Abstract and Applied Analysis

Volume 2014 (2014), Article ID 801531, 9 pages

http://dx.doi.org/10.1155/2014/801531

Research Article

Interpolation of Gentle Spaces

Mourad Ben Slimane¹ and Hnia Ben Braiek²

¹Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

²Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire de Recherche Équations aux Dérivées Partielles et Applications, 2092 Tunis, Tunisia

Received 1 November 2013; Revised 21 February 2014; Accepted 28 February 2014; Published 6 May 2014

Academic Editor: Paul W. Eloe

Copyright © 2014 Mourad Ben Slimane and Hnia Ben Braiek. 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.

Abstract.

The notion of gentle spaces, introduced by Jaffard, describes what would be an “ideal” function space to work with wavelet coefficients. It is based mainly on the separability, the existence of bases, the homogeneity, and the γ-stability. We prove that real and complex interpolation spaces between two gentle spaces are also gentle. This shows the relevance and the stability of this notion. We deduce that Lorentz spaces and spaces are gentle. Further, an application to nonlinear approximation is presented.

1. Introduction

Interpolation is a powerful technique for proving continuity of linear operators. Let us recall some basic notions concerning interpolation between Banach spaces. Let and be two Banach spaces. We say that is a compatible couple if and are continuously embedded in a common Hausdorff topological vector space (we write for ).

Let be a compatible couple. Then and are Banach spaces under the norms

where the infimum extends over all representations of with and .

If is a compatible couple, then a Banach space is said to be an intermediate space between and if is continuously embedded between and .

Let be a second compatible couple. A linear operator defined on and taking values in is said to be admissible with respect to couples and if, for each , the restriction of to is a linear continuous operator from into (if , then is said to be admissible with respect to ). One looks for intermediate spaces and of the couples and , respectively, such that every admissible operator maps into . The pair is called interpolation pair relative to and . If for and , then is called an interpolation space between and .

Note that , , , and are examples of interpolation spaces between and . Other examples can be constructed by several methods.

In 1926, Riesz found the first interpolation method for . A generalized version was given by Thorin in 1939/1948 and is known as the Convexity Theorem of Riesz and Thorin or the Riesz-Thorin interpolation theorem. There are many extensions of this theorem. In this connection, we mention the Marcinkiewicz interpolation theorem (in 1939) which extends the Riesz-Thorin interpolation theorem to couples of weak -spaces and which was proved by Zygmund in 1956. In 1958, Stein and Weiss generalized the method for couples with different measures and . At the end of 1958, Lions gave the first proof of the interpolation theorem for quadratic interpolation between Hilbert spaces. Since then several authors have introduced and developed different interpolation methods for couples of general Banach spaces. We mention here essentially two methods: the real interpolation method introduced by Lions and Peetre and the complex method developed by Lions, Calderón, and Krejn. In general, these methods lead to different interpolation spaces. Let us quote the example of usual complex spaces of -integrable functions and homogeneous Bessel-potential spaces . If we denote by and , with , the interpolation spaces obtained, respectively, by the complex and the real method, then we have (see for instance [1])

(i), , and .(ii), , , and .(iii), , ,   and .(iv), , and ,  .(v), , and .

Here, and are, respectively, Lorentz and homogenous Besov spaces (see [1–3]). The spaces for , where and , and with and , are gentle spaces (see [4]). Nonetheless, inhomogeneous Besov spaces are not gentle because the homogeneity property (the second requirement in Definition 1) is not verified. Recall that

This means that the interpolation space between two gentle spaces is not always gentle (recall that, above, we said that , , , and are examples of interpolation spaces between and ). However, in the third section of this paper, we will prove that gentleness is stable by real and complex interpolation methods. In the next section, we give all the necessary recalls concerning these methods.

The notion of gentleness was introduced by Jaffard [5]. It describes what would be an “ideal” function space to work with wavelet coefficients, in “any” wavelet basis. This is the case for Sobolev spaces in PDEs (see [6, 7] for instance), and for Besov spaces in statistics, see [8]. Moreover, many signals and images are stored, denoised, or transmitted by their wavelet coefficients (see [9]). One often needs to obtain local or global information on signals or images by conditions bearing on the moduli of their wavelet coefficients. These conditions should be independent of the chosen wavelet basis.

Gentleness is based mainly on separability, existence of bases, homogeneity, and -stability; let be the Schwartz space of all complex valued rapidly decreasing functions on . Set

(where for all and all ). By we denote its topological dual (also called the space of tempered distributions modulo polynomials).

Definition 1. Let . A function space is gentle of order if we have the following.

(i) is a Banach or a quasi-Banach space of distributions.(ii) is homogeneous of order ; that is, there exists a constant such that for all , all and all

where and are the shift and the dilation operators defined by

(iii).(iv)If is separable, then is dense in , and if is the dual of a separable space , then is dense in .(v)There exists such that is -stable.

The first requirement is explained in the following definition introduced by Bourdaud in [10]. Denote by the dual space of the space of all complex valued compactly supported functions on .

Definition 2. A Banach (resp., quasi-Banach) space of distributions is a vector subspace of endowed with a complete norm (resp., quasi-norm) such that the embedding is continuous.

Recall that a quasi-Banach space is a complete topological vector space endowed with a quasi-norm. A quasi-norm (see [1] page 59) satisfies the requirements of a norm except for the triangular inequality which is replaced by the weaker condition

We say that is a -norm where if in addition

Note that a quasi-norm is always equivalent to a -norm (see [11]). The real Hardy spaces and Besov spaces , with and , are quasi-Banach spaces.

We should point out that the definitions of intermediate and interpolation spaces carry over without change for a compatible couple of quasi-Banach spaces (see [12–14]).

In the second requirement in Definition 1, the shift invariance implies that the definition of pointwise regularity introduced by Jaffard in [5] (which describes how the norm of (properly renormalized by substracting a polynomial) behaves in small neighbourhoods of a given point ) is the same at every point, and the dilation invariance is implicit in pointwise regularity through scaling invariance. This requirement and the third one imply that Meyer-Lemarié wavelets [15] belong to since they are obtained by translations and dyadic dilations of a basic function (mother wavelet) in .

The fourth requirement shows that wavelet bases are either unconditional bases or unconditional -weak bases of gentle spaces. Examples of nonseparable spaces for which wavelets are -weak bases include homogeneous Hölder spaces and, more generally, homogeneous Besov spaces with or .

The last requirement implies that the characterization of gentle space by wavelet coefficients does not depend on the particular -smooth wavelet basis (see [15]) which is chosen for ; let be the space of infinite matrices indexed by dyadic cubes and (where and ) and satisfying

where

(here denotes the Euclidean norm). Meyer proved that is algebra. Besides, he defined as the algebra of bounded operators on whose matrices on a -smooth wavelet basis (for a ) belong to and showed that this definition does not depend on the chosen wavelet basis. In particular, we can use compactly supported wavelet bases.

Definition 3. Let . A Banach or a quasi-Banach space of distributions is -stable if the operators of are continuous on .

In [16], we extended the notion of gentle spaces to include anisotropic homogeneous Besov spaces.

In the fourth section of this paper, we will apply our results to exhibit new examples of gentle spaces, namely, Lorentz spaces (see [2]) and spaces (see [17]). We will also prove that if the Jackson and Bernstein inequalities are valid, then “nonlinear approximation space” as defined in [18] associated to a gentle space is gentle. Note that there are different types of nonlinear approximations. The -term approximation is one of the dominant types. We can mention, for example, approximation by splines with free knots or by rational functions of degree ; see DeVore and Popov [19]. Approximation by a linear combination with -term of -function was developed by DeVore et al. in [20]. A generalization of -term approximation (called restricted approximation) by a linear combination of compactly supported biorthogonal wavelets was presented by Cohen et al. in [21]. In this paper, we consider approximation by a linear combination of Lemarié-Meyer wavelets as was done by Kyriazis in [18]. This form of approximation occurs in several applications including image processing, statistical estimation, and numerical solutions of differential equations.

2. Real and Complex Interpolation Methods

Originally, real and complex interpolation methods were developed for Banach spaces. The extension of real interpolation for quasi-Banach spaces causes no serious problem (see [12, 13, 22]). However, for the complex method the situation is quite different (see [23–25]). Let us recall briefly some basic definitions and notations related to these two methods. For more details see [1, 12, 13, 22, 26–28].

Definition 4. Let be a compatible couple of Banach or quasi-Banach spaces.

(1)The -functional is defined for each and by

where the infimum extends over all representations of with and .(2)The -functional is defined for each and by

The -functional and -functional, introduced By Peetre, are nonnegative concave and increasing functions.

Definition 5. Let be a compatible couple of Banach or quasi-Banach spaces.

(1)Let and or let and . The space consists of all such that

is finite.(2)Let and . The space consists of all that are represented by Bochner-integral

where is measurable with values in and such that

is finite, where the infimum is taken over all representations (13) of .

Remark 6. There is a discrete representation of the space (see [1]); in fact if and only if there exists a sequence in such that

or

Moreover,

where the infimum extends over all sequences satisfying (15).

The following result is given in [1] (see also [29]).

Theorem 7. Let be a compatible couple of Banach (resp., quasi-Banach) spaces. Then, spaces , with and or and , and , with and , equipped, respectively, by the norms (resp., quasi-norms) (12) and (14) are Banach (resp., quasi-Banach) spaces and are interpolation spaces between and .

Furthermore, if and , then with equivalence of norms (resp., quasi-norms).

Real interpolation method means either the - or -method. We will write instead of or , if . If or 1 and , then denotes . By we denote the norm or quasi-norm on depending whether this space is Banach or quasi-Banach.

In the complex case, we will first restrict ourselves to Banach spaces. There are two interpolation spaces whose norms are equivalent under some conditions. Set

Let be a compatible couple of Banach spaces. We denote by the space of all functions that are bounded, continuous on , and analytic on , such that functions , , from into , are continuous and tend to zero as .

By we denote the space of all functions that are continuous on and analytic on , satisfying

such that has values in , , for any .

Spaces and provided, respectively, with

are Banach spaces.

Definition 8. Let be a compatible couple of Banach spaces. For all (resp. ) we define (resp., ) as the space of all such that (resp., ) for some (resp., ).

Theorem 9. Let be a compatible couple of Banach spaces. Then, spaces , where and , where , equipped respectively with

are Banach spaces and are interpolation spaces between and .

Remark 10. We have . In general, and are not equal. However, if either or is reflexive and if , then and , for all .

The extension of this method to quasi-Banach spaces is not routine; one cannot use duality as was done for Banach spaces. The duality theorem is not true in general in the quasi-Banach case, and the maximum principle fails for functions taking values in a quasi-Banach space (see [30]).

There are several possible ways to define complex interpolation spaces. For example, in [23] complex interpolation was defined in the framework of Fourier analysis, while in [24] complex interpolation was defined as in the Banach setting but by adding in (which was given in (21)) a third term . In [25], the authors described a new approach to interpolate by the complex method some quasi-Banach spaces; let be the space of all scalar valued functions continuous and bounded on and analytic on . Let be a compatible couple of quasi-Banach spaces. Denote by the collection of all functions that can be written as a finite sum where and . We put

and for all and let

This functional is a semi-quasi-norm. Let

then, is a quasi-normed space. We define as the completion of (see [31]).

3. Real and Complex Interpolation between Gentle Spaces

Assume that and are two gentle spaces. Let be an interpolation space between them. Then, is a Banach or a quasi-Banach space of distributions and .

On the other hand, there exist two constants and such that is -stable, . Set . If , then . So is continuous on and therefore continuous on . Thus, is -stable.

We will now prove the homogeneity property.

Proposition 11. Let and be two gentle spaces of order and , respectively. Then, spaces

(1), with and or and ,(2), with (in the Banach setting),(3), with (in the Banach setting),(4), with (in the quasi-Banach setting)

are homogeneous spaces of order .

Proof. (1) Let and or let and . Let , , and such that . Let . From the homogeneity of and we get

By taking the infimum over all such decompositions of , we obtain

(i)If and , then

It follows, by a simple change of variable, that

(i)If and , we obtain

Therefore, the space is homogeneous of order .

Now, let ; there exists such that . Clearly

Thus, and are analytic functions.

From (23), (21), and the homogeneity of and , we can easily see that

Put

Therefore,

Hence,

It follows from (35) and the homogeneity of and that

This implies that

Taking the infimum over all such that , we obtain

If , then there exists such that . So as previously and are analytic functions. Clearly, from (24), (22), and the homogeneity of and , we have

Set

Hence,

Thus,

Let . From (42) and the fact that is homogeneous of order , we get

Similarly

Therefore,

By taking the infimum over all such that , we obtain

Now, let . There exists a sequence such that . Let such that and is written as a finite sum where and . We define

where

and and are analytic functions. By arguing similarly as in , by taking instead of , we get the desired result.

Let us now prove the density property.

Proposition 12. If , then is dense in .

Proof. Since , then is dense in (see e.g., [1]). Therefore it suffices to prove that every function can be approached by a sequence in . The assumption implies that , so by the equivalence between real interpolation spaces we will prove this approach in .

On the other hand, since and are gentle spaces, then wavelets are either unconditional bases or unconditional -weak bases, depending on the separability of these spaces. Using the -stability, we can take Lemarié-Meyer wavelet basis; let and be, respectively, the Lemarié-Meyer [32] father and mother wavelets; that is, and are in the Schwartz class such that all moments of vanish, , and the collection of the union of and is an orthonormal basis of . Let be the set of nonzero vertices of the unit cube in , for each vertex we set

and we define

indexed by dyadic cube , where and . Denote by (resp., ) the set of all dyadic cubes (resp., dyadic cubes at scale ). Then, the collection

is an orthonormal basis of . Thus any function can be written as

Let . Put ; then, belongs to , so to and . Since , we therefore obtain from (15) and (16) that

Set now

It follows from (18) that

which tends to 0 as tends to infinity.

Proposition 13. Suppose that is dense in and in .

(1)Assume that and are Banach spaces. Then

(a)If , then is dense in .(b)If , then dense in .

(2)Assume that and are quasi-Banach spaces. Then is dense in , where .

Proof . Let . There exists such that .

Since is dense in both and , then, for all , there exist two functions and in such that for all ,

Let such that takes values in , , and for all . Put . Thus, from (23), (21), and (58), we obtain

Since , then this implies the density of in .

Now let . Arguing similarly as previously but by replacing by , by , and (58) by

Taking such that takes values in , and for all . To finish the proof, it suffices to put .

For the proof of the density of in , we argue similarly as in by taking instead of , where is a sequence in such that .

Consequence. Suppose that is dense in both and , where , with . Let . It follows from the previous proposition that is dense in both and . On the other hand, since

then is dense in , . From the duality theorem (see [1]), it follows that

Furthermore, if either or is reflexive (then either or is reflexive), then

4. Applications

As a first application we show that Lorentz spaces are gentle. This follows from the fact that (see [1]), where , , and and and (where ) are gentle spaces (see [4]).

In the second application, we consider the real interpolation between Hardy spaces and spaces. In [4], we investigated the gentleness property of some functional spaces and we proved the stability of this property by duality. We also proved that homogeneous Lizorkin-Triebel spaces (see [33]), where , and are gentle spaces. In particular, , where are gentle. As a consequence, since BMO is the topological dual of , then BMO is gentle. In [34], it was shown that, for , , , and , , where is the space of all tempered distributions on such that

where and is a smooth function with (see [17]). Consequently are gentle spaces. Note that for .

Our last application refers to some nonlinear approximation spaces introduced in [18]. Take the Lemarié-Meyer orthonormal basis (53). Let be the set of all functions

where and is a subset of with cardinality . For a given distribution and any quasi-normed subspace , we define

For and , we define the approximation class as the set of all such that

is finite.

In [18], Kyriazis proved that if and are quasi-normed spaces that verify, for some , Jackson inequality () and Bernstein inequality (), then for each and

This result was also proved in [21], where Cohen et al. have used compactly supported biorthogonal wavelets, and in [35], by DeVore and Lorentz, where is embedded in .

In the literature complete characterizations are known for the cases , ; , ; and , , , and . DeVore et al. have shown, in [20], that for any and any

This result was proved by simpler techniques in [21], where the authors proved also that for any and any

and for any and any

Approximation in was investigated in [18]. More precisely, it has been shown that, for every , ,  , , and ,

where and .

Since , , and are gentle spaces, then their approximation spaces (which are real interpolation spaces) are also gentle. Moreover, the pairs , , and , under the above conditions on all parameters , , , , , and , verify the Jackson and Bernstein inequalities. More generally, we deduce that if and are gentle spaces and if the Jackson and Bernstein inequalities are valid, then the approximation space is also gentle.

In [18, 21], the authors restricted themselves to particular wavelets. They remarked that all of their theorems hold in more generality. Using our result, this remark is now confirmed using the gentleness stability by real interpolation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group No. RGP-350.

References

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, New York, NY, USA, 1976.

R. A. Hunt, “On spaces,” L'Enseignement Mathématique, vol. 12, pp. 249–276, 1966.

J. Peetre, New Thoughts on Besov Spaces, Duke University Mathematics Series I, Duke University, 1976.

H. Ben Braiek, Espaces gentils et Analyse de régularité [Ph.D. thesis], Université Tunis El Manar-Université Paris, 2013.

S. Jaffard, “Wavelet techniques for pointwise regularity,” Annales de la Faculté des Sciences de Toulouse. Mathématiques, vol. 15, no. 1, pp. 3–33, 2006.

A. Cohen, W. Dahmen, and R. deVore, “Adaptive wavelet methods for elliptic operator equations: convergence rates,” Mathematics of Computation, vol. 70, no. 233, pp. 27–75, 2001.

S. Jaffard, “Wavelet methods for fast resolution of elliptic problems,” SIAM Journal on Numerical Analysis, vol. 29, no. 4, pp. 965–986, 1992.

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: asymptopia?” Journal of the Royal Statistical Society B, vol. 57, no. 2, pp. 301–369, 1995.

S. Mallat, Wavelet Tour of Signal Processing, Academic Press, 1998.

G. Bourdaud, Analyse Fonctionnelle Dans l'espace Euclidien, Publications Mathématiques de l'Université Paris VII, Paris, France, 1995.

S. Rolewicz, Metric Linear Spaces, vol. 20, PWN. Warsaw, Dordrecht, Germany, 2nd edition, 1985.

P. Krée, “Interpolation d'espaces vectoriels qui ne sont ni normés, ni complets. Applications,” Université de Grenoble. Annales de l'Institut Fourier, vol. 17, pp. 137–174, 1967.

T. Holmstedt, “Interpolation of quasi-normed spaces,” Mathematica Scandinavica, vol. 26, pp. 177–199, 1970.

Y. Sagher, “Interpolation of -Banach spaces,” Studia Mathematica, vol. 41, pp. 45–70, 1972.

Y. Meyer, Ondelettes et Opérateurs. I-II, Hermann, Paris, France, 1990.

M. Ben Slimane and H. Ben Braiek, “On the gentle properties of anisotropic Besov spaces,” Journal of Mathematical Analysis and Applications, vol. 396, no. 1, pp. 21–48, 2012.

C. Fefferman, N. M. Rivière, and Y. Sagher, “Interpolation between spaces: the real method,” Transactions of the American Mathematical Society, vol. 191, pp. 75–81, 1974.

G. Kyriazis, “Non-linear approximation and interpolation spaces,” Journal of Approximation Theory, vol. 113, no. 1, pp. 110–126, 2001.

R. A. deVore and V. A. Popov, “Interpolation spaces and nonlinear approximation,” Function Spaces and Applications, vol. 1302 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988.

R. A. deVore, B. Jawerth, and V. Popov, “Compression of wavelet decompositions,” American Journal of Mathematics, vol. 114, no. 4, pp. 737–785, 1992.

A. Cohen, R. A. deVore, and R. Hochmuth, “Restricted nonlinear approximation,” Constructive Approximation, vol. 16, no. 1, pp. 85–113, 2000.

M. Mastyło, “On interpolation of some quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 403–419, 1990.

A.-P. Calderón and A. Torchinsky, “Parabolic maximal functions associated with a distribution. II,” Advances in Mathematics, vol. 24, no. 2, pp. 101–171, 1977.

S. Janson and P. W. Jones, “Interpolation between spaces: the complex method,” Journal of Functional Analysis, vol. 48, no. 1, pp. 58–80, 1982.

M. Cwikel, M. Milman, and Y. Sagher, “Complex interpolation of some quasi-Banach spaces,” Journal of Functional Analysis, vol. 65, no. 3, pp. 339–347, 1986.

C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, Mass, USA, 1988.

A.-P. Calderón, “Intermediate spaces and interpolation, the complex method,” Studia Mathematica, vol. 24, pp. 113–190, 1964.

J. Peetre, “A theory of interpolation of normed spaces,” in Notas de Matematica, vol. 39, pp. 1–86, Rio de Janeiro, Brazil, 1963.

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, The Netherlands, 1978.

J. Peetre, “Locally analytically pseudoconvex topological vector spaces,” Studia Mathematica, vol. 63, no. 3, pp. 253–269, 1982.

F. Cobos, J. Peetre, and L. E. Persson, “On the connection between real and complex interpolation of quasi-Banach spaces,” Bulletin des Sciences Mathématiques, vol. 122, no. 1, pp. 17–37, 1998.

P. G. Lemarié and Y. Meyer, “Ondelettes et bases hilbertiennes,” Revista Matemática Iberoamericana, vol. 2, no. 1-2, pp. 1–18, 1986.

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983.

R. Hanks, “Interpolation by the real method between BMO, and ,” Indiana University Mathematics Journal, vol. 26, no. 4, pp. 679–689, 1977.

R. A. deVore and G. G. Lorentz, Constructive Approximation, vol. 303, Springer, Berlin, Germany, 1993.