of the best approximation of the function, we have
If \(2^{1/2}< \alpha _{\psi }, \beta _{\psi } < 2\) and \(1< \tau _{2} \le 2\), then using Lemma 2.6 we get
Further, taking into account that \(\frac{\psi _{2}}{\psi _{1}} \in SVL\) we get
Now, from equality (4.10) and inequality (4.11) it follows that
in the case \(2^{1/2}< \alpha _{\psi _{2}}, \beta _{\psi _{2}} < 2\) and \(1< \tau _{2} \le 2\).
If \(1< \alpha _{\psi _{2}}, \beta _{\psi _{2}} < 2^{1/2}\) and \(2\le \tau _{2} < \infty \), then using Lemma 2.4 and after similar reasoning we get
Hence,
in the case \(2^{1/2}< \alpha _{\psi _{2}}, \beta _{\psi _{2}} < 2\) and \(1< \tau _{2} \le 2\) and
in the case \(1< \alpha _{\psi _{2}}, \beta _{\psi _{2}} < 2^{1/2}\) and \(2< \tau _{2} < \infty \), \(\theta < \infty \).
If \(\theta = \infty \), then we will consider the function
where \({\bar{x}} \in {\mathbf {I}}^{m}\). Then taking into account (3.12) we get
Hence, the function \(F_{4}=C_{4}^{-1}f_{4}\in {\mathbf {B}}_{\psi _{1}, \tau _{1}, \infty }^{0, \alpha }\).
If \(1< \alpha _{\psi _{2}}, \beta _{\psi _{2}} < 2^{1/2}\) and \(2\le \tau _{2} < \infty \), then using Lemma 2.4 and in the case \(2^{1/2}< \alpha _{\psi }, \beta _{\psi } < 2\) and \(1< \tau _{2} \le 2\), using Lemma 2.6 we get
where \(\gamma =\max \{\tau _{2}, 2\}\). Hence,
in the case \(\theta = \infty \). \(\square \)
Theorem 4.2
Let \(1< \alpha _{\psi }\le \beta _{\psi } < 2\) and \(1<\tau \le 2\) or \(1< \alpha _{\psi }\le \beta _{\psi } < 2^{1/2}\), \(2 \le \tau < \infty \), \(1 \le \theta \le \infty \), \(\tau _{0} = \min \{\tau , 2\}\). If \(\alpha > (\frac{1}{\tau _{0}} - \frac{1}{\theta })_{+}\), then
where \(a_{+} = \max \{a, 0\}\).
Proof
Let \(f \in {\mathbf {B}}_{\psi , \tau , \theta }^{0, \alpha }\) and a positive integer n such that \(2^{n-1} \le M <2^{n}\). It follows from Lemma 2.1 and Lemma 2.4 that
Now applying this inequality to the function \(f - \sum \limits _{s=0}^{n}\sigma _{s}(f)\in L_{\psi ,\tau }({\mathbf {T}}^{m})\) we will have
If \(\theta \le \tau _{0}\), then applying Jensen’s inequality (see [24, Lemma 3.3.3]) from (4.13) we obtain
for any function \(f \in {\mathbf {B}}_{\psi , \tau , \theta }^{0, \alpha }\) in the case \(\theta \le \tau _{0}\). Hence,
in the case \(\theta \le \tau _{0}\).
Let \(\tau _{0} < \theta \). Then applying the Hölder’s inequality \((\beta = \frac{\theta }{\tau _{0}} > 1, \frac{1}{\beta } + \frac{1}{\beta ^{'}} = 1)\) and taking into account the inequality \(\alpha > \frac{1}{\tau _{0}} - \frac{1}{\theta }\) from (4.13) we have
Therefore,
in the case \(\tau _{0} < \theta \).
If \(\theta =\infty \), then taking into account the inequality \(\alpha > \frac{1}{\tau _{0}}\) from (4.13) we have
This proves the upper bound.
Let us prove the lower bounds. Let \(\tau _{0} <\theta \). We will consider the function
for \({\bar{x}} \in {\mathbf {I}}^{m}\), \(n \in {\mathbf {N}}_{0}\).
By the estimate of the norm of the Dirichlet kernel in the generalized Lorentz space (3.2), we have
Thus, the function \(F_{5}=C_{5}^{-1}f_{5} \in {\mathbf {B}}_{p, \tau , \theta }^{0, \alpha }\) for \( 1<p, \tau <\infty \), \( 1 \le \theta <\infty \).
Let \(1< \alpha _{\psi }\le \beta _{\psi } < 2\), \(1 < \tau \le 2\), i.e. \(\tau _{0} = \tau \). We select the number \(q > (\log _{2}\alpha _{\psi })^{-1}\) i.e. \(2^{1/q} < \alpha _{\psi }\). Then, using Theorem 2.3 and the method of proving Lemma 2.6, we can prove that
for \(f\in L_{\psi , \tau }({\mathbf {T}}^{m}), \,\, 1< \tau < \infty \).
Now we apply this inequality to the function \(F_{5}=C_{5}^{-1}f_{5} \in {\mathbf {B}}_{\psi , \tau , \theta }^{0, \alpha }\). Then, given the estimate of the norm of the Dirichlet kernel (see relation (3.2)), we obtain
Thus, \( E_{2^{n}}(F_{5})_{\psi ,\tau }>>(n+1)^{-\alpha + \frac{1}{\tau } -\frac{1}{\theta }} \) for \(1< \alpha _{\psi }\le \beta _{\psi } < 2\), \(1< \tau < \infty \). Hence,
for \(1< \alpha _{\psi }\le \beta _{\psi } < 2\), \(1< \tau < \infty \). This inequality shows the exactness of the estimate in Theorem 4.2 for \(1 < \tau \le 2\) , \(\tau _{0}=\min \{\tau ,2\} < \theta \), \(1< \alpha _{\psi }\le \beta _{\psi } < 2\).
If \(\theta = \infty \), then we will consider the function
for \(1< \alpha _{\psi }\le \beta _{\psi } < 2\), \(1 < \tau \le 2\).
Using relation (3.2) and inequality (4.12), it is easy to verify that \(f_{6}\in L_{\psi , \tau }({\mathbf {T}}^{m})\) and
Hence, the function \(F_{6}=C_{6}^{-1}f_{6} \in {\mathbf {B}}_{p, \tau , \infty }^{0, \alpha }\).
Further, using inequality (4.14), we can verify that
Hence,
for \(1< \alpha _{\psi }\le \beta _{\psi } < 2\), \(1 < \tau \le 2\).
Now we prove lower bounds for \(\beta _{\psi } < 2^{1/2}\), \(2\le \tau < \infty \) and \(2=\tau _{0} < \theta \). We will consider the function
where \(R_{s}(x_{j})=\sum \limits _{k=2^{s-1}}^{2^{s} - 1}\varepsilon _{k}e^{ik2\pi x_{j}}\) is the Rudin–Shapiro polynomial and \(\varepsilon _{k} = \pm 1\). It is known that \(\Vert R_{s}\Vert _{\infty }<<2^{s/2}\) (see [19, p. 146]). Therefore,
Hence,
i.e. the function \(F_{7}=C_{7}^{-1}f_{7} \in {\mathbf {B}}_{\psi, \tau , \theta }^{0, \alpha }\). Since \(\beta _{\psi } < 2^{1/2}\), \(2\le \tau < \infty \), then \(L_{\psi , \tau }({\mathbf {T}}^{m}) \subset L_{2}({\mathbf {T}}^{m})\) and \(\Vert f\Vert _{2} \le C \Vert f\Vert _{\psi , \tau }\), for \(f \in L_{\psi , \tau }({\mathbf {T}}^{m})\). Therefore, according to Parseval’s equality, we obtain
for \(\beta _{\psi } < 2^{1/2}\). Hence,
in the case \(\beta _{\psi } < 2^{1/2}\), \(2\le \tau < \infty \) and \(2=\tau _{0} < \theta \).
Now we prove the lower bound for \(\theta \le \tau _{0}\). We will consider the function
Then, taking into account relation (3.2), we have
Therefore, the function \(F_{8}=C_{8}^{-1}f_{8} \in {\mathbf {B}}_{p, \tau , \theta }^{0, \alpha }\). Now using relation (3.2) will have
Hence,
in the case \(\theta \le \tau _{0}\), for \(1<\alpha _{\psi }\le \beta _{\psi } < 2\) \(1< \tau < \infty \).
If \(\theta = \infty \), then we will consider the function
for \({\bar{x}} \in {\mathbf {I}}^{m}\).
By the property of the Rudin-Shapiro polynomial, we obtain \( \sup _{s\in {\mathbf{N}}_{0}}(s+1)^{\alpha}\Vert \sigma _{s}(f_{9})\Vert _{\psi, \tau}\le C_{9}. \) Hence, the function \(F_{9}=C_{9}^{-1}f_{9}\in {\mathbf {B}}_{\psi , \tau, \infty }^{0, \alpha }\). Further, as in the proof of (4.15), one can verify that
Hence,
for \(\beta _{\psi } < 2^{1/2}\), \(2\le \tau < \infty \). \(\square \)
Remark 4.1
In the case \(\psi (t) = t^{1/p}\) and \(1<\tau = p < \infty \), \(1\le \theta \le \min \{2, p\}\) from Theorem 4.2 we obtain [30, Theorem, item (i)]. For the function \(\psi (t) = t^{1/p}\) and \(1<\tau , p < \infty \), \(1\le \theta \le \infty \) Theorem 4.2 is proved in [4, Theorem 2.1].
In the case \(\psi _{1}(t) = \psi _{2}(t) = t^{1/p}\), \(t\in [0, 1]\), \(1< p < \infty \) Theorem 4.1 was proved in [4, Theorem 3.4].
Remark 4.2
In the case, \(1< \alpha _{\psi _{2}} \le \beta _{\psi _{2}}< \alpha _{\psi _{1}}\le \beta _{\psi _{1}}< 2\) and \(1< \tau _{1}, \tau _{2} < \infty \) using Theorem 2.4, we can study the estimate of the quantity \(E_{M}({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r})_{\psi _{2}, \tau _{2}}\). In particular, for \(\psi _{1}(t)=t^{1/p}\), \(\psi _{2}(t)=t^{1/q}\) and \(\tau _{1}=p,\) \(\tau _{2}=q\), \(1<p, q< \infty \) this problem was investigated by A. S. Romanyuk [25, Theorem 1].
Remark 4.3
We note that the results of this paper can be applied to estimate the best \(M-\) term approximations, trigonometric widths, linear widths of classes \({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{r}\) and \({\mathbf {B}}_{\psi _{1}, \tau _{1}, \theta }^{0, \alpha }\) in the Lorentz space \(L_{\psi _{2}, \tau _{2}}({\mathbf {T}}^{m})\) (see special cases for example in [3]).
Remark 4.4
Let \(1<\tau _{2}< \tau _{1}<\infty \), \(1< \alpha _{\psi _{1}}=\beta _{\psi _{2}}<2\) and
then \(L_{\psi _{1}, \tau _{1}}({\mathbf {T}}^{m})\subset L_{\psi _{2}, \tau _{2}}({\mathbf {T}}^{m})\) (see [29, p. 916]). Therefore, we can consider an analogue of Theorem 2.2, Theorem 3.2, Theorem 4.1, and Theorem 4.2 in this case.
References
Akishev, G.: On the imbedding of certain classes of functions of several variables in the Lorentz space. Izvestya AN KazSSR, ser. fiz.-mat. 3, 47–51 (1982)
Akishev, G.A.: On degrees of approximation of some classes by polynomials with respect to generalized Haar system. Sib. Electron. Math. Rep. 3, 92–105 (2006)
Akishev, G.: On the orders \(M\)-terms approximations of classes of functions of the symmetrical space. Mat. Zh. 14(4), 44–71 (2014)
Akishev, G.: Estimates for best approximations of functions from the logarthmic smoothness class in the Lorentz space. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 23(3), 3–21 (2017)
Akishev, G.: An inequality of different metrics in the generalized Lorentz space. Trudy Instituta Matematiki i Mekhaniki UrO RAN 24(4), 5–18 (2018)
Akishev, G.: On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz spaces. Trudy Instituta Matematiki i Mekhaniki UrO RAN 25(2), 9–20 (2019)
Besov, O.V.: Investigation of a class of function spaces in connection with imbedding and extension theorems. Tr. Mat. Inst. Steklov. 60, 42–81 (1961)
Burenkov V.I.: Imbedding and extension theorems for classes of differentiable functions of several variables defined on the entire spaces, In Itogi Nauki i Tekhniki. Seriya “Matematicheskii Analiz“ pp.71-155, Moscow (1966)
Cobos, F., Dominguez, O.: On Besov spaces of logarithmic smoothness and Lipschitz spaces. J. Math. Anal. Appl. 425, 71–84 (2015)
Cobos, F., Milman, M.: On a limit class of approximation spaces. Numer. Funct. Anal. Optimiz. 11, 11–31 (1990)
DeVore, R.A., Riemenschneider, S.D., Sharpley, R.C.: Weak interpolation in Banach spaces. J. Funct. Anal. 33, 58–94 (1979)
Ditzian, Z., Tikhonov, S.: Ul’yanov and Nikol’skii-type inequalities. J. Approx. Theory 133(1), 100–133 (2005)
Dominguez, O., Tikhonov, S.: Function spaces of logarithmic smoothness: embeding and characterizations. Preprint (2018). arXiv:1811.06399v [math.FA]. Mem. Amer. Math. Soc. (accepted)
Dung, D., Temlyakov, V., Ullrich, T.: Hyperbolic cross approximation. Adv. Courses Math, CRM Barselona (2018)
Dzhafarov, A.S.: Embedding theorems for classes of functions with differential properties in the norms of special spaces. Dokl. AN Azerb. SSR. 21(2), 10–14 (1965)
Gol’dman, M.L.: On the inclusion of generalized Hölder classes. Math. Notes. 12(3), 626–631 (1972)
Janson, S.: On the interpolation of sublinear operators. Stud. Math. 75, 51–53 (1982)
Kashin B.S., Temlyakov V.: On a norm and approximation characteristics of classes of functions of several variables. Metric theory of functions and related problems in analysis (Russian). Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra (AFTs), Moscow (1999)
Kashin, B.S., Saakyan, A.A.: Orthogonal series. Aktuarno-Finans, Tsentra (AFTs), Moscow (1999)
Kokilashvili, V., Yildirir, Y.E.: Trigonometric polynomials in weighted Lorentz spaces. J. Funct. Spaces Appl. 8(1), 67–86 (2010)
Krein, S.G., Petunin, YuI, Semenov, E.M.: Interpolation of linear operators. Nauka, Moscow (1978)
Lapin S.V.: Some embedding theorems for products of functions, Manuscript N 1036-80Dep, deposited at VINITI (Russian). pp. 31 (1980)
Lizorkin, P.I.: Generalized Holder spaces \(B_{p, \theta }^{(r)}\) and their relations with the Sobolev spaces \(L_{p}^{(r)}\). Sib. Mat. Zhur. 9(5), 1127–1152 (1968)
Nikol’skii, S.M.: Approximation of functions of several variables and embedding theorems. Nauka, Moscow (1977)
Romanyuk, A.S.: The approximation of the isotropic classes \(B_{p, \theta }^{r}\) of periodic functions of many variables in the space \(L_{q}\). Tr. Inst. Mat. Ukrain. 5(1), 263–278 (2008)
Semenov, E.M.: Interpolation of linear operators in symmetric spaces. Sov. Math. Dokl. 6(), 1294–1298 (1965)
Sharpley, R.: Space \(\Lambda _{\alpha }(X)\) and interpolation. J. Funct. Anal. 11, 479–513 (1972)
Sherstneva, L.A.: On the properties of best Lorentz approximations and certain embedding theorems. Izvestiya Vysshikh Uchebnykh Zavedenii. Matem. 10, 48–58 (1987)
Simonov, B.V.: Embedding Nikol’skii classes into Lorentz spaces. Sib. Math. J. 51(4), 728–744 (2010)
Stasyuk, S.A.: Approximating characteristics of the analogs of Besov classes with logarithmic smoothness. Ukr. Math. J. 66(4), 553–560 (2014)
Stasyuk, S.A.: Kolmogorov widths for analogs ofthe Nikol’skii - Besov classes with logarithmic smoothness. Ukr. Math. J. 67(11), 1786–1792 (2015)
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton (1971)
Temirgaliev, N.: On the embedding of the classes \(H_{p}^{\omega }\) in Lorentz spaces. Sib. Mat. Zh. 24(2), 160–172 (1983)
Temlyakov, V.N.: Approximation of functions with bounded mixed derivative. Tr. Mat. Inst. Steklov. 178, 3–112 (1986)
Temlyakov, V.: Multivariate approximation. Cambridge University Press, Cambridge (2018)
Acknowledgements
The author is grateful for the support of this work given by the Russian Academic Excellence Project (Agreement No. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University). I also thank the referee for some good suggestions, which have improved the final version of this paper.
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Akishev, G. Estimates of the best approximations of the functions of the Nikol’skii–Besov class in the generalized space of Lorentz. Adv. Oper. Theory 6, 15 (2021). https://doi.org/10.1007/s43036-020-00108-z
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DOI: https://doi.org/10.1007/s43036-020-00108-z