Nikol’skii–Type Inequalities for Trigonometric Polynomials for Lorentz–Zygmund Spaces

Doktorski, Leo R. Ya.

doi:https://doi.org/10.1155/2020/6853723

Journal of Function Spaces

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Abstract Introduction Preliminaries Applications Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6853723 | https://doi.org/10.1155/2020/6853723

Nikol’skii–Type Inequalities for Trigonometric Polynomials for Lorentz–Zygmund Spaces

Leo R. Ya. Doktorski¹

Academic Editor: Shanhe Wu

Received18 Jul 2019

Revised04 Oct 2019

Accepted12 Oct 2019

Published03 Feb 2020

Abstract

Nikol’skii–type inequalities, that is inequalities between different metrics of trigonometric polynomials on the torus for the Lorentz–Zygmund spaces, are obtained. The results of previous paper “Nikol’skii inequalities for Lorentz–Zygmund spaces” are extended. Applications to approximation spaces in Lorentz–Zygmund spaces and to Besov spaces are given.

1. Introduction

The classical Nikol’skii (or Jackson–Nikol’skii) inequality for the trigonometric polynomials on [0,1] of degree at most can be written as [1, 2]

where and is the usual norm on the Lebesgue spaces . For , this estimate has been proved by Jackson [3]. The proofs in [1–3] are based on Bernstein’s inequality.

Inequalities between different (quasi-)norms of the same function are known as Nikol’skii–type (or Jackson–Nikol’skii–type) inequalities. They play a crucial role in many areas of mathematics, e.g., theory of approximation, theory of functions of several variables, and functional analysis (embedding theorems for Besov spaces).

Nessel and Wilmes [4] extended inequality (1) for . They also observed that in inequality (1) one may in fact take into account the spectrum of the function involved for or certain gaps in the spectrum for . Sherstneva [5] extended inequality (1) in the Lorentz spaces and showed that they are exact relative to the order . Moreover, she investigated the limiting case when Lorentz spaces on both sides have the same value of the main parameter. She proved that if and , then

Some Nikol’skii–type inequalities for the Lorentz–Zygmund spaces are considered in [6–8] and for the generalized Lorentz space in [9]. In other investigations, different sets of functions, domains, and measures were explored. For further information about these results and applications, we refer to [1–19] and references therein.

In [19], the results of [5] were improved in two directions. First, the functions of the form were considered, where is an orthonormal system in uniformly bounded in (with ). No assumptions about smoothness of were made. Secondly, the inequalities (1) and (2) were extended to the Lorentz–Zygmund spaces. However, in [19], only the case is obtained.

The principal aim of this paper is to extend the results of [19] for the case for the trigonometric polynomials. The technique we apply relies on the observation that the power of a trigonometric polynomial is also a trigonometric polynomial [4, 10, 17, 18]. This paper is organized as follows. Section 2 contains necessary notations and definitions. Main results of this contribution are Theorems 4, 6, 8, and 9. They are formulated and proved in Section 3. Note that Theorems 8, and 9 deal with the limiting case. In Section 4, we reformulate Theorems 4, 6, 8, and 9 for trigonometric polynomials of degree at most . In Section 5, we give some applications to embeddings between approximation spaces in Lorentz–Zygmund spaces and between Besov spaces.

2. Preliminaries

We write for two quasi-normed spaces and to indicate that is continuously embedded in . The notation means that and . If and are positive functions, we write if , where the constant is independent on all significant quantities. We put if and . We adopt the convention that . We use the abbreviation and .

Throughout the paper, let be the -dimensional torus with Lebesgue measure. We consider (equivalence classes of) complex-valued measurable functions on and bounded complex-valued sequences . As usual [20–23], and are the nonincreasing rearrangements of a function and of a sequence , respectively. Because the measure of the torus equals 1, if .

Definition 1. Let and . The Lorentz–Zygmund space can be defined as follows:

where is the usual (quasi-)norm on the Lebesgue space (0,1). Similarly,

where is the usual (quasi-)norm on the Lebesgue sequence space .

We use the same notation for both (quasi-)norms. If , then coincides with the Lorentz space . If in addition , then the space becomes the Lebesgue space . The same is valid for sequence spaces, too. Note that is not trivial if and only if either or and . For the detailed information about these spaces we refer to [21–24]. It is known that all spaces are complete. Moreover [23, Theorem 7.4], on there exists a norm, equivalent to , iff , where

In this case, it is a Banach function space. For details see [20, 23]. In addition, let

The following statement must surely be known. (see Acknowledgments).

Lemma 2. Let . Then, the space is -normed, that is, there exists such that

for each sequence such that the series converse in .

Proof. It is enough to prove that (7) holds for finite sums . Because , for , we have

Let be as usual the maximal function of given by . If or , , then [21, Theorem 9.1 and Theorem 9.3 (i)] . Hence, for each function , it holds and therefore is well-defined. In virtue of the monotonicity and subadditivity of (see [20, Chapter 2, Proposition 3.2 and Theorem 3.4]), we conclude:

which together with [20, Chapter 2, (1.20)] yields that

Applying now Hardy’s lemma [20, Chapter 2, Proposition 3.6], we infer that

for any nonnegative decreasing function . Let , , or , . In this case, we can find a decreasing function such that (see, e.g., [25]). Therefore, by (11),

The case , is trivial. This completes the proof.

Lemma 3. Let , , , and . Then,

3. Main Results

Let be a finite set of lattice points. We denote the number of elements of the set by . Everywhere below . So, for the th Fourier coefficient of the function it holds . Hence, .

Our goal is to obtain Nikol’skii–type inequalities of the form

where the constant does not depend on . Or, writing it shorter

Obviously, only the situations are of interest, otherwise .

For any triple , we define a corresponding natural number in the following way. Let . For all triples , we set . For each other triple , we define as the smallest integer such that . We set . Note that .

Theorem 4. Let , , , and . For the triple , we assume that either , , , or , . Then

Proof. For the case , from [19, Theorem 1] we immediately have

Consider the case . Note that for the triple the corresponding -value is 1. Hence, due to (13) and (17) we have

This completes the proof.

Remark 5. Theorem 4 covers Theorems 1 and 2 in [4]. See also [10].

Theorems 6, 8, and 9 can be proved following similar approach as proof of Theorem 4, using [19, Theorems 2, 3, and 4] respectively.

Theorem 6. Let , , , , and . Then

Remark 7. According to [21, Theorem 9.3], the conditions of Theorems 4 and 6 imply that.

Both Theorems 4 and 6 deal with the case . The next two theorems examine the limiting case .

Theorem 8. Let , , , and .

(i) If , then, (ii) If , then

Theorem 9. Let , , , and . Then

Remark 10. According to [22, Proposition 3.1], Theorem 8 (i) and Theorem 9 deal with comparable as well as incomparable pairs of Lorentz–Zygmund spaces. Theorem 8(ii) only handles incomparable pairs.

4. Corollaries for Trigonometric Polynomials of Degree at Most

Let be the set of all trigonometric polynomials of degree at most n , i. e.

where .

Corollary 11. Let , , and . For the triple , we assume that either , , , or , . Then for all .

Proof. Because in Theorem 4 depends only on , , and , we have

For , , the sequence is either increasing or equivalent to an increasing sequence (see, e.g., [25]). In both cases, because of (25)

The last estimate and Theorem 4 imply (24).

Analogously, Theorems 6, 8, and 9 imply next three corollaries.

Corollary 12. Let , , , and . Then for all .

Corollary 13. Let , , and .

(i) If , then for all , (ii) If , then for all ,

Corollary 14. Let , , and . Then for all .

Remark 15. Corollary 13(i) covers [7, (3.16)], [8, Lemma 5.4], and [9, Corollary 2]. Corollary 14 covers [7, Lemma 3.4]. In [26, Lemma 3] a variant of (29) was obtained. Inequality given in [26, Lemma 2] provides a limiting counterpart (i.e., ) of (30). Both Lemmas in [26] involve generalized Lorentz–Zygmund spaces with two iterations of logarithm.

5. Applications

Let be the set of all trigonometric polynomials of degree at most described above. The sequence allows construction of an approximation family in all Lorentz–Zygmund spaces, which produces approximation spaces. In Subsection 5.1 we start with some necessary definitions and auxiliary results dealing with approximation spaces. In Subsection 5.2 we present some corollaries of the statements from Section 4 dealing with embeddings of the approximation spaces into Lorentz–Zygmund spaces and between these approximation spaces. In Subsections 5.3 and 5.4 these corollaries will be reformulated in terms of Besov spaces.

5.1. Basic Approximation Constructions

A quasi-norm on a quasi-Banach space is denoted by . Let be a quasi-Banach space and be the constant from the triangle inequality in the space . A sequence of subsets of is called an approximation family in if the following conditions are satisfied:(1),(2) for all scalars and ,(3) for , .

For and , the th approximation number is defined by

Let , , and . The approximation space is formed by all those , for which with the quasi-norm . Note that coincides with if . Approximation spaces were investigated in many works. For more information about such spaces, we refer to [8, 27–30] and references therein. We will use some common statements from the approximation theory. We begin with the two representation theorems.

Theorem 16 [8, 27, 28]. Let . An element belongs to if and only if there is a representation

with

Moreover,

where the infimum is taken over all possible representations (32), defines an equivalent quasi-norm on with equivalence constants depending only on , , , and . The usual modification shall be made when .

Theorem 17 ([29], Theorem 1). Let and . Denote . An element belongs to if and only if there is a representation

with

Moreover,

where the infimum is taken over all possible representations (35), defines an equivalent quasi-norm on with equivalence constants depending only on , , and . The usual modification shall be made when .

These representation theorems are useful to prove the following two lemmas.

Lemma 18 [8, 30]. Let , be quasi-Banach spaces that are continuously embedded in a Hausdorff topological vector space. Let be an approximation family such that . Assume that there are constants , such that

Then, for and , we have

Lemma 19 (Cf. [8, Lemma 2.1], [28, Theorem 4.3], and [27, Theorem 3.4]). Let , be quasi-Banach spaces which are continuously embedded in a Hausdorff topological vector space. Let be an approximation family such that . Assume that there are constants , , and such that

Then, for , , and , we have

We omit the proof since it can be carried out as in [8, Lemma 2.1]. We will also use some interpolation formulae for approximation spaces. Let , , and . By , we denote the classical real interpolation functor [20, 31] and by the real interpolation functor involving logarithmic factor for ordered couples with integration over (0,1) (see, e.g., [30, 32, 33]). Note that .

Lemma 20 ([30, Proposition 2.7]). Suppose that , , , , and , then

The reiteration relation (see, e.g., [32]) leads to the following result.

Corollary 21. Suppose that , , , , and , , then

5.2. Approximation Spaces in Lorentz–Zygmund Spaces

Everywhere below, we consider the following sequence of subsets: , be the set of all trigonometric polynomials of degree at most . It builds an approximation family in all nontrivial Lorentz–Zygmund spaces. Now we will apply Corollaries 11–14 to the approximation spaces . First, we will investigate embeddings of these approximation spaces into Lorentz–Zygmund spaces. Next, we will investigate embeddings between different approximation spaces for and between different limiting approximation spaces . From Corollary 11, we get the following result.

Corollary 22. Let , , , , . Then

Proof. By Theorem 16, given any , we can find a representation , (convergence in ), such that

Because all , it is not hard to show that the series converges to in . Since is a Banach space, using Corollary 11 and (45), we derive that

Therefore,

Since , we can define and by formulae

From Corollary 21, we have

Furthermore, it is known [32, Theorem 7] that

Using now (47) and [32, Theorem 2], we obtain (44).

From Corollary 12, we immediately get the following result.

Corollary 23. Let , , , , and . Then

Proof. Consider the case . By Theorem 16, given any , we can find a representation , (convergence in ), such that

Because all , it is not hard to show that the series converges to in . Since , using Corollary 12 and (52), we derive that

For the case , by Theorem 16, given any , we can find a representation , (convergence in ), such that

Due to Lemma 2, Corollary 12 and (54), we obtain

This completes the proof.

Using some real interpolation technique formulae we obtain the following result, which complements Corollary 23.

Corollary 24. Let , , , , and . Then

Proof. Using Lemma 20, we get

We consider . From (57), [32, Theorem 6 ,and Lemma 4], and Lemma 20, we have

To complete the proof, we have only to use (47) and [32, Corollary 7]:

Remark 25. Recall that the spaces and are formed by all those for which belongs to and , respectively. However, for , these sequence spaces are incomparable [24]. Note that for , formulae (51) and (56) coincide.

Corollary 26. Let , , , and . If then

Proof. Let . In this case . By Theorem 17, given any , we can find a representation , (convergence in ), such that

Because all , it is not hard to show that the series converges to in . Since, using Corollary 13(i), we derive that

Let now. In this case, . By Theorem 17, given any , we can find a representation , (convergence in ), such that

Due to Lemma 2 and Corollary 13(i), we obtain

This completes the proof.

The following result can be proved analogously based on Corollary 14.

Corollary 27. Let , , , and . Then

The next four corollaries investigate embeddings between different approximation spaces for . They follow immediately from Corollaries 11, 12, 13(i), and 14 and from Lemma 19.

Corollary 28. Let , , . For the triple , we assume that either , , , or , . Then, for , , and , we have

Corollary 29. Let , , , , and . Then, for , , and , we have

Corollary 30. Let , , , , and . Then, for , , and , we have

Corollary 31. Let , , and . Then, for , , and , we have

The two next corollaries deal with embeddings between different limiting approximation spaces . They follow immediately from Lemma 18 and Corollaries 13(i) and 14, respectively.

Corollary 32. Let , , , , and . Then, for 0<u≤∞ and , we have

Corollary 33. Let , , and . Then, for and , we have

5.3. Embeddings of Besov Spaces

Here we give some applications of previous results to embeddings of Besov spaces into Lorentz–Zygmund spaces and between Besov spaces.

There are several definitions of Besov spaces. The Besov space is based on and has classical smoothness and additional logarithmic smoothness with exponent . It is formed by all those such that

with an obvious modification when . Here is the modulus of smoothness of order with respect to the quasi-norm on . It makes sense to consider the spaces with zero classical smoothness only for . All we need for our application is the characterization of the Besov spaces by approximation. An important result in approximation theory states that

For details see, e.g., [8, 30] and the references given there. Using this characterization with Corollaries 22–24 and 26–28, we obtain the following embeddings. Due to similarity, we will give only the proof of Corollary 37.

Corollary 34 (Cf. [8, Theorem 5.2]). Let , , and . Then

Corollary 35. Let , , and . Then

Corollary 36 (Cf. [8, Theorem 5.3]). Let 0<q<∞, 0<b≤∞, and α <−1/b. Then

Corollary 37 (Cf. [8, Theorems 5.5, 5.6, and 5.7]). Let , , and . Then

Proof. Using the characterization (73) and Corollary 26, we obtain

Corollary 38. Let . In addition, let , or and . Then

Corollary 39 (Cf. [34, Corollary 5.3 (i)] and [31, Theorem 2.8.1]). Let . Then, for , , and , we have

5.4. Embeddings of Besov–Type Spaces

In [7], the Besov–type spaces (with zero classical smoothness) based on are introduced, so that . The space is formed by all those such that

with an obvious modification when . Here is the modulus of smoothness of order with respect to the quasi-norm on . The definition is independent of . The spaces have the following characterization by approximation [7, (3.7)]:

Using this characterization and Corollaries 26, 27, 32, and 33, we obtain the following embeddings.

Corollary 40 (Cf. [7, (3.9)]). Let , , , and , . If , then

Corollary 41 (Cf. [7, (3.8)]). Let , , and . Then

Corollary 42. Let , , , , and. Then, for and , we have

Corollary 43. Let , , and . Then, for and , we have

Remark 44. Some other limiting embeddings between Besov spaces based on generalized Lorentz–Zygmund spaces with two iterations of logarithm were obtained in [26, Theorem 2].

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

The author thanks all referees for the helpful comments. In particular, the second anonymous referee provided valuable advices. In the first version of the paper, Corollaries 23, 26, 27, 35, 37, and 39 were formulated under the restriction . The second referee wrote “I think that” these results “can be extended to ” and he formulated the estimate (7) for the case . Furthermore, this referee wrote, “This estimate must surely be known, but since I lack precise references, I include a detailed proof below.” The author could also not find a precise reference and has included, therefore, the formulation and the proof of Lemma 2. In addition, the second referee have pointed out of the paper [26] to me. The author would also like to thank Doctor Dimitri Bulatov for their help during the preparation of the manuscript.

References

S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables,” American Mathematical Society Translations, vol. 80, no. 2, pp. 1–38, 1969.
View at: Google Scholar
S. M. Nikol’skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin, 1975.
D. Jackson, “Certain problems of closest approximation,” Bulletin of the American Mathematical Society, vol. 39, no. 12, pp. 889–907, 1933.
View at: Publisher Site | Google Scholar
R. J. Nessel and G. Wilmes, “Nikolskii–type inequalities for trigonometric polynomials and entire functions of exponential type,” Journal of the Australian Mathematical Society, vol. 25, no. 1, pp. 7–18, 1978.
View at: Publisher Site | Google Scholar
L. A. Sherstneva, “Nikol’skii’s inequalities for trigonometric polynomials in Lorentz spaces,” Moscow University Mathematics Bulletin, vol. 39, no. 4, pp. 75–81, 1984.
View at: Google Scholar
G. Akishev, “Similar to orthogonal system and inequality of different metrics in Lorentz–Zygmund space,” Sibirskii Matematicheskii Zhurnal, vol. 13, no. 1, pp. 5–16, 2013, (Russian).
View at: Google Scholar
A. Gogatishvili, B. Opic, S. Tikhonov, and W. Trebels, “Ulyanov–type inequalities between Lorentz–Zygmund spaces,” Journal of Fourier Analysis and Applications, vol. 20, no. 5, pp. 1020–1049, 2014.
View at: Publisher Site | Google Scholar
O. Domínguez, “Approximation spaces, limiting interpolation and Besov spaces [Ph.D. Thesis],” Madrid, 2017.
View at: Google Scholar
G. Akishev, “An inequality of different metrics in the generalized Lorentz space,” Trudy Instituta Matematiki Mekhaniki. UrO RAN, vol. 24, no. 4, pp. 5–18, 2018.
View at: Publisher Site | Google Scholar
H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, NY, USA, 1987.
O. V. Besov, V. A. Sadovnichii, and S. A. Telyakovskii, “On the scientific work of S. M. Nikol’skii,” Russian Mathematical Surveys, vol. 60, no. 6, pp. 1005–1020, 2005.
View at: Publisher Site | Google Scholar
H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
Z. Ditzian and S. Tikhonov, “Ul’yanov and Nikol’skii–type inequalities,” Journal of Approximation Theory, vol. 133, no. 1, pp. 100–133, 2005.
View at: Publisher Site | Google Scholar
Z. Ditzian and A. Prymak, “Nikol’skii inequalities for Lorentz spaces,” Rocky Mountain Journal of Mathematics, vol. 40, no. 1, pp. 209–223, 2010.
View at: Publisher Site | Google Scholar
Z. Ditzian and A. Prymak, “On Nikol’skii Inequalities for domains in R^d,” Constructive Approximation, vol. 44, no. 1, pp. 23–51, 2016.
View at: Publisher Site | Google Scholar
E. OstrovskyL. Sirota, “Nikolskii–type inequalities for rearrangement invariant spaces,” Functional Analysis, 2008.
View at: Google Scholar
A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, 1983.
E. Nursultanov, M. Ruzhansky, and S. Tikhonov, “Nikolskii inequality and Besov, Triebel–Lizorkin, Wiener, and Beurling spaces on compact homogeneous manifolds,” Annali della Scuola Normale Superiore di Pisa-Classe di Sciene, vol. 16, no. 3, pp. 981–1017, 2016.
View at: Google Scholar
L. R. Ya. Doktorski and D. Gendler, “Nikol’skii inequalities for Lorentz–Zygmund spaces,” Boletín de la Sociedad Matemática Mexicana, 2018.
View at: Publisher Site | Google Scholar
C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
C. Bennett and K. Rudnick, “On Lorentz–Zygmund spaces,” Dissertationes Mathematicae, vol. 175, pp. 1–72, 1980.
View at: Google Scholar
R. Sharpley, “Counterexamples for classical operators in Lorentz–Zygmund spaces,” Studia Mathematica, vol. 68, no. 2, pp. 141–158, 1980.
View at: Google Scholar
B. Opic and L. Pick, “On generalized Lorentz–Zygmund spaces,” Mathematical Inequalities & Applications, vol. 2, no. 3, pp. 391–467, 1998.
View at: Publisher Site | Google Scholar
R. Ya. Doktorskii, “A multiparametric real interpolation method III. Lorentz–Zygmund spaces,” 1988, Manuscript No. 6070–B88, deposited at VINITI (Russian).
View at: Google Scholar
A. Gogatishvili, B. Opic, and W. Trebels, “Limiting reiteration for real interpolation with slowly varying functions,” Mathematische Nachrichten, vol. 278, no. 1–2, pp. 86–107, 2005.
View at: Publisher Site | Google Scholar
O. Domínguez, “Ul’yanov-type inequalities and embeddings between Besov spaces: the case of parameters with limit values,” Mathematical Inequalities & Applications, vol. 20, no. 3, pp. 755–772, 1998.
View at: Publisher Site | Google Scholar
A. Pietsch, “Approximation spaces,” Journal of Approximation Theory, vol. 32, no. 2, pp. 115–134, 1981.
View at: Publisher Site | Google Scholar
E. Pustylnik, “Ultrasymmetric sequence spaces in approximation theory,” Collectanea Mathematica, vol. 57, pp. 257–277, 2006.
View at: Google Scholar
F. Fehér and G. Grässler, “On an extremal scale of approximation spaces,” Journal of Computational Analysis and Applications, vol. 3, no. 2, pp. 95–108, 2001.
View at: Publisher Site | Google Scholar
F. Cobos and O. Domínguez, “Approximation spaces, limiting interpolation and Besov spaces,” Journal of Approximation Theory, vol. 189, pp. 43–66, 2015.
View at: Publisher Site | Google Scholar
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth Verlag, North-Holland, Amsterdam, 1978.
R. Ya. Doktorskii, “Reiteration relations of the real interpolation method,” Soviet Mathematics Doklady, vol. 44, no. 3, pp. 665–669, 1992.
View at: Google Scholar
L. R. Ya. Doktorski, “Limiting reiteration for real interpolation with logarithmic functions,” Boletín de la Sociedad Matemática Mexicana, vol. 22, no. 2, pp. 679–693, 2016.
View at: Publisher Site | Google Scholar
R. A. DeVore, S. D. Riemenschneider, and R. C. Sharpley, “Weak interpolation in Banach spaces,” Journal of Functional Analysis, vol. 33, no. 1, pp. 58–94, 1979.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Leo R. Ya. Doktorski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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