Tensor products of Sobolev-Besov spaces and applications to ...

Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross Dedicated to our dear colleague and friend G´erard Bourdaud on occasion of his 60th birthday

Winfried Sickel ∗ Friedrich-Schiller-University Jena, Mathematical Institute, Ernst-Abbe-Platz 3, D-07740, Germany

Tino Ullrich 1 Hausdorff-Center for Mathematics, Endenicher Allee 60, D-53115 Bonn, Germany

Abstract Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R. Based on this we derive several useful characterizations from the the one-dimensional case to the ddimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed. Key words: Tensor products, Besov spaces, fractional Sobolev spaces, Besov spaces of dominating mixed smoothness, Sobolev spaces of dominating mixed smoothness, p-nuclear, projective and injective norm, wavelet decompositions, approximation from hyperbolic crosses, tensor product splines, best n-term approximation. 1991 MSC: 41A25, 41A63, 42B99, 46E35.

∗ Corresponding author Email addresses: [email protected] (Winfried Sickel), [email protected] (Tino Ullrich). 1 Supported by the German Academic Exchange Service DAAD (D/06/48125)

Preprint submitted to Elsevier

19 December 2008

1

Introduction

The present paper deals with characterizations of iterated tensor products of Sobolev and Besov spaces on R. Our main results are the identities r1 rd r1 ,...,rd Bp,p (R) ⊗α . . . ⊗α Bp,p (R) = Sp,p B(Rd )

for 0 < p < ∞, ri ∈ R, i = 1, . . . , d, and Hpr1 (R) ⊗α . . . ⊗α Hprd (R) = Spr1 ,...,rd H(Rd ) for 1 < p < ∞ and ri ∈ R, i = 1, . . . , d. Here α denotes a specific tensor (quasi-)norm depending on p. Based on these formulas we will discuss a new approach to estimates of certain best approximations related to hyperbolic crosses. Here we concentrate on tensor product splines and derive results parallel to the well-known estimates for trigonometric polynomials with frequencies from hyperbolic crosses. In addition we consider best m-term approximation with respect to tensor product wavelet systems oriented on some recent results of Nitsche [22]. Function spaces of dominating mixed smoothness have been introduced by Nikol’skij around 1962. They are systematically studied by Amanov [1], Schmeisser [29] and Vybiral [48]. Connections to hyperbolic cross approximation (in the context of periodic functions) may be found in Temlyakov [37], Sprengel [35] as well as in [34] and [45]. Definitions will be recalled in Appendix A. There is a well-developed abstract theory for tensor products of Banach spaces, cf. the monograph Defant and Floret [8]. Since we are dealing with function spaces and sequence spaces only we do not need the abstract theory in its full generality. Our approach is based on the treatment of Light and Cheney [20]. Basic concepts of the abstract theory of tensor products of Banach spaces are collected in Appendix B. There we also discuss an extension of the projective norm to tensor products of certain quasi-Banach spaces. This is parallel to the approach suggested by Nitsche [22]. Acknowledgement: We would like to thank Markus Hansen for a critical reading of a preliminary version of this manuscript and for several valuable hints to improve it. Notation The symbols R, C, N, N0 and Z denote the real numbers, complex numbers, natural numbers, natural numbers including 0 and the integers. The natural number d is reserved for the dimension of the considered Euclidean space Rd . The Euclidean distance of x ∈ Rd to the origin is given by |x|2 , whereas the 2

`d1 -norm is denoted by |x|1 . We often need further vector-type quantities like ¯ k, ¯ ¯j and r¯ with numbered indices and parameters. They are denoted by `, ¯ components . As usual we put r¯ + ` = (r1 + `1 , ..., rd + `d ), λ · `¯ = (λ · `1 , ..., λ · `d ) , λ ∈ R , and k¯ · r¯ = k1 r1 + ... + kd rd . For a multi-index α ¯ we define the differential operator Dα¯ by ¯1 ∂ |α| D = . ∂xα1 1 · · · ∂xαd d α ¯

Let X and Y be quasi-Banach spaces. Then L(X, Y ) denotes the class of all linear and bounded operators P : X → Y equipped with the usual quasi-norm. We also use the notation a ³ b if there exists a constant c > 0 (independent of the context dependent relevant parameters) such that c−1 a ≤ b ≤ c a . Constants will change their value from line to line, sometimes indicated by adding subscripts.

2

Main results

To begin with we describe our results on the representation of certain Besov as well as Sobolev spaces of dominating mixed smoothness as iterated tensor products of Besov and Sobolev spaces defined on R. As an application we obtain characterizations by means of wavelets under improved conditions. Then we continue with a discussion of some applications to hyperbolic cross approximation. 2.1 Tensor products of Sobolev and Besov spaces Recall the definitions of the distribution spaces in Appendix A. Basic notions of tensor products can be found in Appendix B. Theorem 2.1 (Tensor products of Sobolev spaces) Let 1 < p < ∞, d ∈ N, and r1 , r2 , . . . , rd+1 ∈ R. Then Hpr1 (R) ⊗αp Spr2 ,...,rd+1 H(Rd ) = Spr1 ,...,rd H(Rd ) ⊗αp Hprd+1 (R) = Spr1 ,r2 ,...,rd+1 H(Rd+1 ) . Remark (i) Here and in what follows we identify Spr1 ,...,rd H(Rd ) and Spr2 ,...,rd+1 H(Rd ) with Hpr1 (R) and Hpr2 (R) if d = 1. 3

(2.1)

(ii) The norms on the left-hand side and on the right-hand side in (2.1) coincide. That could be of some interest for the investigation of the d-dependence of some operators, in particular for large d. (iii) Defining for m > 2 µ

X1 ⊗αp X2 ⊗αp . . . ⊗αp Xm := X1 ⊗αp

¶

. . . Xm−2 ⊗αp (Xm−1 ⊗αp Xm )

we obtain an interpretation of Spr1 ,...,rd H(Rd ) as an iterated tensor product of the Sobolev spaces of fractional order. In all cases of occurence of iterated tensor products of quasi-Banach spaces considered in this paper the resulting space will not depend on the order of the tuples which are formed during the process of calculating X1 ⊗αp X2 ⊗αp . . . ⊗αp Xm , i.e., (X1 ⊗αp X2 ) ⊗αp X3 = X1 ⊗αp (X2 ⊗αp X3 ) . (iv) In the periodic case with d = 1 formula (2.1) is already known, see Sprengel [35]. Theorem 2.2 (Tensor products of Besov spaces) Let d ≥ 1 and let r1 , . . . , rd+1 ∈ R. (i) Let 1 < p < ∞. Then the following r1 r2 ,... ,rd+1 r1 ,...,rd rd+1 Bp,p (R) ⊗αp Sp,p B(Rd ) = Sp,p B(Rd ) ⊗αp Bp,p (R) r1 ,r2 ,...,rd+1 = Sp,p B(Rd+1 )

(2.2)

holds true in the sense of equivalent norms. (ii) Let p = ∞. Then we have r1 r2 ,...,rd+1 r1 ,...,rd rd+1 ˚∞,∞ ˚∞,∞ ˚∞,∞ ˚∞,∞ B (R) ⊗λ S B(Rd ) = S B(Rd ) ⊗λ B (R) r1 ,r2 ,...,rd+1 ˚∞,∞ =S B(Rd+1 )

in the sense of equivalent norms. (iii) Let 0 < p ≤ 1. Then the following formula rd+1 r1 ,...,rd r2 ,... ,rd+1 r1 (R) B(Rd ) ⊗γp Bp,p B(Rd ) = Sp,p (R) ⊗γp Sp,p Bp,p r1 ,r2 ,...,rd+1 = Sp,p B(Rd+1 )

holds true in the sense of equivalent quasi-norms. r1 ,...,rd B(Rd ) and Remark (a) Similarly as above we identify Sp,p r2 r1 r2 ,...,rd+1 (R) if d = 1. (R) and Bp,p B(Rd ) with Bp,p Sp,p (b) The fact (iii) in the remark after Theorem 2.1 applies, mutatis mutandis, r1 ,...,rd r1 ,...,rd ˚∞,∞ B(Rd ), respectively. B(Rd ), 0 < p < ∞, and S for the spaces Sp,p

4

(c) Nitsche [22] has introduced tensor products of Besov spaces with 0 < p < 1 in connection with best m-term approximation for tensor product wavelets. But in his paper, no relation to spaces of dominating mixed smoothness is given. (d) In a periodic context (2.2) (with d = 1) has been proved earlier in [33]. 2.2 Wavelets and spaces of dominating mixed smoothness Since many years it is well-known that isotropic Besov spaces as well as isotropic Sobolev spaces allow a discretization. That means, there exist isomorphisms onto sequence spaces, see Subsection A.3.1 for details. The following sequence spaces play a major role in our investigations. Definition 2.3 Let 0 < p ≤ ∞ and let r, r1 , . . . , rd ∈ R. (i) We put ½

brp

:= (aj,k )j,k ⊂ C :

k a |brp k

:=

µX ∞ X ∞

j(r+ 12 − p1 )p

2

|aj,k |

p

¶1/p

¾

1 and r1 , . . . , rd ∈ R. i )j,k be an unconditional basis of Hpri (R) such that (i) Let 1 < p < ∞. Let (ψj,k k f |Hpri (R)k

³ 6

i k hf, ψj,k i |fpri k

holds for all f ∈ Hpri (R) and all i = 1, . . . , d. Then the system of functions Ψ is an unconditional basis of Spr1 ,...,rd H(Rd ) such that k f |Spr1 ,...,rd H(Rd )k

³

k hf, ψ¯j,k¯ i |spr1 ,...,rd f k

holds for all f ∈ Spr1 ,...,rd H(Rd ). i ri (ii) Let 0 < p < ∞. Let (ψj,k )j,k be an unconditional basis of Bp,p (R) such that ri k f |Bp,p (R)k

³

i k hf, ψj,k i |brpi k

ri holds for all f ∈ Bp,p (R) and all i = 1, . . . , d. Then the system of functions Ψ r1 ,...,rd B(Rd ) such that is an unconditional basis of Sp,p r1 ,...,rd k f |Sp,p B(Rd )k

³

k hf, ψ¯j,k¯ i |spr1 ,...,rd bk

r1 ,...,rd holds for all f ∈ Sp,p B(Rd ). i ˚ri (R) such that (iii) Let (ψj,k )j,k be an unconditional basis of B ∞,∞ ri k f |B∞,∞ (R)k

³

i k hf, ψj,k i |br∞i k

˚ri (R) and all i = 1, . . . , d. Then Ψ is an unconditional holds for all f ∈ B p,p r1 ,...,rd d ˚ basis of S∞,∞ B(R ) such that r1 ,...,rd k f |S∞,∞ B(Rd )k

³

r1 ,...,rd k hf, ψ¯j,k¯ i |s∞ bk

r1 ,...,rd ˚∞,∞ holds for all f ∈ S B(Rd ).

Remark Observe that we only need to know the restrictions for the wavelet isomorphism for d = 1. No further conditions enter.

2.3 Nonlinear approximation and spaces of dominating mixed smoothness Let 0 < p < 2. Let ϕ be an orthonormal scaling function and ψ be an associated wavelet such that the system (ψj,k )j,k given by ψ0,k (t) := ϕ(t − k)

ψj+1,k (t) := 2j/2 ψ(2j t − k) ,

and

1/p−1/2 (R) where t ∈ R, k ∈ Z and j ∈ N0 , yields an unconditional basis of Bp,p satisfying 1

−1

p 2 k f |Bp,p (R)k

1

³

k hf, ψj,k i |bpp 7

− 12

k = k hf, ψj,k i |`p k

1/p−1/2 for all f ∈ Bp,p (R). By Ψ, see (2.3), we denote the corresponding tensor product system. Further, let

Σm :=

½ X

¾

a¯j,k¯ ψ¯j,k¯ :

Λ⊂

¯ (¯ j,k)∈Λ

Nd0

d

×Z ,

|Λ| ≤ m ,

m ∈ N,

(here |Λ| denotes the cardinality of the set Λ). The best m-term approximation of f ∈ L2 (Rd ) by the tensor product wavelet system Ψ with respect to the L2 -norm is the quantity ¾

½

σm (f )2 := inf k f − g |L2 (Rd )k :

g ∈ Σm .

For s > 0 and p as above we consider the approximation space Asp (L2 (Rd )), see e.g. [9], as the collection of all f ∈ L2 (Rd ) such that kf

|Asp (L2 (Rd ))k

d

:= k f |L2 (R )k +

µX ∞ 1 m=1

m

s

p

¶1/p

[m σm (f )2 ] 1

< ∞.

−1

Mainly as a corollary of the characterization of App 2 (L2 (Rd )), see e.g. Pietsch [25] or DeVore [9, Thm. 4], and of Corollary 2.6 we obtain the following. Theorem 2.7 (Nonlinear approximation spaces) Let 0 < p < 2. Then we have 1

App

− 12

1

− 12 ,..., p1 − 12

p (L2 (Rd )) = Sp,p

B(Rd )

in the sense of equivalent quasi-norms. 1/p−1/2,...,1/p−1/2 Remark With the right-hand side Sp,p B(Rd ) replaced by the 1/p−1/2 1/p−1/2 tensor product Bp,p (R)⊗p . . . ⊗p Bp,p (R) Theorem 2.7 has been proved in Nitsche [22]. Our treatment of tensor products, see Theorem 2.2, allows to identify this approximation space with a Besov space of dominating mixed smoothness. Recall, that these spaces can be described by means of differences and derivatives, cf. [31] and [44].

2.4 Splines and spaces of dominating mixed smoothness A case of particular importance is given by choosing J1 = . . . = Jd to be an isomorphism associated to an orthonormal spline wavelet system. Let m ∈ N. Let X be the characteristic function of the interval (0, 1). Then the normalized cardinal B-spline of order m + 1 is given by Nm+1 (x) := Nm ∗ X (x) , 8

x ∈ R,

m ∈ N,

beginning with N1 = X . Let F denote the Fourier transform and F −1 its inverse transform. We normalize these transformations by 1 Z ∞ −ixξ Ff (ξ) := √ e f (x) dx , 2π −∞

ξ ∈ R,

f ∈ L1 (R) .

By ·

1 ϕm (x) := √ F −1 µ ∞ P 2π k=−∞

¸

FNm (ξ) |FNm (ξ +

2πk)|2

¶1/2 (x) ,

x ∈ R,

we obtain an orthonormal scaling function which is again a spline of order m. Finally, by ∞ X

ψm (x) :=

h ϕm (t/2), ϕm (t − k)i (−1)k ϕm (2x + k + 1)

k=−∞

we obtain the generator of an orthonormal wavelet system. For m = 1 it is easily checked that −ψ1 (x − 1) is the Haar wavelet. In general these functions ψm have the following properties: (i) ψm restricted to intervals [ k2 , k+1 ], k ∈ Z, is a polynomial of degree at most 2 m − 1. (ii) ψm ∈ C m−2 (R) if m ≥ 2. (m−2) (iii) ψm is uniformly Lipschitz continuous on R if m ≥ 2. (iv) There exist positive numbers τm and sequences (ck )k and (dk )k such that ψm (x) =

∞ X

ck Nm (2x − k) ,

Nm (x) =

k=−∞

∞ X

dk ϕm (x − k) ,

x ∈ R,

k=−∞

and sup (|ck | + |dk |) eτm |k| < ∞

and

k∈Z

(`) max sup |ψm (x)| eτm |x| < ∞ .

0≤`≤m−2 x∈R

(v) The functions ψm satisfy a moment condition of order m − 1, i.e. Z ∞ −∞

x` ψm (x) dx = 0 ,

` = 0, 1, . . . , m − 1 .

It will be convenient for us to use the following abbreviations: m ψ0,k (x) := ϕm (x − k)

and

m ψj+1,k (x) := 2j/2 ψm (2j x − k) ,

where x ∈ R, k ∈ Z and j ∈ N0 . Let us now focus on the mapping Rm :

m i)j,k . f 7→ (hf, ψj,k

9

(2.4)

Proposition 2.8 Let m ∈ N. (i) Let 1 < p < ∞ and suppose −m + 1 < r < m − 1. The mapping Rm generates an isomorphism of Hpr (R) onto fpr . (ii) Let either 1 ≤ p < ∞ and −m + 1/p < r < m − 1 + 1/p or 0 < p < 1 and −m + p1 < r < m . Then the mapping Rm generates an isomorphism of r Bp,p (R) onto brp . (iii) Let −m < r < m − 1. The mapping Rm generates an isomorphism of ˚r (R) onto ˚ B br∞ . ∞,∞ Remark (a) The content of this proposition is essentially well-known, see Bourdaud [5], Frazier and Jawerth [12], Meyer [21], Lemarie and Kahane [19] or DeVore [9]. More details will be given later on. However, we wish to mention that the case m = 1 has its own history. The characterization of Besov spaces by means of the Haar basis attracted some attention in the literature, see e.g. Ropela [27], Oswald [23] and Triebel [39,40] for earlier papers with this respect. (b) Wavelet characterizations of Besov spaces with p < 1 are investigated e.g. in Bourdaud [5], Cohen [6], Kyriazis and Petrushev [18] and Triebel [42]. Cohen and Triebel concentrate on biorthogonal (orthogonal) wavelets with compact support. (c) Observe that the interval −m + 1 < r < m − 1 is empty if m = 1, i.e. for the Haar basis. It still seems to be an open question whether the spaces Hps (R), p 6= 2, −1 + 1/p < s < 1/p, can be characterized by means of the s Haar basis. For p = 2 this is guaranteed by H2s (R) = B2,2 (R) in the sense of equivalent norms. As an immediate conclusion of the preceeding proposition and Corollary 2.6 we obtain assertions on the characterization of the spaces Spr1 ,...,rd H(Rd ) and r1 ,... ,rd Sp,p B(Rd ), respectively. Of course, Ψm denotes the system Ψ, cf. (2.3), generated by ϕm and ψm . Theorem 2.9 (Spline wavelet isomorphisms) Let d > 1. (i) Let 1 < p < ∞ and −m + 1 < r1 , . . . , rd < m − 1. The spline system Ψm is an unconditional basis of Spr1 ,...,rd H(Rd ). The quantity k hf, ψ¯j,mk¯ i |srp1 ,...,rd f k represents an equivalent norm in Spr1 ,...,rd H(Rd ). (ii) Let either 1 ≤ p < ∞ and −m + 1/p < r < m − 1 + 1/p or 0 < p < 1 and −m + p1 < r < m. The spline system Ψm is an unconditional basis r1 ,...,rd B(Rd ). The quantity k hf, ψ¯j,mk¯ i |srp1 ,...,rd bk represents an equivalent of Sp,p r1 ,...,rd B(Rd ). quasi-norm in Sp,p (iii) Let −m < r1 , . . . , rd < m − 1. The spline system Ψm is an uncondi˚r1 ,...,rd B(Rd ). The quantity k hf, ψ¯m¯ i |sr1 ,...,rd bk represents an tional basis of S ∞ ∞,∞ j,k d r1 ,... ,rd ˚ equivalent quasi-norm in S∞,∞ B(R ). 10

Remark Kamont [17] has proved a similar result for the spaces r1 ,...,rd Sp,p B([0, 1]d ), 1 < p < ∞, r1 = r2 = . . . = rd > 0, but using a different system of splines, at least if m > 1. The case p = 2 and m = 1 may be found in Oswald [24] as well. 2.5 Approximation from the hyperbolic cross We concentrate on the case r1 = r2 = . . . = rd > 0. Here we make use of the r r,...,r conventions Spr H(Rd ) = Spr,...,r H(Rd ) and Sp,p B(Rd ) = Sp,p B(Rd ). In [10] the following type of approximation related to hyperbolic crosses is considered. Define Pnm f :=

X

X

¯ d |¯ j|1 ≤n k∈Z

hf, ψ¯j,mk¯ i ψ¯j,mk¯

,

n ∈ N.

We are interested in determining the asymptotic behaviour of the quantities r k I − Pnm |L(Sp,p B(Rd ), Lp (Rd ))k

k I − Pnm |L(Spr H(Rd ), Lp (Rd ))k

and

as n tends to infinity. Mainly as a consequence of Theorem 2.9 one obtains the following. Proposition 2.10 Let d > 1. (i) Let 1 < p < ∞ and 0 < r < m − 1. Then k I − Pnm |L(Spr H(Rd ), Lp (Rd ))k ³ 2−rn ,

n ∈ N,

holds. (ii) Let 0 < p < ∞ and max(0, p1 − 1) < r < m − 1 + min(1, 1/p). Then we have for n ∈ N    2−rn

r k I − Pnm |L(Sp,p B(Rd ), Lp (Rd ))k ³  

1

1

n(d−1)( 2 − p ) 2−rn

if

0 < p ≤ 2,

if

2 < p < ∞.

(iii) Let p = ∞ and 0 < r < m − 1. Then we have r k I − Pnm |L(S∞,∞ B(Rd ), L∞ (Rd ))k ³ nd−1 2−rn ,

n ∈ N.

Next we want to define the error of best approximation of a function f ∈ Lp (Rd ), 0 < p ≤ ∞, by splines of degree less than m related to the hyperbolic cross. For this purpose it will be convenient to introduce some further notation first. Let ½

Vjm

¾ j

:= span Nm (2 · −k) :

11

k∈Z ,

j ∈ N0 ,

and ½

Vnm

:= span Nm (2j1 · −k1 ) ⊗ . . . ⊗ Nm (2jd · −kd ) : ¯j ∈ Nd0 ,

|¯j|1 = n ,

¾

k¯ ∈ Zd ,

n ∈ N0 . Sometimes spaces of this type are called sparse grid ansatz spaces. Since “span” contains finite sums only we have Vnm ⊂ Lp (Rd ). We put ¾

½

Enm (f, Lp (Rd ))

d

:= inf k f − g |Lp (R )k :

g∈

Vnm

,

n ∈ N0 .

m , Some comments are necessary. The classes Vnm are nested, i.e. Vnm ⊂ Vn+1 since à ! m X m −m+1 Nm (x) = 2 Nm (2x − k) , x ∈ R. k k=0 Furthermore, the spaces Vnm do not contain our basis functions ψ¯j,mk¯ . However, it becomes obvious from Section 2.4/(iv) that ψ¯j,mk¯ belongs to the closure of Vnm , |¯j|1 = n, in Lp (Rd ). Alternatively to the quantity Enm (f, Lp (Rd )) one could consider the following

½

Eenm (f, Lp (Rd )) := inf a¯j,k¯

k f − g |Lp (Rd )k : g ∈ Lp (Rd ), ∃(a¯j,k¯ )¯j,k¯ X X

Lp = 0 if |¯j|1 > n and g =

¯ j∈Nd0 k∈Zd

s.t. ¾

a¯j,k¯ ψ¯j,mk¯

,

a concept which is related to the definition of the Pnm . Fortunately we have Enm (f, Lp (Rd )) = Eenm (f, Lp (Rd )) . This can be seen by using (iv) in our list of properties of the ψm given above. To have a compact formulation we shall use the following quantity: Enm (F )p := sup Enm (f, Lp (Rd )) kf kF ≤1

where F ,→ Lp denotes an arbitrary quasi-Banach space. Theorem 2.11 (Error of best approximation) Let d > 1 and let m ∈ N. (i) Let 1 < p < ∞ and 0 < r < m − 1. Then it holds Enm (Spr H(Rd ))p ³ 2−rn ,

n ∈ N.

(ii) Let 1 ≤ p < ∞ and 0 < r < m − 1 + 1/p. Then it holds r B(Rd ))p ³ Enm (Sp,p

   2−rn  n

(d−1)( 12 − p1 )

2−rn

12

if

1 ≤ p ≤ 2,

if

2 < p < ∞,

(2.5)

n ∈ N. (iii) Let 0 < p < 1 and that

1 p

− 1 < r < m. Then there exists a constant c such r Enm (Sp,p B(Rd ))p ≤ c 2−rn

holds for all n ∈ N. (iv) Let p = ∞ and 0 < r < m − 1. Then there exists a constant c such that r B(Rd ))∞ ≤ c nd−1 2−rn , Enm (S∞,∞

n ∈ N.

r B(Rd ) ,→ Spr H(Rd ) if 1 < p ≤ 2 and Remark (a) Observe Sp,p r d r d Sp H(R ) ,→ Sp,p B(R ) if 2 ≤ p < ∞, see [46]. All the inclusions are strict except for p = 2 where we have coincidence. (b) DeVore, Konyagin, Temlyakov [10] have also proved the upper bound in (i). But we would like to mention a difference between the results in part (i) and part (ii), respectively. The assumptions in (i) are more restrictive than in part (ii). E.g., if m = 1, i.e. for the Haar system, part (i) is not applicable whereas in (ii) the natural restriction r < 1/p shows up. (c) In case 1 < p < ∞ we refer to Kamont [17] for corresponding estimates in terms of a certain modulus of smoothness for functions defined on [0, 1]d . (d) Bazarkhanov [3,4] derived similar estimates by using Meyer wavelets instead of splines. (e) All estimates stated in Theorem 2.11 have counterparts in the classical periodic context of best approximation by polynomials with frequencies taken from a hyperbolic cross. We refer to Dinh Dung [11], Galeev [13], Romanyuk [26], Temlyakov [37] and [34], [45]. However, the Littlewood-Paley theory for the spline system Ψm differs from the Littlewood-Paley theory of the trigonometric system. So, at least partly, our test functions are not the same as used in the quoted literature.

At least for the most interesting case p = 2 there is a partial inverse of the inequality (2.5). In fact we have the following equivalence. Corollary 2.12 (Characterization via approximation) r Let 0 < r < m − 1/2. A function f ∈ L2 (Rd ) belongs to S2,2 B(Rd ) if and only d rn m if the sequence (2 En (f, L2 (R )))n belongs to `2 . Moreover, we have kf

r |S2,2 B(Rd )k

³

d

k f |L2 (R )k +

µX ∞ µ

rn

2

Enm (f, L2 (Rd ))

¶2 ¶1/2

.

n=0

Remark For m = 1 we refer to Oswald [24]. Let us further mention that there is not much hope to generalize Corollary 2.12 to p 6= 2. In a slightly different setting (approximation by entire analytic functions with frequencies in the hyperbolic cross) it has been shown in [30] that the approximation 13

spaces Arp,q (Rd ) characterized by a condition d

k f |Lp (R )k +

µX ∞ µ

rn

d

¶q ¶1/q

2 En (f, Lp (R ))

M or if ` > N . Then h=

M X N X

hk,` (ek ⊗ e` ) =

k=1 `=1

N µX M X `=1

¶

hk,` w2 (`) ek

⊗ (1/w2 (`)) e` ,

where ek denotes the elements of the canonical basis. Let a` := It follows kh|`p (w1 , N) ⊗αp `p (w2 , N)kp µ

≤ w2 (`)

p

N X

=

N X M X

¶ p

k a` |`p (w1 , N)k

`=1

(3.5)

k=1

sup kλ|`p k≤1

° N ° X ° λ` ° `=1

¯

PM

k=1

hk,` ek .

°

°p e` ¯¯ ° ` (w , N) p 2 ° w2 (`) ¯

w2p (`) |hk,` w1 (k)|p .

`=1 k=1

Hence k h |`p (w1 , N) ⊗αp `p (w2 , N)k ≤ k h |`p (w1 ⊗ w2 , N2 )k . A density argument completes the proof. 2 Proposition 3.4 Let 0 < p ≤ 1. Then `p (w1 , N) ⊗γp `p (w2 , N) = `p (w1 ⊗ w2 , N2 ) .

(3.6)

The quasi-norms on the left-hand side and on the right-hand side coincide. 19

Remark (i) Formula (3.6) with p = 1 is well-known. We refer to [20, Cor. 1.16]. (ii) In the framework of a more general concept of tensor products of quasiBanach spaces, Nitsche [22] has proved a similar result for the unweighted case.

Proof of Proposition 3.4. Step 1. We shall prove `p (w1 , N) ⊗γp `p (w2 , N) ,→ `p (w1 ⊗ w2 , N2 ) . Let h ∈ `p (w1 , N) ⊗ `p (w2 , N) be given by h = (hk,` )k,` ,

hk,` =

n X

aik bi` ,

k, ` ∈ N ,

i=1

where (aik )k ∈ `p (w1 , N), (bi` )` ∈ `p (w2 , N), i = 1, . . . , n. Then the elementary P P inequality ( i |ci |)p ≤ i |ci |p yields 2

p

kh|`p (w1 ⊗ w2 , N )k = ≤

∞ X ∞ X k=1 `=1 ∞ X ∞ X

¯ n ¯ ¯ X i i ¯p w1 (k) w2 (`) ¯ ak b` ¯¯ p

i=1

w1 (k)p w2 (`)p

n X

|aik |p · |bi` |p

i=1

k=1 `=1

=

p¯

n X ∞ X ∞ X

w1 (k)p w2 (`)p |aik |p · |bi` |p

i=1 k=1 `=1

=

n X

kai |`p (w1 , N)kp · kbi |`p (w2 , N)kp .

i=1

Since this is true for all representations of h we conlude kh|`p (w1 ⊗ w2 , N2 )k ≤ γp (h, `p (w1 , N), `p (w2 , N)) . Step 2. It remains to prove `p (w1 ⊗ w2 , N2 ) ,→ `p (w1 , N) ⊗γp `p (w2 , N) . We follow the arguments from Step 2 in the proof of the previous Proposition. By the same density argument it will be enough to deal with finite sequences. Therefore, let h=

M X N X k=1 `=1

hk,` (ek ⊗ e` ) =

N µX M X `=1

k=1

Then we obtain 20

¶

hk,` w2 (`) ek

⊗ (1/w2 (`)) e` .

k h |`p (w1 , N) ⊗γp `p (w2 , N)kp

¯ °p N ° M X ° X ¯ ° ° ¯ ≤ hk,` w2 (`) ek ¯`p (w1 , N)°° k e` /w2 (`) |`p (w2 , N)kp °

=

`=1 k=1 N X M X

|w2 (`)|p |w1 (k)|p |hk,` |p

`=1 k=1

= k h |`p (w1 ⊗ w2 , N2 )kp . This proves the claim. 2 Proposition 3.5 Let p = ∞. Then c0 (w1 , N) ⊗λ c0 (w2 , N) = c0 (w1 ⊗ w2 , N2 ) .

(3.7)

The norms on the left-hand side and on the right-hand side coincide. Proof. Let h ∈ c0 (w1 , N) ⊗ c0 (w2 , N) be given by h = (hk,` )k,` ,

hk,` =

n X

aik bi` ,

k, ` ∈ N ,

i=1

where ai := (aik )k ∈ c0 (w1 , N), bi := (bi` )` ∈ c0 (w2 , N) and i = 1, . . . , n. Here we suppose aik = bi` = 0 if k, ` > n. Let X = c0 (w1 , N). Obviously, ¯ n ¯ ¯ X i i¯ ¯ k h |c0 (w1 ⊗ w2 , N )k = sup w1 (k) w2 (`) ¯ ak b` ¯¯ 2

k,`∈N

= sup `=1,...,n

i=1

°µ X ¶ ° n i i w2 (`) °° ak b`

= sup w2 (`) `=1,...,n

=

sup

k

i=1

sup kψ|X 0 k≤1

sup

kψ|X 0 k≤1 `=1,...,n

¯ ° ¯ ° ¯c0 (w1 , N)° ¯ °

¯ µµ X ¶ ¶¯ n ¯ ¯ i i ¯ψ ¯ ak b` ¯ ¯ i=1

¯ n ¯ ¯X ¯ w2 (`) ¯¯ ψ(ai ) bi` ¯¯

k

i=1

= λ(h, c0 (w1 , N), c0 (w2 , N)) . Vice versa, if

h=

M X N X

hk,` (ek ⊗ e` ) =

k=1 `=1

N µX M X `=1

¶

hk,` w2 (`) ek

⊗ (1/w2 (`)) e` .

k=1

(we put hk,` = 0 if either ` > N or k > M ) and using the abbreviations `

a :=

M X

hk,` w2 (`) ek

and

k=1

21

b` := (1/w2 (`)) e` ,

we obtain

k h |c0 (w1 , N) ⊗λ c0 (w2 , N)k = =

sup kψ|X 0 k≤1

°X ¯ ° ° N ¯ ° ` `¯ ° ° ψ(a ) b c (w , N) ° ¯ 0 2 ° `=1

sup

sup

kψ|X 0 k≤1

`=1,...,N

| ψ(a` ) |

° M ¯ ° ° X ¯ ° = sup °° hk,` w2 (`) ek ¯¯c0 (w1 , N)°° `∈N

k=1

= sup w2 (`) sup w1 (k) |hk,` | `∈N

k∈N

= k h |c0 (w1 ⊗ w2 , N2 )k . A density argument completes the proof of (3.7). 2 We formulate a few consequences of Propositions 3.3, 3.4, 3.5 for more specialized sequence spaces, see Definition 2.3. Switching in these propositions from N to N0 × Z and from N to Nd0 × Zd (simply by renumbering) and selecting the appropriate weights we obtain the following. Corollary 3.6 Let r1 , . . . , rd , rd+1 ∈ R. (i) Let 1 < p < ∞. Then brp1 ⊗αp srp2 ,...,rd+1 b = srp1 ,...,rd b ⊗αp brpd+1 = spr1 ,r2 ,...,rd+1 b . (ii) Let 0 < p ≤ 1. Then brp1 ⊗γp srp2 ,...,rd+1 b = srp1 ,...,rd b ⊗γp brpd+1 = spr1 ,r2 ,...,rd+1 b . (iii) Let p = ∞. Then rd+1 r1 ,r2 ,...,rd+1 ˚ br∞1 ⊗λ ˚ sr∞2 ,...,rd+1 b = ˚ sr∞1 ,...,rd b ⊗λ ˚ b∞ =˚ s∞ b.

In (i)-(iii) all quasi-norms coincide. Now we investigate the counterpart of Corollary 3.6 for the spaces fpr . Lemma 3.7 Let 1 < p < ∞ and r1 , . . . , rd+1 ∈ R. Then fpr1 ⊗αp srp2 ,...,rd+1 f = srp1 ,...,rd f ⊗αp fprd+1 = spr1 ,r2 ,...,rd+1 f in the sense of equivalent norms. Proof. We argue similar as in the proof of Theorem 2.2 below. We combine Propositions A.8, A.9 with the observation that the isomorphism Jd used in 22

Proposition A.9 is the tensor product of the isomorphisms J used in Proposition A.8. This is a simple consequence of the fact that Jd−1 and J −1 ⊗ ... ⊗ J −1 coincide on the space fpr1 ⊗. . .⊗fprd . Hence, they coincide on fpr1 ⊗αp . . .⊗αp fprd . From Proposition A.9 we know Jd :

Spr1 ,...,rd H(Rd ) → spr1 ,...,rd f

and by general properties of tensor product operators and Proposition A.8 we conclude Jd :

Hpr1 (R) ⊗αp . . . ⊗αp Hprd (R) → fpr1 ⊗αp . . . ⊗αp fprd .

The notation has to be understood as the iterated tensor product, see the remark after Theorem 2.1. Since Spr1 ,...,rd H(Rd ) = Hpr1 (R) ⊗αp . . . ⊗αp Hprd (R), see Theorem 2.1, we have done. 2

3.3 Tensor products of Besov spaces

The heart of the matter consists in the assertions on tensor products of weighted sequence spaces of the previous paragraph. Proof of Theorem 2.2 As in proof of Lemma 3.7 we first combine Propositions A.7, A.9 with the observation that the isomorphism Jd used in Proposition A.9 is the tensor product of the isomorphisms J used in Proposition A.7. This follows from the coincidence of Jd−1 and J −1 ⊗ ... ⊗ J −1 on the space brp1 ⊗ . . . ⊗ brpd . Hence, they coincide on brp1 ⊗δp . . . ⊗δp brpd = spr1 ,...,rd b (Corollary 3.6(i),(ii)), where δp := αp if 1 < p < ∞ and δp := γp if 0 < p ≤ 1. Therefore, r1 ,...rd J −1 ⊗ ... ⊗ J −1 yields an isomorphism of brp1 ⊗δp . . . ⊗δp brpd onto Sp,p B(Rd ). r1 On the other hand we know that J ⊗ . . . ⊗ J is an isomorphism of Bp,p (R) ⊗δp rd . . .⊗δp Bp,p (R) onto brp1 ⊗δp . . .⊗δp brpd using Lemma B.1, Lemma B.6 and Propor1 ,...,rd r1 rd sition A.7. But this means that Sp,p B(Rd ) and Bp,p (R) ⊗δp . . . ⊗δp Bp,p (R) coincide (up to equivalent quasi-norms). This proves Theorem 2.2(i),(iii). To prove (ii) we use a similar argument, now in connection with Corollary 3.6(iii) and Subsection 3.6. 2

3.4 Sequence space isomorphisms

Proof of Theorem 2.5. It is enough to combine Theorem 2.1, Theorem 2.2 with Lemma B.1 and Lemma B.6. 2 23

3.5 Nonlinear approximation spaces 1

Proof of Theorem 2.7. By means of [9, Thm. 4] we know that f ∈ App if and only if (hf, ψ¯j,k¯ i)¯j,k¯ ∈ `p and 1

k f |App

− 12

− 12

(L2 (Rd ))

(L2 (Rd ))k ³ k (hf, ψ¯j,k¯ i)¯j,k¯ |`p k

In view of Corollary 2.6(ii) we see 1

k (hf, ψ¯j,k¯ i)¯j,k¯ |`p k ³ k (hf, ψ¯j,k¯ i)¯j,k¯ |spp

− 21 ,..., p1 − 12

1

− 12 ,..., p1 − 12

p bk ³ k f |Sp,p

B(Rd )k .

The proof is complete. 2 3.6 Spline wavelet isomorphisms Proof of Proposition 2.8. Step 1. The assertion in part (i) is covered by Theorems 3.5 and 3.7 in Frazier and Jawerth [12] (see also the remarks on the top of page 132 concerning the inhomogeneous counterparts), since the m functions ψj,k are smooth molecules. Step 2. For part (ii) we refer to Bourdaud [5], for p ≥ 1 also to Lemarie and Kahane [19, Part II, Chapt. 6, Thm. 5] and DeVore [9]. Only the case p = ∞ r requires an additional comment. The references cover Rm ∈ L(B∞,∞ (R), br∞ ) as well as ° ∞ ∞ ¯ ° ° X X ¯ ° m ¯ r r ° ° aj,k ψj,k ° ¯B∞,∞ (R)° ≤ c k (aj,k )j,k |b∞ k j=0 k=−∞

with c independent of (aj,k )j,k and −m < r < m − 1. Furthermore, observe that the space ˚ br∞ consists of all sequences a = (aj,k )j,k ∈ br∞ such that lim

j+|k|→∞

2j(r+1/2) |aj,k | = 0 .

(3.8)

Obviously, the image of f ∈ C0∞ (R) under the mapping Rm belongs to bs∞ for any s < m − 1. This and the fact that m lim h f, ψj,k i=0 ,

|k|→∞

j ∈ N0

,

m br∞ with i. Therefore, we have Rm (C0∞ (R)) ⊂ ˚ imply (3.8) with aj,k := h f, ψj,k r < m − 1. Vice versa, the functions ϕm , ψm belong to C m−2 (R) and the (m−2) are uniformly Lipschitz continuous and rapidly de, ψm derivatives ϕ(m−2) m ∞ be functions such that supp %, supp ω are compact, caying. Let %, ω ∈ C0 (R) R %(x) = 1 if |x| ≤ 1, and ω(x) dx = 1. Then it is not difficult to see that

µ

¶

1 · %(λ ·) ω( ) ∗ ϕ −−−→ ϕ λ,ε→0 ε ε

µ

as well as

24

%(λ ·)

¶

· 1 ω( ) ∗ ψ −−−→ ψ λ,ε→0 ε ε

r holds with respect to k · |B∞,∞ (R)k for r < m − 1. Assume now {aj,k }j,k ∈ ˚ br∞ and define

fN :=

N X X

m ˚r (R) , aj,k ψj,k ∈B ∞,∞

N ∈ N.

j=0 |k|n k∈Z

°µ X ° ≤ c °°

¯ ¯

° °

hf, ψ¯j,mk¯ i ψ¯j,mk¯ ¯¯Lp (Rd )°°

X

|hf, ψ¯j,mk¯ i|2

X¯j,2k¯

¯ d |¯ j|1 >n k∈Z −nr ≤c2 k hf, ψ¯j,mk¯ i |srp f k .

° ¶1/2 ¯ ° ¯ ¯Lp (Rd )° ° ¯

This proves the estimate from above in (i). Now we turn to the estimate from below. Our test functions are 1

1

for some |¯j|1 = n + 1 .

gn := 2−(n+1)(r+ 2 − p ) ψ¯j,¯0

Because of k gn |Spr H(Rd )k ³ 1, k gn |Lp (Rd )k ³ 2−(n+1)r , see Theorem 2.9, and Pnm gn = 0 we obtain 2−nr ≤ c k I − Pnm |L(Spr H(Rd ), Lp (Rd ))k for some positive constant c independent of n. This completes the proof of (i). Step 2. Let 1 < p ≤ 2. We shall use the elementary inequality k (f` )` |Lp (`2 )k ≤ k (f` )` |`min(p,2) (Lp )k together with Lemma 3.8. Then kf −

Pnm f

° ° X |Lp (R )k = °° d

X

¯ d |¯ j|1 >n k∈Z

°µ X ° ≤ c °°

hf, ψ¯j,mk¯ i ψ¯j,mk¯

X

¯ d |¯ j|1 >n k∈Z

≤c

µ X

X

¯ d |¯ j|1 >n k∈Z

=c

µ X

X

¯ ° ¯ ° ¯Lp (Rd )° ¯ °

|hf, ψ¯j,mk¯ i|2

|hf, ψ¯j,mk¯ i|p |hf, ψ¯j,mk¯ i|p

X¯j,2k¯

° ¶1/2 ¯ ¯ ° ¯Lp (Rd )° ¯ ° d

p

¶1/p

k X¯j,k¯ |Lp (R )k |¯ j|1 ( 21 − p1 )p

¶1/p

2

¯ d |¯ j|1 >n k∈Z −nr ≤c2 k hf, ψ¯j,mk¯ i |srp bk .

This proves the estimate from above in part (ii) under the given restrictions. P P Step 3. Let 0 < p ≤ 1. Using the elementary inequality ( j |cj |)p ≤ j |cj |p we find 26

° ° X ° °

¯ ¯

X

¯ d |¯ j|1 >n k∈Z

°p °

hf, ψ¯j,mk¯ i ψ¯j,mk¯ ¯¯Lp (Rd )°° ≤ ³

X

X

¯ d |¯ j|1 >n k∈Z

X

X

¯ d |¯ j|1 >n k∈Z −nrp

≤2

|hf, ψ¯j,mk¯ i|p k ψ¯j,k¯ |Lp (Rd )kp ¯

1

1

2|j|1 ( 2 − p )p |hf, ψ¯j,mk¯ i|p

k (hf, ψ¯j,mk¯ i)¯j,k¯ |srp bkp .

P

P

m m d Hence, the sequence ( |¯j|1 ≤n k∈Z ¯ d hf, ψ¯ ¯ i ψ¯ ¯ )n converges in Lp (R ). But j,k j,k the same sequence converges in Lq (Rd ), where

1 1 < −r q p

and

1 < q < ∞,

r This follows from Sp,p B(Rd ) ,→ Lq (Rd ), see e.g. [31, 2.4.1] or [14], and Step 2. Hence, also with respect to the Lp -norm the limit of this sequence is f ∈ r Sp,p B(Rd ) itself. This implies the estimate from above in part (i) with 0 < p ≤ 1. Step 4. Let 2 < p < ∞. We argue as in Step 1. In addition we shall apply P m 2 2 1/2 H¨older’s inequality with 1/2 = 1/p+1/u. With f¯j = ( k∈Z ¯ d |hf, ψ¯ ¯ i| X¯ ¯) j,k j,k it follows

° ° X

k f − Pnm f |Lp (Rd )k = °° °µ X ° ≤ c1 °°

X

|¯ j|1 >n k∈Zd

¯ ¯

X

¯ d |¯ j|1 >n k∈Z

|hf, ψ¯j,mk¯ i|2

° °

hf, ψ¯j,mk¯ i ψ¯j,mk¯ ¯¯Lp (Rd )°°

X¯j,2k¯

° ¶1/2 ¯ ¯ ° ¯Lp (Rd )° ¯ °

°µ ° ¶1/2 ¯ X ° ¯ ° ¯Lp (Rd )° = c1 °° f¯j2 ¯ ° |¯ j|1 >n

≤ c1

µ X

d

k f¯j |Lp (R )k

2

¶1/2

|¯ j|1 >n

≤ c1

µ X

¯

2|j|1 rp k f¯j |Lp (Rd )kp

|¯ j|1 >n (d−1)/u

≤ c2 n

−nr

2

µ X

¶1/p µ X

¯

2−|j|1 ru

|¯ j|1 >n |¯ j|1 rp

2

d

k f¯j |Lp (R )k

p

¶1/u

¶1/p

.

|¯ j|1 >n

Since ¯

1

1

k f¯j |Lp (Rd )kp = 2|j|1 ( 2 − p )p

X ¯ d k∈Z

the estimate from above in (ii) follows. Step 5. It remains to deal with p = ∞. We find 27

|hf, ψ¯j,mk¯ i|p

° X °

k f − Pnm f |L∞ (Rd )k = °°

¯ ¯

X

¯ d |¯ j|1 >n k∈Z

° °

hf, ψ¯j,mk¯ i ψ¯j,mk¯ ¯¯L∞ (Rd )°°

¯ ° X ° ° X ¯ ° m m ¯ d ° ° ≤ hf, ψ¯j,k¯ iψ¯j,k¯ ¯L∞ (R )° ° |¯ j|1 >n

≤ c1

X

k∈Zd |¯ j|1 /2

2

|¯ j|1 >n

≤ c1

µ X

sup |hf, ψ¯j,mk¯ i|

¯ d k∈Z −r|¯ j|1

¶

2

|¯ j|1 >n

k(hf, ψ¯j,mk¯ i)¯j,k¯ |sr∞ bk , P

¯

where we used again Section 2.4/(iv) in the third line. Since |¯j|1 >n 2−r|j| ≤ c2 nd−1 2−nr the estimate in part (iii) follows. Step 6. Estimate from below in (ii). Substep 6.1. Let 0 < p ≤ 2. We shall use the same test functions as in Step 1. r Because of k gn |Sp,p B(Rd )k ³ 1, k gn |Lp (Rd )k ³ 2−(n+1)r if |¯j|1 = n + 1, see Theorem 2.9, and Pnm gn = 0 we obtain r 2−nr ≤ c k I − Pnm |L(Sp,p B(Rd ), Lp (Rd ))k

for some positive constant c independent of n. Substep 6.2. Let 2 < p < ∞. We put ½

¾ d

I¯j := k ∈ Z : and

j`

0 ≤ k` < 2 , ` = 1, . . . , d X

1

fn := n−(d−1)/p 2−(n+1)(r+ 2 )

X

¯ ¯ |¯ j|1 =n+1 k∈I j

ψ¯j,mk¯ ,

n ∈ N.

Then Theorem 2.9 implies µ r k fn |Sp,p B(Rd )k ³ n−(d−1)/p

X

¯

2−|j|1 |I¯j |

¶1/p

³ 1.

|¯ j|1 =n+1

Furthermore, by means of Lemma 3.8,

d

k fn |Lp (R )k ³ n

−(d−1)/p

−(n+1)(r+ 21 )

2

°µ ° −(d−1)/p −(n+1)r ° ³n 2 °

³n

(d−1)( 21 − p1 )

°µ ° ° °

X

X

¯ ¯ |¯ j|1 =n+1 k∈I j

X |¯ j|1 =n+1

X¯j,2k¯

° ¶1/2 ¯ ¯ ° ¯Lp (Rd )° ¯ °

° ¶1/2 ¯ ¯ ° d ° ¯ 1 ¯Lp ([0, 1] )°

2−(n+1)r .

This completes the proof of (ii). Step 7. Estimate from below in (iii). Substep 7.1. Preparations. We claim that there is at least one integer k such 28

that ψm (k) 6= 0, m > 1. P We need a few more facts about wavelets. Let Rm (ξ) := ∞ k=−∞ |FNm (ξ + 2πk)|2 be the autocorrelation function of Nm . These functions Rm are trigonometric polynomials (apply Poisson’s Summation formula) bounded away from zero. Our scaling functions ϕm are given by (see Section 2.4) FNm (ξ) Fϕm (ξ) = q , 2πRm (ξ)

ξ ∈ R,

(3.9)

where µ

1 sin(ξ/2) FNm (ξ) = √ e−imξ/2 ξ/2 2π

¶m

,

ξ ∈ R,

(3.10)

and satisfy a refinement equation given by q

Rm (ξ) Fϕm (2ξ) = Mm (ξ) Fϕm (ξ) with Mm (ξ) := q cosm (ξ/2) e−imξ/2 . Rm (2ξ) Obviously, the functions Mm are 2π periodic C ∞ functions satisfying Mm (0) = 1 and Mm (π) = 0. The Fourier transform of ψm is then given by Fψm (2ξ) = eiξ Mm (π + ξ) Fϕm (ξ) ,

ξ ∈ R.

Let us further introduce the 2π-periodic functions Θm (ξ) :=

∞ X

FNm (ξ + 2π`) ,

Φm (ξ) :=

`=−∞

∞ X

Fϕm (ξ + 2π`)

`=−∞

and Ψm (ξ) :=

∞ X

Fψm (ξ + 2π`) ,

ξ ∈ R.

`=−∞

The decay of FNm , m > 1, guarantees absolute convergence of these three series (taking into account the boundedness of Rm and Mm as well) and continuity of the limit functions. The Fourier series of Ψm (ξ) is given by √ P∞ (1/ 2π) k=−∞ ψm (k) eikξ , see Poisson’s summation formula in [36, Cor. 7.2.6]. From the decay properties of ψm , see Section√ 2.4, it follows that this series also P ikξ for all ξ. converges uniformly and hence Ψm (ξ) = (1/ 2π) ∞ k=−∞ ψm (k) e q Observe Φm (ξ) = Θm (ξ)/ 2π Rm (ξ), see (3.9). Let us calculate the value of Ψm at some particular points. We obtain 29

∞ X

Ψm (ξ) = eiξ/2

(−1)k Mm ((k + 1)π + ξ/2) Fϕm (πk + ξ/2)

k=−∞

= eiξ/2

µ X ∞

− µ

`=−∞ ∞ X

Mm ((2` + 1)π + ξ/2) Fϕm (2π` + ξ/2) ¶

Mm ((2` + 2)π + ξ/2) Fϕm (π(2` + 1) + ξ/2)

`=−∞ ∞ X

= eiξ/2 Mm (π + ξ/2)

Fϕm (2π` + ξ/2)

`=−∞

− Mm (ξ/2)

∞ X

¶

Fϕm (π(2` + 1) + ξ/2) .

`=−∞

Using the functions defined above we arrive at µ

Ψm (ξ) = e

iξ/2

¶

Mm (π + ξ/2) Φm (ξ/2) − Mm (ξ/2) Φm (π + ξ/2)

(3.11)

An easy calculation using equation (3.10) gives us ∞ X 1 1 Θm (ξ) = √ 2m e−imξ/2 sinm (ξ/2) , m 2π `=−∞ (ξ + 2π`)

ξ ∈ R.

We introduce a further abbreviation for the Eisenstein series εm (ξ) :=

∞ X

1 m `=−∞ (ξ + 2π`)

,

ξ ∈ R.

This yields in particular 1 Θm (π) = √ 2m (−i)m εm (π) 2π 1 m −imπ/4 sinm (π/4) εm (π/2) Θm (π/2) = √ 2 e 2π 1 m Θm (3π/2) = √ 2 (−i)m e−imπ/4 sinm (π/4) εm (3π/2) . 2π

(3.12)

If m is even then we have Θm (π) 6= 0. This and (3.11) yield Θm (π)

Ψm (0) = −Φm (π) = − q

2π Rm (π)

6= 0 .

(3.13)

In case m > 1 is odd we claim Ψm (π) 6= 0. We argue by contradiction. Assume µ

¶

0 = Ψm (π) = i Mm (3π/2) Φm (π/2) − Mm (π/2) Φm (3π/2) ,

30

see (3.11), then it follows Θm (π/2) Θm (3π/2) Mm (3π/2) q = Mm (π/2) q . Rm (π/2) Rm (3π/2) Using our formula for Mm this turns out to be equivalent to Rm (3π/2) (−i)m Θm (π/2) = Rm (π/2) Θm (3π/2) . With the help of (3.12) this identity can be transformed into Rm (3π/2) εm (π/2) = Rm (π/2) εm (3π/2) . √ Next we use Rm (ξ) = (1/ 2π) eimξ Θ2m (ξ) which gives ε2m (3π/2) εm (π/2) = ε2m (π/2) εm (3π/2) ,

(3.14)

(see (3.12)). Therefore, (3.14) can not be true since ε2m (ξ) > 0 for all ξ, εm (π/2) > 0 and εm (3π/2) < 0. This and (3.13) imply that for all m > 1 the continuous function Ψm is not identically zero and consequently 0 < k Ψm |L2 (−π, π)k2 = 2π

∞ X

|ψm (k)|2 .

k=−∞

Substep 7.2. The shifts ψm ( · − k), k ∈ Z, are, of course, also wavelets and satisfy the same list of properties as ψm itself. The spaces Vnm and the operators Pnm remain unchanged if we replace the generator ψ by ψ( · − k) for some k ∈ Z. In what follows we assume that ψ denotes a shift of ψm (ignoring m) such that ψ(0) 6= 0. Consequently there exists a positive constant c such that for all n ∈ N ° ¯ ° °

c nd−1 2n/2 ≤ °°

0 X

|¯ j|1 =n+1

¯

°

ψ¯j,¯0 ¯¯L∞ (Rd )°°

P0

indicates that we sum only over those vectors ¯j satisfying j` > 0 holds. Here for all `. Furthermore ° ° ° °

0 X |¯ j|1 =n+1

¯ ° ¯ r ° d ° ¯ ψ¯j,¯0 ¯S∞,∞ B(R )° ³ 2n(r+1/2) ,

n ∈ N.

Hence r B(Rd ), L∞ (Rd ))k nd−1 2−nr ≤ c k I − Pnm |L(S∞,∞

for some positive constant c independent of n. 2 Proof of Theorem 2.11. The Littlewood-Paley argument in Lemma 3.8 yields the uniform boundedness of k Pnm |L(Lp (Rd ), Lp (Rd ))k, n ∈ N0 if 1 < p < ∞. Since Pnm is a projection a standard argument leads to 31

Enm (f, Lp (Rd )) ≤ k f − Pnm f |Lp (Rd )k ≤ (1 + k Pnm |L(Lp (Rd ), Lp (Rd ))k) Enm (f, Lp (Rd )) . Hence Enm (F )p ³ k I − Pnm |L(F, Lp (Rd ))k for any space F . This proves the theorem if 1 < p < ∞. If p = 1 we use the test functions gn = 2−(n+1)(r−1/2) ψ¯j,0 for a certain |¯j|1 = n + 1 (see Step 6 in the proof of Proposition 2.10). This r gives kgn |S1,1 B(Rd )k ³ 1, kgn |L2 (Rd )k = 2−(n+1)(r−1/2) and kgn |L∞ (Rd )k ≤ −(n+1)(r−1) c2 . Let h ∈ Vnm . Then, by an argument we learned from [37, p. 252], we obtain 2−2(n+1)(r−1/2) = k gn |L2 (Rd )k2 = h gn , gn i = h gn , gn − h i ≤ k gn − h |L1 (Rd )k kgn |L∞ (Rd )k ≤ c 2−(n+1)(r−1) kgn − h|L1 (Rd )k , r which implies immediately Enm (S1,1 B(Rd ))1 ≥ c 2−rn for some c independent r of n. Finally, for all values of p the quantity k I − Pnm |L(Sp,p B(Rd ), Lp (Rd ))k r yields an upper bound for Enm (Sp,p B(Rd ))p since Enm (f, Lp (Rd )) = Eenm (f, Lp (Rd )). 2

3.8 Characterization via approximation Proof of Corollary 2.12. For p = 2 we have the identity Enm (f, L2 (Rd )) = k f − Pnm f |L2 (Rd )k To continue we shall use some standard arguments. Observe r k f |S2,2 B(Rd )k ³ k P0m f |L2 (Rd )k +

µX ∞

m 22nr k Pn+1 f − Pnm f |L2 (Rd )k2

¶1/2

.

n=0

From this equivalence the inequality µ

kf

r B(Rd )k |S2,2

d

≤ c k f |L2 (R )k +

µX ∞

2nr

2

Enm (f, L2 (Rd ))2

¶1/2 ¶

n=0

follows by triangle inequality and the uniform boundedness of kPnm |L(L2 (Rd ))k. Vice versa, using lim Pnm f = f

n→∞

for all f ∈ L2 (Rd ) ,

we derive 32

Enm (f, L2 (Rd ))2 = k

∞ X

m P`+1 f − P`m f |L2 (Rd )k2

`=n

=

∞ X

m k P`+1 f − P`m f |L2 (Rd )k2 .

`=n

Hence k (2nr Enm (f, L2 (Rd )))n |`2 k2 =

∞ ∞ X X

m m f |L2 (Rd )k2 f − P`+n 22nr k P`+n+1

`=0 n=0 ∞ X −2`r

≤c

2

r k f |S2,2 B(Rd )k2 .

`=0

Since r > 0 the claim follows. 2

A

Appendix - Distribution spaces

As usual, S(Rd ) denotes the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on Rd . Its locally convex topology is generated by the (semi-)norms kϕkk,` = sup (1 + |x|)k x∈Rd

X

|Dα¯ ϕ(x)| ,

k, ` ∈ N0 .

|α| ¯ 1 ≤`

In other words, a sequence {ϕj }j ⊂ S(Rd ) converges to ϕ ∈ S(Rd ) in S(Rd ) if and only if kϕ − ϕj kk,` −−−→ 0 holds for all pairs (k, `) ∈ N20 . Then we shall j→∞

write ϕj − → ϕ. The space S 0 (Rd ) denotes the topological dual of S(Rd ). We S

equip S 0 (Rd ) with the weak topology. The Fourier transform on S 0 (Rd ) will be denoted by F and its inverse transform by F −1 . A.1 Classes of distributions on R Here we recall the definition and a few properties of Besov and Sobolev spaces defined on R. We shall use the Fourier analytic approach, see e.g. [41]. Let ϕ ∈ C0∞ (R) be a function such that ϕ(t) = 1 in an open set containing the origin. Then by means of ϕ0 (t) = ϕ(t) ,

ϕj (t) = ϕ(2−j t) − ϕ(2−j+1 t) ,

t ∈ R,

j ∈ N,

we get a smooth dyadic decomposition of unity. First we deal with Besov spaces. 33

s Definition A.1 Let 0 < p ≤ ∞ and s ∈ R. The Besov space Bp,p (R) is then 0 the collection of all tempered distributions f ∈ S (R) such that

s k f |Bp,p (R)k :=

µX ∞

2jsp k F −1 [ϕj Ff ]( · ) |Lp (R)kp

¶1/p

j=0 s ˚p,p is finite (modification if p = ∞). By B (R) we denote the closure of C0∞ (R) s with respect to the quasi-norm k · |Bp,p (R)k.

In a similar way one could introduce Sobolev spaces of fractional order. However, here we prefer the interpretation as potential spaces. Definition A.2 Let 1 < p < ∞ and s ∈ R. The fractional Sobolev space Hps (R) is the collection of all tempered distributions f ∈ S 0 (R) such that k f |Hps (R)k := kF −1 [(1 + |ξ|2 )s/2 Ff (ξ)]( · )|Lp (R)k is finite. s Remark (i) These quasi-Banach spaces Bp,p (R) and Hps (R) can be characterized in various ways, e.g. by differences and derivatives, whenever s > max(0, 1/p − 1). We refer to [41] for details. s (ii) The spaces Bp,p (R) and Hps (R) do not coincide as sets except the case s p = 2. For p = 2 we have B2,2 (R) = H2s (R) in the sense of equivalent norms.

Let us recall the definition σp := max(0, p1 − 1). Lemma A.3 Let s ∈ R. (i) Let 0 < p < ∞ and 1/p + 1/p0 = 1, where we put p0 = ∞ if p ≤ 1. Then s s S(R) is dense in Bp,p (R) and the dual space of Bp,p (R) can be identified with −s+σp Bp0 ,p0 (R). −s s ˚∞,∞ (ii) The dual space of B (R) can be identified with B1,1 (R). 0 (iii) Let 1 < p < ∞, 1/p + 1/p = 1 and s ∈ R. Then S(R) is dense in Hps (R) and the dual space of Hps (R) can be identified with Hp−s 0 (R). Remark We refer to [41, Thm. 2.3.3] for the density assertions and to [41, Thm. 2.11.2, 2.11.3] and the references given there for the assertions concerning duality. A.2 Spaces of dominating mixed smoothness on Rd Detailed treatments of Besov as well as Sobolev spaces of dominating mixed smoothness are given at various places, we refer to the monographs [1,31], the survey [29] as well as to the booklet [48]. 34

If ϕj , j ∈ N0 , is a smooth dyadic decompositon of unity as introduced in Subsection A.1, then by means of ¯j = (j1 , . . . , jd ) ∈ Nd0 ,

ϕ¯j := ϕj1 ⊗ . . . ⊗ ϕjd ,

we obtain a smooth decompositon of unity on Rd . Definition A.4 Let 0 < p ≤ ∞ and r1 , . . . , rd ∈ R. Then the Besov space r1 ,...,rd Sp,p B(Rd ) is the collection of all tempered distributions f ∈ S 0 (Rd ) such that kf

r1 ,...,rd |Sp,p B(Rd )k

:=

µ X

(j1 r1 +...+jd rd )p

2

kF

−1

d

[ϕ¯j Ff ]( · )|Lp (R )k

p

¶1/p

¯ j∈Nd0 r1 ,...,rd ˚p,p is finite (modification if p = ∞). By S B(Rd ) we denote the closure of ∞ r1 ,...,rd C0 (R) with respect to the quasi-norm k · |Sp,p B(Rd )k.

Again we introduce Sobolev type spaces as potential spaces. Definition A.5 Let 1 < p < ∞ and r1 , . . . , rd ∈ R. The fractional Sobolev space with dominating mixed smoothness Spr1 ,...,rd H(Rd ) is then the collection of all tempered distributions f ∈ S 0 (Rd ) such that kf

|Spr1 ,...,rd H(Rd )k

° ¯ ° · d ¸ ° −1 Y ¯ ° 2 ri /2 d ° ° ¯ := ° F (1 + |ξi | ) Ff (ξ) ( · )¯Lp (R )° i=1

is finite. r1 ,...,rd Remark (i) These classes Sp,p B(Rd ) as well as Spr1 ,...,rd H(Rd ) are quasiBanach spaces. If min(r1 , . . . , rd ) > max(0, (1/p) − 1) then they can be characterized by differences, we refer to [31] and [44] for details. r1 ,...,rd (ii) Again the spaces Sp,p B(Rd ) and Spr1 ,...,rd H(Rd ) do not coincide as sets r1 ,...,rd except the case p = 2. For p = 2 it holds S2,2 B(Rd ) = S2r1 ,...,rd H(Rd ) in the sense of equivalent norms.

Also here we need to know about density of S(Rd ) and duality. Recall σp := max(0, 1/p − 1). Lemma A.6 Let r ∈ R. (i) Let 0 < p < ∞ and 1/p + 1/p0 = 1, where we put p0 = ∞ if p ≤ 1. Then r r B(Rd ) can be identified B(Rd ) and the dual space of Sp,p S(Rd ) is dense in Sp,p −r+σ with Sp0 ,p0 p (Rd ). −r ˚r B(R) can be identified with S1,1 B(Rd ). (ii) The dual space of S ∞,∞ (iii) Let 1 < p < ∞, 1/p + 1/p0 = 1 and r ∈ R. Then S(R) is dense in d Spr H(Rd ) and the dual space of Spr H(Rd ) can be identified with Sp−r 0 H(R ). Remark For the proof we refer to Vybiral [48, p. 42].

35

A.3 Discretization of Besov and Sobolev spaces There are different ways to discretize Besov or Sobolev spaces. Most convenient for us will be the use of Daubechies wavelets. In connection with the characterization of function spaces we refer to [5–7,21,42,48,49]. A.3.1 Wavelet bases of Besov and Sobolev spaces on the real line Let ϕ be an orthonormal compactly supported scaling function belonging to C N (R). Let ψ be an associated compactly supported orthonormal wavelet. Then this function satisfies a moment condition of order N , see [7, Thm. 5.5.1] or [49, Prop. 3.1]. We shall use the same abbreviations as done in (2.4). Proposition A.7 Let 0 < p < ∞ and suppose µ

¶

1 max r, max(0, − 1) − r < N . p

(A.1)

(i) The mapping J defined by f 7→ (hf, ψj,k i)j,k r generates an isomorphism of Bp,p (R) onto brp . r (ii) In case p = ∞ the mapping J : B∞,∞ (R) → br∞ is continuous. Furr thermore, if a = (aj,k )j,k ∈ b∞ then the tempered distribution f given by P P∞ r f := ∞ j=0 k=−∞ aj,k ψj,k belongs to B∞,∞ (R) and there exists a constant c such that ∞ ∞

k

X X

r aj,k ψj,k |B∞,∞ (R)k ≤ c k a |br∞ k

j=0 k=−∞

holds for all such sequences a. Remark (i) Proofs in case p ≥ 1 may be found in many places, see e.g. Meyer [21] and Wojtaszczyk [49]. For p < 1 we refer to Bourdaud [5], Cohen [6] (r > max(0, 1/p − 1)), Kyriazis and Petrushev [18] and to Triebel [42] (general case). See also the survey DeVore [9] and the references given there. (ii) Since ψ does not belong to S(Rd ) the interpretation of the symbol hf, ψj,k i needs some care. However, because of the compact support of ϕ and ψ we have N ϕ , ψ ∈ Bp,∞ (R)

for all 0 < p ≤ ∞ .

Now our restriction (A.1) allows an interpretation of hf, ψj,k i by Lemma A.3. Proposition A.8 Let 1 < p < ∞ and suppose |r| < N . The mapping J defined by f 7→ (hf, ψj,k i)j,k 36

generates an isomorphism of Hpr (R) onto fpr . Remark This result, even in a more general form, can be found in Triebel [42]. We also refer to Frazier and Jawerth [12] and Kyriazis and Petrushev [18], where corresponding estimates for more general systems (not only orthonormal) are treated.

A.3.2 Wavelet bases of Besov and Sobolev spaces on Rd An extension to spaces of dominating mixed smoothness has been given in Vybiral [48]. Let ϕ and ψ be as in the preceding subsection. Defining ψ¯j,k¯ (x1 , . . . , xd ) := ψj1 ,k1 (x1 ) · . . . · ψjd ,kd (xd ) r1 ,...,rd we end up with the following characterization of Sp,p B(Rd ).

Proposition A.9 Let d > 1 and r1 , . . . , rd ∈ R. (i) Let 1 < p < ∞. Let N = N (r1 , . . . , rd , p) be sufficiently large. The mapping Jd defined by f 7→ (hf, ψ¯j,k¯ i)¯j,k¯ generates an isomorphism of Spr1 ,...,rd H(Rd ) onto spr1 ,...,rd f . (ii) Let 0 < p < ∞. Let N = N (r1 , . . . , rd , p) be sufficiently large. The mapping Jd defined by f 7→ (hf, ψ¯j,k¯ i)¯j,k¯ r1 ,...,rd generates an isomorphism of Sp,p B(Rd ) onto spr1 ,...,rd b. (iii) Let p = ∞. Let N = N (r1 , . . . , rd ) be sufficiently large. The mapping r1 ,...,rd Jd : S∞,∞ B(Rd ) → sr∞1 ,...,rd b is continuous. Furthermore, if a = (a¯j,k¯ )¯j,k¯ ∈ P P sr∞1 ,...,rd b then the tempered distribution f := ¯j∈Nd0 k∈Z ¯ d a¯ ¯ ψ¯ ¯ belongs to j,k j,k r1 ,...,rd d S∞,∞ B(R ) and there exists a constant c such that r1 ,...,rd r1 ,...,rd k f |S∞,∞ B(Rd )k ≤ c k a |s∞ bk

holds for all such sequences a. Remark (i) Of some importance will be the fact that the mapping Jd is the tensor product of the isomorphism discussed in Propositions A.7, A.8. (ii) Again the interpretation of hf, ψ¯j,k¯ i needs some care. In view of Lemma A.6 we can argue as in the remark after Proposition A.7. (iii) In the periodic setting, i.e. on the d-dimensional torus Td , Schmeisser r1 ,...,rd B(Td ) consisting of [28] constructed unconditional Schauder bases of Sp,p trigonometric polynomials. 37

B

Appendix - Tensor products

We shall deal with tensor products in several different situations.

B.1 Tensor products of Banach spaces We follow [20], but see also [8]. Let X and Y be Banach spaces. X 0 denotes the dual of X. Consider the set P of all formal expressions ni=1 fi ⊗ gi , n ∈ N, fi ∈ X and gi ∈ Y . We introduce an equivalence relation by means of n X

fi ⊗ gi ∼

i=1

m X

uj ⊗ v j

j=1

if both expressions generate the same operator A : X 0 → Y , i.e. n X

ϕ(fi ) gi =

i=1

m X

ϕ(uj ) vj

for all ϕ ∈ X 0 .

(B.1)

j=1

Then the algebraic tensor product X ⊗ Y of X and Y is defined to be the set of all such equivalence classes. One can equip this set with several different norms. We are interested in so-called uniform norms only. Let X1 , X2 , Y1 , Y2 be Banach spaces. For Ti ∈ L(Xi , Yi ), i = 1, 2, we define their tensor product by (T1 ⊗ T2 )h :=

n X

(T1 fi ) ⊗ (T2 gi ) ,

h=

i=1

n X

fi ⊗ gi ∈ X1 ⊗ X2 .

(B.2)

i=1

We call a norm α(·, X, Y ) on X ⊗ Y a uniform tensor norm if it satisfies ³

´

³

´

α (T1 ⊗ T2 )h, Y1 , Y2 ≤ kT1 |L(X1 , Y1 )k · kT2 |L(X2 , Y2 )k α h, X1 , X2 . for all h =

n P j=1

fj ⊗ gj ∈ X1 ⊗ X2 and all T1 ∈ L(X1 , Y1 ), T2 ∈ L(X2 , Y2 ). The

completion of X ⊗ Y with respect to the tensor norm α will be denoted by X ⊗α Y . If α is uniform then T1 ⊗ T2 has a unique extension to X1 ⊗α X2 which we again denote by T1 ⊗ T2 . Simple, but important, is the next property we need. Lemma B.1 Let X1 , X2 , Y1 , Y2 be Banach spaces and let α(·, X, Y ) be a uniform tensor norm. Further we suppose that T1 ∈ L(X1 , Y1 ) and T2 ∈ L(X2 , Y2 ) are linear isomorphisms. Then the operator T1 ⊗ T2 is a linear isomorphism from X1 ⊗α X2 onto Y1 ⊗α Y2 . 38

Proof. Obviously, T1 ⊗ T2 ∈ L(X1 ⊗α X2 , Y1 ⊗α Y2 ) and T1−1 ⊗ T2−1 ∈ L(Y1 ⊗α Y2 , X1 ⊗α X2 ). So it remains to show that (T1 ⊗α T2 )−1 = T1−1 ⊗α T2−1 in the algebraical sense, which is a simple consequence of (T1 ⊗ T2 ) ◦ (T1−1 ⊗ T2−1 ) = I

on Y1 ⊗ Y2

,

on X1 ⊗ X2 (T1−1 ⊗ T2−1 ) ◦ (T1 ⊗ T2 ) = I and a limit argument. Here I denotes the identity on the corresponding space. 2 Next we recall three well-known constructions of tensor norms, namely the injective, the projective and the p-nuclear norm. Definition B.2 Let X and Y be Banach spaces. (i) Let h ∈ X ⊗ Y be given by h=

n X

fj ⊗ gj ,

fj ∈ X ,

gj ∈ Y .

j=1

Then the injective tensor norm λ(·, X, Y ) is defined as ½°X ° n

λ(h, X, Y ) = sup °°

¯ ° ¯ °

¾

ψ ∈ X 0 , kψ|X 0 k ≤ 1 .

ψ(fj ) · gj ¯¯Y °° :

j=1

(ii) The projective tensor norm γ(·, X, Y ) is defined by γ(h, X, Y ) = inf

½X n

kfj |Xk kgj |Y k :

fj ∈ X, gj ∈ Y, h =

j=1

n X

¾

fj ⊗ gj .

j=1

(iii) Let 1 ≤ p ≤ ∞ and let 1/p + 1/p0 = 1. Then the p-nuclear tensor norm αp (·, X, Y ) is given by αp (h, X, Y ) := inf

(µ n X

kfi |Xk

p

¶1/p

· sup

½µ X n

i=1

¯p0 ¶1/p0 ¾) ¯ 0 0 ¯ |ψ(gi )¯ : ψ ∈ Y , kψ|Y k ≤ 1 ,

i=1

where the infimum is taken over all representations of h (as in (ii)). Remark (i) All three expressions define norms, we refer to [20, Chapt. 1]. In particular, λ is independent of the representation of h. (ii) In Definition B.2(iii) one can replace sup

½µ X n

|ψ(gi )|

p0

¶1/p0

¾ 0

0

: ψ ∈ Y , kψ|Y k ≤ 1

i=1

39

(B.3)

by

¯ ° µX ½° X ¶1/p ¾ n ° n ¯ ° p ° ¯ ° sup ° λi gi ¯Y ° : |λi | ≤1 , i=1

(B.4)

i=1

see [20, Lem. 1.44].

B.2 Tensor products of distributions As usual, we put D(Rd ) = C0∞ (Rd ). It is equipped with a topology: a sequence {ϕj }j ⊂ D(Rd ) converges to a ϕ ∈ D(Rd ) if supp ϕj ⊂ K, j = 1, 2, ..., where K ⊂ Rd is a compact subset and {Dα¯ ϕj }j converges uniformly to Dα¯ ϕ for every multi-index α ¯ = (α1 , ..., αd ) ∈ Nd0 . The topological dual of D(Rd ) 0 d is denoted by D (R ). Tensor products of distributions is a well-developed subject, mainly in the framework of D0 (Rd ), we refer e.g. to [32, Chapt. IV], [38, III.13] as well as [16, Chapt. X]. We need the following. Lemma B.3 Let T ∈ S 0 (Rd1 ) and S ∈ S 0 (Rd2 ). Then there exists a unique distribution U ∈ S 0 (Rd1 +d2 ), called the tensor product of T and S and denoted by T ⊗D S, such that for all functions ϕ ∈ S(Rd1 ) and ψ ∈ S(Rd2 ) U (ϕ(x) · ψ(y)) = T (ϕ(x))S(ψ(y)) holds true. Furthermore, U is given explicitly by the formula U (ρ(x, y)) = Sy (Tx (ρ(x, y))) = Tx (Sy (ρ(x, y))) ,

ρ ∈ S(Rd1 +d2 ) .

Proof. In [16, Chapt. X] it is proved that (T ⊗D S)(ρ(x, y)) = Sy (Tx (ρ(x, y))) = Tx (Sy (ρ(x, y))) ,

ρ ∈ S(Rd1 +d2 ) ,

defines a tempered distribution belonging to S 0 (Rd1 +d2 ). The uniqueness of this distribution can be proved following the lines of the proof for the D0 counterpart of Lemma B.3 (see e.g. [38, III.13]) making use of the facts: D(Rd ) ,→ S(Rd ) (topological embedding); D(Rd ) is dense in S(Rd ); and the set ½

ρ=

N X j

ϕ1 ⊗ . . . ⊗

¾

ϕjd

:

ϕjk

∈ D(R) , N ∈ N, j = 1, ..., N, k = 1, ..., d

j=1

is dense in D(Rd ). The proof is complete. 2

By means of the linearity of distributions this implies the following. 40

Proposition B.4 Let Ti ∈ S 0 (Rd1 ) and Si ∈ S 0 (Rd2 ), i = 1, ..., n. Then there exists a unique distribution U ∈ S 0 (Rd1 +d2 ) such that for all functions ϕ ∈ S(Rd1 ) and ψ ∈ S(Rd2 ) n X

U (ϕ(x) · ψ(y)) =

Ti (ϕ(x)) Si (ψ(y))

i=1

holds true. Furthermore, U is given explicitely by U=

n X

Ti ⊗ D S i .

i=1

B.3 Tensor products of spaces of distributions Tensor products of functions and sequences have been the original source for the introduction of the abstract tensor product. However, since we are dealing with quasi-Banach spaces of distributions and sequences, we prefer to make a few comments to the coincidence of these tensor products. First we recall the notion of a reasonable cross-norm. A norm α on X ⊗ Y is called a cross-norm, if α(f ⊗ g, X, Y ) = k f |Xk k g |Y k for all f ∈ X and all g ∈ Y . Further, a cross-norm α is called reasonable if, for all ϕ ∈ X 0 and all ψ ∈ Y 0 , the linear form ϕ ⊗ ψ is bounded on X ⊗α Y and has norm k ϕ |X 0 k k ψ |Y 0 k. Let X and Y be spaces of tempered distributions such that S(Rd1 ) ,→ X ,→ S 0 (Rd1 ) and S(Rd2 ) ,→ Y ,→ S 0 (Rd2 ), respectively. Similar to Subsection B.1 we can introduce the following set ½

X ⊗D Y := h =

n X

¾

fj ⊗D gj : fj ∈ X, gj ∈ Y, n ∈ N, j = 1, . . . , n .

j=1

Because of X ,→ S 0 we have S(Rd1 ) ⊂ X 0 in the sense that a fixed function ϕ ∈ S(Rd1 ) defines a continuous linear functional on X via f 7→ f (ϕ) ,

f ∈X,

(B.5)

(analogously S(Rd2 ) ⊂ Y 0 ). Let us additionally assume a dense embedding k·|X 0 k

S(Rd1 )

= X0

with the above interpretation. Under these assumptions the set X ⊗D Y equipped and completed with a reasonable cross-norm α is isomorphic to X ⊗α Y (see Subsection B.1). Indeed, let us suppose n X

fi ⊗ gi =

i=1

m X j=1

41

uj ⊗ g j

(B.6)

in the sense of Subsection B.1. This implies n X

m X

η(fi ) · β(gi ) =

i=1

η(uj ) · β(gj )

j=1

for all η ∈ X 0 and β ∈ Y 0 and therefore (in the sense of (B.5)) n X

m X

fi (ϕ) · gi (ψ) =

i=1

uj (ϕ) · gj (ψ)

j=1

for all ϕ ∈ S(Rd1 ) and ψ ∈ S(Rd2 ). Then Proposition B.4 implies n X

fi ⊗D gi =

i=1

m X

uj ⊗ D g j .

(B.7)

j=1

Vice versa, if we assume (B.7), then arguing backwards, we find that n X

fi (ϕ) · gi =

i=1

m X

uj (ϕ) · gj

j=1

holds in Y for every ϕ ∈ S(Rd1 ). Finally, because of the dense inclusion S(Rd1 ) ⊂ X 0 , we conclude the equality (B.6) (in the sense of Subsection B.1) by a limit argument. Hence, in the case of Banach spaces X and Y of distributions with the properties above we have the coincidence of both approaches. We therefore write X ⊗α Y for both constructions. Remark (i) All the spaces under consideration in Section 2 have the properties required in the previous consideration, see Lemma A.3 and Lemma A.6. (ii) The injective, the projective and the p-nuclear norm are reasonable crossnorms, cf. [20, Lem. 1.6, 1.8, 1.46]. B.4 Tensor products of sequence spaces Let I and J denote countable index-sets. Then F (I) is the class (C-vector space) of all functions f : I 7→ C. Similar as above we consider subspaces X ⊂ F (I) and Y ⊂ F (J) and define the tensor product f ⊗ g ∈ F (I × J) of f ∈ X and g ∈ Y by (f ⊗s g)(i, j) := f (i) g(j) ,

i∈I,

j∈J.

B.5 Tensor products of certain quasi-Banach spaces of distributions We want to generalize the concept of the projective tensor-norm γ (see Definition B.2 (ii)) in order to include also special quasi-Banach spaces either of 42

type X ,→ S 0 (Rd ) or of type `p (w). Let us mention that there is no hope for a general abstract theory of tensor product for quasi-Banach spaces. At least one of the reasons consists in the somehow “poor” dual spaces of some quasi-Banach spaces. This has to be compared with (B.1). So we concentrate on those situations where the tensor product has a meaning from the very beginning, i.e. for distributions and sequences of complex numbers. Definition B.5 Let 0 < p < 1. (i) Let X and Y be quasi-Banach spaces such that X ,→ S 0 (Rd1 ) and Y ,→ S 0 (Rd2 ). Then we define the projective tensor p-norm γp by γp (h, X, Y ) := inf

½µ X n

kfj |Xkp kgj |Y kp

¶1/p

: fj ∈ X, gj ∈ Y, h =

j=1

n X

¾

fj ⊗D gj .

j=1

(ii) Let X and Y be quasi-Banach spaces such that X = `q1 (w1 ) and Y = `q2 (w2 ) for some q1 , q2 ∈ (0, ∞]. Then the projective tensor p-norm is defined as γp (h, X, Y ) := inf

½µ X n

p

kfj |Xk kgj |Y k

j=1

p

¶1/p

: fj ∈ X, gj ∈ Y, h =

n X

¾ s

fj ⊗ gj .

j=1

Remark (i) γp defines a uniform quasi-norm (p-norm) on X ⊗ Y . The inequality γp (h1 + h2 , X, Y )p ≤ γp (h1 , X, Y )p + γp (h2 , X, Y )p as well as the uniformness are obvious. (ii) Different attempts to introduce tensor products of quasi-Banach spaces have been undertaken by Turpin [43] and Nitsche [22]. In particular the approach of Nitsche applies to so-called placid q-Banach spaces. Let us mention r r1 ,...,rd that `q , Bq,q (R) as well as Sq,q (Rd ) are placid q-quasi-Banach spaces if 0 < q < 1. We can carry over the definition of the tensor product of operators to the present case, see (B.2). One has to check that this definition does not depend on the chosen representation of h ∈ X ⊗ Y . But this can be done as in case of Banach spaces, cf. [20, p. 19]. Now we are in position to formulate a supplement to Lemma B.1. Lemma B.6 Let X1 , X2 , Y1 , Y2 be quasi-Banach spaces such that the pairs (X1 , X2 ) and (Y1 , Y2 ) are admissible in Definition B.5. Let 0 < p ≤ 1. Further we suppose that T1 ∈ L(X1 , Y1 ) and T2 ∈ L(X2 , Y2 ) are linear isomorphisms. Then the operator T1 ⊗ T2 is a linear isomorphism from X1 ⊗γp X2 onto Y1 ⊗γp Y2 . 43

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