The Spectral Radius of Matrix Continuous

Advances in Computational Mathematics manuscript No. (will be inserted by the editor)

The Spectral Radius of Matrix Continuous Refinement Operators Victor Didenko · Wee Ping Yeo

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Abstract A simple analytic formula for the spectral radius of matrix continuous res finement operators is established. On the space Lm 2 (R ), m ≥ 1 and s ≥ 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and the matrix function c defining the corresponding refinement operator. A similar representation is valid for the continuous refinement operators considered on spaces Lp for p ∈ [1, ∞), p ̸= 2. However, additional restrictions on the kernel c are imposed in this case. Keywords Matrix continuous refinement operator · spectral radius Mathematics Subject Classification (2000) 34K99 · 41A30 · 41A63 · 47A10

1 Introduction s Given p ∈ [1, ∞), let Lm p (R ), m ≥ 1, s ≥ 1 refer to the set of vector functions φ = (φ1 , φ2 , . . . , φm ) with entries from the Lebesgue space Lp (Rs ) and the norm

 ||φ||p := 

m ∑

1/p ||φj ||pL

p (R

s)



j=1

where

(∫ ||φj ||Lp (Rs ) :=

Rs

)1/p |φj |p dx

,

j = 1, . . . , m.

This research was supported in part by the Universiti Brunei Darussalam, Grant PNC2/2/RG/1(72) Victor Didenko Universiti Brunei Darussalam, Department of Mathematics, Bandar Seri Begawan, BE1410 Brunei E-mail: [email protected] Wee Ping Yeo Universiti Brunei Darussalam, Department of Mathematics, Bandar Seri Begawan, BE1410 Brunei E-mail: [email protected]

2

Further, let M ∈ Rs×s , s ≥ 1 be a non-singular real matrix and let c = (cij )m i,j=1 be a s complex-valued matrix function with entries from L1 (Rs ). On the space Lm p (R ), 1 ≤ M p ≤ ∞, the matrices M and c define a bounded linear operator Wc ,

∫ (WcM φ)(x)

:= Rs

c(M x − y)φ(y) dy,

which is called continuous refinement operator. The operator WcM and the associated equation WcM φ = g have important applications in various fields of mathematics, including subdivision processes, dynamical systems, wavelet analysis, so they have been studied thoroughly [1, 4, 5, 11–14, 18–21]. One particular point of interest is the evaluation of the spectral radius of the operator WcM in various functional spaces. The first step in this direction was undertaken in [11]. Assuming that m = 1 and the kernel c is a compactly supported function from L1 (Rs ), the authors show that in the space Lp (Rs ), 1 ≤ p ≤ ∞ the spectral radius of the operator TcM ,

∫ (TcM φ)(x) :=

R

c(x − M y)φ(y) dy,

(1)

can be computed as the spectral radius of the associated compact operator TΩ : Lp (Ω) 7→ Lp (Ω), where

∫ M φ)(x) := (Tc,Ω

c(x − M y)φ(y) dy,

(2)

Ω −1

and Ω is a compact set in Rs . Note that TcM = WceM with e c(x) := c(M x), x ∈ Rs , M so all results concerning the spectral radius of operators Tc can be reformulated in terms of the operators WcM and vice versa. Moreover, among many other representations, Goodman et al [11] established a remarkable analytic formula for the spectral radius ρp (TcM ) of the operator TcM : Lp (Rs ) 7→ Lp (Rs ) for non-negative compactly supported kernels c. Thus if c ≥ 0 almost everywhere on Rs , then ρp (TcM ) =

1 | det M |1−1/p

∫ c(y) dy,

(3)

Rs

cf. [11, Corollary 2.1]. It is worth noting that both the finiteness of the support and non-negativity of the function c are important for the proof of (3) given in [11]. On the other hand, for the operators WcM in the one-dimensional univariate case, i.e. if m = 1 and s = 1, the spectra of the associated compact operators similar to the operators M Tc,Ω from (2), can be fully described [10]. This allows to obtain an effective formula for the spectral radius ρp (WcM ) of the operator WcM . Although it went unnoticed, the spectral radius formula presented in [10] is in fact similar to the formula (3). However, the kernel c in [10] is not assumed to be non-negative. Finally, for m = 1 and s ≥ 1 the formula ∫ 1 M c(y) dy ρ2 (Wc ) = √ s | det M | R

3

was established in [8] under the assumption that c ∈ L1 (Rs ) and M is an expansive matrix, i.e., all its eigenvalues λj satisfy the inequality |λj | > 1,

j = 1, . . . , s.

Little is still known about the spectral radius of matrix continuous refinement operators. In fact, the authors are not aware of any non-trivial results concerned with analytic formulas for ρp (WcM ) in the case where m > 1. In the present paper, the spectral radius of the operators WcM is studied quite generally. Thus if c ∈ Lm×m (Rs ) and if M is an expansive or a contractive dilation 1 matrix, we present a simple analytic formula for the spectral radius of the operators s WcM on the space Lm 2 (R ), where m ≥ 1 and s ≥ 1. However, if p ̸= 2 the results are less complete. In particular, one of the problems arising is the absence of exact formulas for the norms of Fourier multipliers. Nevertheless, it is still possible to obtain a simple expression of the spectral radius ρp (WcM ) for some classes of kernels c.

s 2 A general formula for spectral radius of operator WcM on space Lm 2 (R ).

Let a ∈ Cm×m . The spectral norm of the matrix a is defined by √

||a|| := max

1≤j≤m

µj

where µj , j = 1, . . . , m are the eigenvalues of the matrix a∗ a. It is useful to note that ||a|| = ρ(a) = max |λj |, 1≤j≤m

where λj , j = 1, . . . , m and ρ(a) are the eigenvalues and the spectral radius of the matrix a, respectively. s On the other hand, on the space Lm 2 (R ) the norm of the operator of multiplication m×m s by matrix a ∈ L∞ (R ) can be evaluated as ||a||∞ := max ess sup

√

µj (x),

1≤j≤m x∈Rs

(4)

where µj (x), x ∈ Rs are the eigenvalues of the matrix function a∗ a [9]. Thus the set Lm×m (Rs ) can be provided with the norm (4), and for constant matrix a it is clear ∞ that ||a||∞ = ||a||. Let F and F−1 denote the direct and inverse Fourier transforms, i.e., (Fφ)(x) :=

1 (2π)s/2

∫

and (F−1 φ)(x) :=

1 (2π)s/2

e−ixy φ(y) dy

Rs

∫ eixy φ(y) dy. Rs

If c is a matrix function with entries from L1 (Rs ), then one can introduce the matrix a(x) := (2π)s/2 (F−1 c)(x),

x ∈ Rs .

(5)

4

Recall that the entries of the matrix a are continuous functions on Rs , and by the Riemann-Lebesgue Lemma [16, p.77], [17, Theorem IX.7] they vanish at infinity, i.e. limx→∞ aij (x) = 0 [17]. s m s Consider a linear operator BM : Lm p (R ) 7→ Lp (R ) defined by (BM φ)(x) = φ(M x). By the Convolution Theorem [17, Theorem IX.3], the continuous refinement operator WcM admits the representation WcM = BM FaF−1 . If D ∈ Rs×s is a non-singular real matrix and a = a(x), x ∈ Rs is a continuous matrix function, then by definition aD (x) := a((DT )−1 x),

x ∈ Rs

where DT denotes the transpose matrix. It is easily seen that the continuous refinement operator WcM can be rewritten in the form

( WcM = F

aM B T −1 | det M | (M )

)

F−1 .

(6)

Now we can establish a general formula for the spectral radius of the operator WcM s considered on the space Lm 2 (R ). (Rs ) and let M ∈ Rs×s be a non-singular matrix. Then Theorem 1 Let c ∈ Lm×m 1 M s m s the spectral radius ρ2 (Wc ) of the operator WcM : Lm 2 (R ) 7→ L2 (R ) can be evaluated as ρ2 (WcM ) = √

1 | det M |

lim ||

n→∞

n−1 ∏

1/n

a((M T )n−k−1 ·)||∞ ,

(7)

k=0

where n−1 ∏

a((M T )n−k−1 x) := a((M T )n−1 x)a((M T )n−2 x) · · · a(x),

x ∈ Rs .

k=0

Proof. Let I refer to the identity operator, VM be the operator defined by VM := √

1 | det M |

B(M T )−1 ,

and let λ ∈ C. Using representation (6) and invertibility of the operator F on the space s Lm 2 (R ) one obtains

( λI

− WcM

=F

) aM

λI − √ VM | det M |

F−1 ,

so 1 ρ2 (WcM ) = √ ρ2 (aM VM ) . | det M |

5

The operator aM VM is known as the composition or weighted shift operator. There is vast literature devoted to such a kind operators, and for more detail, the reader may consult [2]. One can for example evaluate the powers of the operator aM VM as n

(aM VM ) = (VM )

n

(n−1 ∏

) T n−k−1

a((M )

·)

;

k=0 s m s m s and since VM : Lm 2 (R ) 7→ L2 (R ) is an isometry, for any φ ∈ L2 (R )

||(aM VM )n φ||2 = ||

n−1 ∏

a((M T )n−k−1 ·)φ||2 .

k=0 n Thus the norm of the operator (aM V∏ M ) is equal to the norm of the operator of n−1 multiplication by the matrix function k=0 a((M T )n−k−1 x). Therefore, to derive expression (7) one can use general formulas for the spectral radius of bounded linear operators [9] and formula (4). ⊔ ⊓

3 An explicit formula for spectral radius of operator WcM on space s Lm 2 (R ). In this section, we present a simple analytic formula for the spectral radius of the matrix continuous refinement operator. Note that the general formula (7) contains the limit of the norms of matrix sequences. Consequently, to obtain a suitable expression for ρ2 (WcM ), one has to evaluate the norms of the corresponding matrices. Usually this is a very demanding task; and as mentioned in [2], at present there are only a few analytic formulas for the spectral radii of general weighted shift operators with matrix coefficients. Moreover, even for discrete refinement operators there are no such formulas and all the results known depend upon the structure of both the dilation and symbol matrices M and a (see [6, 7] and references there). However, in the case at hand, the matrix a has remarkable properties, so the expression in the right-hand side of (7) can be treated effectively. Let us start with a few auxiliary results. s Lemma 1 If a = (aij )m 7 R i,j=1 is the matrix function (5), then the function h : R → defined by h(x) := ||a(x)|| (8)

is continuous on Rs and vanishes at infinity, i.e. limx→∞ h(x) = 0. Proof. To avoid confusion, let us emphasize that ||a(x)|| in (8) means the spectral norm of the matrix a evaluated at a point x ∈ Rs . Now, in addition to the spectral norm, we consider another norm || · ||F on the set Cm×m of m × m matrices. Thus if b = (bij )m i,j=1 , bij ∈ C, then ||b||F :=

m ∑

|bij |.

i,j=1

Recall that any two norms on finite-dimensional space are equivalent, so there are constants k1 , k2 such that for any b ∈ Cm×m k1 ||b||F ≤ ||b|| ≤ k2 ||b||F .

6

Let x, x0 ∈ Rs . Since the entries of matrix (5) are continuous everywhere on Rs and |h(x) − h(x0 )| = |∥a(x)∥ − ∥a(x0 )∥| ≤ ∥a(x) − a(x0 )∥ m ∑

≤ k2 ∥a(x) − a(x0 )∥F = k2

|aij (x) − aij (x0 )|,

i,j=1

the function h is continuous at any point x0 ∈ Rs . Moreover, since h(x) = ||a(x)|| ≤ k2 ||a(x)||F = k2

m ∑

|aij (x)|,

i,j=1

and all aij , i, j = 1, . . . , m vanish at infinity, so does the function h.

⊔ ⊓

Lemma 2 Let Vδ,R ⊂ Rs denote the set Vδ,R := {x ∈ Rs : δ ≤ ||x||2 ≤ R} where δ, R are positive numbers and || · ||2 refers to the Euclidian norm on the space Rs . If M is an expansive matrix, then there is a positive integer k0 such that for any x ∈ Rs the set Vδ,R contains at most k0 members of the sequence x, M x, M 2 x, . . . .

(9)

Proof. It is clear that we can restrict ourselves to the case ||x||2 ≥ δ. Let M be an expansive matrix, and let λj , j = 1, . . . , m be the eigenvalues of the matrix M . Notice that µ := min {|λj |} > 1, 1≤j≤m

and for any x ∈ Rs the inequality ||M x||2 ≥ µ||x||2 holds. Let [y] denote the integral part of the real number y. If x ∈ Vδ,R , then ||x||2 ≥ δ, and the previous inequality shows that the set Vδ,R contains at most [ln(R/δ)/ ln µ] + 1 members of the sequence (9). On the other hand, if x ∈ Rs and ||x||2 > R, then the set Vδ,R does not contain any members of the sequence (9) at all. ⊔ ⊓ Lemma 3 Let h : Rs → R be a continuous function such that limx→∞ h(x) = 0, and let M ∈ Rs×s be an expansive matrix. Then for any ε > 0 there is a number n0 ∈ N such that for all x ∈ Rs and for all n ≥ n0 the inequality

) (n−1 1/n ∏ T n−k−1 x) ≤ |h(0)| + ε h((M ) k=0

holds.

(10)

7

Proof. The function h is bounded on Rs , i.e. there is a number N such that |h(x)| ≤ N for all x ∈ Rs . Moreover, given ε > 0, let δ and R (δ < R) be positive numbers such that for all x ∈ Rs , ||x||2 < δ, |h(x)| < |h(0)| + and for all x ∈ Rs , ||x||2 > R,

ε ; 2

ε . 2

|h(x)|
k0 the product

)1/n (∏ n−1 T n−k−1 h((M ) x)) can be estimated as k=0

( ) n−1 1/n )(n−k0 )/n ( ∏ T n−k−1 ≤ N k0 /n |h(0)| + ε , x) h((M ) 2 k=0

x ∈ Rs ,

and the relation (10) follows. ⊔ ⊓ The results of Lemma 3 allow us to find upper bounds for the spectral radius of the operator WcM . (Rs ) and let M ∈ Rs×s be an expansive matrix. Then Lemma 4 Let c ∈ Lm×m 1 ρ2 (WcM )

(∫ )m



≤ √ cij (x) dx

. | det M | Rs i,j=1 1

(11)

Proof. Consider the function h from (8). This function is continuous on Rs and vanishes at infinity. By Lemma 3, for any ε > 0 there is a number n0 such that for all n ≥ n0 and for all x ∈ Rs the inequality

n−1

1/n (n−1 )

∏

1/n ∏

T n−k−1 T n−k−1 a((M ) x) ≤ x)

a((M )

k=0

k=0

=

(n−1 ∏

)1/n T n−k−1

h((M )

≤ h(0) + ε = ||a(0)|| + ε

x)

k=0

holds. Note that for any continuous matrix function b : Rs 7→ Cm×m , ||b||∞ = sup ||b(x)||. x∈Rs

Therefore, for all n ≥ n0 ,

1/n

n−1

∏

T n−k−1 ≤ ||a(0)|| + ε; a((M ) ·)

k=0

∞

8

and since matrix a(0) has the form

(∫ a(0) = Rs

)m cij (x) dx

,

(12)

i,j=1

the estimate (11) follows from the general formula (7). ⊔ ⊓ Now we are able to establish the spectral radius formula for the operator WcM s considered on the space Lm 2 (R ). Theorem 2 Let c ∈ Lm×m (Rs ) and let M ∈ Rs×s be an expansive matrix. Then 1 ρ2 (WcM )

(∫ )m



= √ cij (x) dx

. | det M | Rs i,j=1 1

(13)

Proof. Taking into account Lemma 4, we only need to verify the estimate

(∫ )m



ρ2 (WcM ) ≥ √ cij (x) dx

.

s | det M | R i,j=1 1

(14)

However, using the inequality ||b||∞ ≥ ||b(0)||, which is valid for any continuous matrix function b = b(x), x ∈ Rs , one obtains



n−1

∏ T n−k−1 a((M ) ·))



(15)

∞

k=0

so that

≥ an (0) ,

n−1

1/n

∏

1/n

T n−k−1 lim a((M ) ·)) . ≥ lim an (0) n→∞ n→∞

(16)

∞

k=0

It remains to observe that the right-hand side of (16) represents the spectral radius of the operator of multiplication by the constant matrix a(0). Thus



1/n = ||a(0)||, lim an (0)

n→∞

(17)

and employing general formula (7) once more, we arrive at inequality (14). Comparing inequalities (11) and (14) leads to formula (13). ⊔ ⊓ The formula (13) actually means that the spectral radius of the matrix continuous refinement operator WcM is the maximal eigenvalue in magnitude of the matrix (12) √ divided by | det M |. If m = 1, the corresponding expression for the spectral radius takes a remarkably simple form, viz. ρ2 (WcM )

∫ = √ | det M | 1

R

c(x) dx . s

(18)

As already mentioned, a similar formula was established in [11] for the operator TcM in the special case of compactly supported non-negative kernels but for arbitrary p ∈ [1, ∞]. For general one-dimensional L1 -kernels formula (18) was proved in [8]. Remark 1 It is easily seen that Lemma 2 remains valid for contractive dilation matrices. Therefore, all results of this and the following section are also valid for the operators WcM with contractive dilation matrix M .

9 s 4 Formulas for spectral radius of operator WcM on spaces Lm p (R ), 1 ≤ p < ∞, p ̸= 2.

For p ̸= 2, the evaluation of the spectral radius of the operator WcM on Lp spaces meets additional difficulties. These difficulties are connected with the irreversibility of the Fourier transform F on the spaces Lp for p ̸= 2 and with the absence of any general formulas for Lp -norms of Fourier operators FaF−1 . Nevertheless, for some classes of kernels c it is still possible to express the spectral radius ρp (WcM ) of the operator WcM via the spectral radius of a constant matrix. Let us start with an auxiliary representation. Lemma 5 Let matrices c and M satisfy the conditions of Theorem 1. Then for any p ∈ [1, ∞),

1

(n) −1 ||(WcM )n ||p = (19) F A F

, a,M p | det M |n/p where (n)

Aa,M (x) :=

n−1 ∏

a((M T )n−k−1 x)),

x ∈ Rs ,

k=0

and the matrix a is defined by (5). Proof. Since L2 (Rs ) ∩ Lp (Rs ) is dense in the space Lp (Rs ), 1 ≤ p < ∞, ||(WcM )n ||p =

sup s f ∈Lm p (R )

=

||(WcM )n f ||p ||f ||p

sup s m s f ∈Lm 2 (R )∩Lp (R )

||(WcM )n f ||p . ||f ||p

If we now employ representation (6), we obtain ||(WcM )n ||p =

1 | det M |n

=

1 | det M |n

=

1 | det M |n

||f ||p

s m s f ∈Lm 2 (R )∩Lp (R )

||F(aM B(M T )−1 )n F−1 f ||p

sup

||f ||p

s m s f ∈Lm 2 (R )∩Lp (R )

||FB((M T )−1 )n Aa,M F−1 f ||p (n)

sup

||f ||p

s m s f ∈Lm 2 (R )∩Lp (R )

1 = | det M |n/p =

||(FaM B(M T )−1 F−1 )n f ||p

sup

1 | det M |n/p

||FAa,M F−1 f ||p (n)

sup s m s f ∈Lm 2 (R )∩Lp (R )

||f ||p

||FAa,M F−1 f ||p (n)

sup s f ∈Lm p (R )

||f ||p

Note that in the above transformations we used the relation

FB((M T )−1 )n = | det M |n BM n F

.

10 s and the fact that | det M |1/p BM is an isometric operator on the space Lm ⊔ ⊓ p (R ). M Thus in order to evaluate the spectral radius of the operator Wc on the space s Lm p (R ), where p ̸= 2, one needs effective estimates for the Lp -norms of Fourier operators FaF−1 . Let us introduce some more notation. If D is a non-singular matrix and c ∈ Lm×m (Rs ), we set 1 1 e cD (x) := c(D−1 x), x ∈ Rs , | det D|

and let (n)

(n−1)

Bc,M := e cM n−1 ∗ Bc,M ,

n = 2, 3, . . .

(20)

(1)

where Bc,M := c and “ ∗ ” denotes the operation of convolution. The entries of the (n)

(n)

(n)

(n)

matrix Bc,M are denoted by bij , hence Bc,M = (bij )m i,j=1 . Notice that Ac,M = (2π)s/2 F−1 (Bc,M ). (n)

(n)

. Corollary 1 Let matrices c and M satisfy the conditions of Theorem 1. Then

(∫

1

| det M |1/p



M cij (x) dx

≤ ρp (Wc )

Rs i,j=1

(∫ )m

1/n 1

(n) lim ≤ |b (x)| dx .

| det M |1/p n→∞ Rs ij i,j=1 )m

(21)

Proof. Let m = 1, b ∈ L1 (Rs ) and a = (2π)s/2 F−1 (b). Then for any p ∈ [1, ∞) the estimate ||a||∞ ≤ ||FaF−1 ||p ≤ ||b||1

(22)

holds [15, Corollary 1.3], [22, Theorem 1.3]. Consider now a Fourier operator FAF−1 m×m (Rs ). The with the matrix symbol A := (2π)s/2 B where B = (bij )m i,j=1 ∈ L1 inequality (22) and standard norm-equivalency arguments lead to the estimate ||A||∞ ≤ d1 ||FAF

−1

(∫ )m



||p ≤ d2 |bi,j (x)| dx

,

Rs i,j=1

where the constants d1 and d2 do not depend on m and p. Then making use of identity (19) one obtains the inequality



(∫

d2 1

(n)

M n

Ac,M ≤ d1 ||(Wc ) ||p ≤

∞ | det M |n/p | det M |n/p

Rs



, i,j=1

)m (n) |bij (x)| dx

so estimate (21) follows from general formulas for the spectral radius of linear operators and relations (15)-(17). ⊔ ⊓

11

Corollary 2 Let matrices c and M satisfy the conditions of Theorem 1. If

(∫ )m )m

(∫





|cij (x)| dx cij (x) dx

=

,

Rs

Rs i,j=1 i,j=1

(∫

then ρp (WcM )

(23)



cij (x) dx

. Rs i,j=1

1

=

| det M |1/p

)m

Proof. Let C denote the matrix

(∫

)m

C := Rs

|cij (x)| dx

We estimate the norms of the matrices

(∫

Bn := Rs

. i,j=1

)m (n) |bij (x)| dx

. i,j=1

It follows from (20) that

∫ Rs

(n) |bij (x)| dx

∫ m ∑ 1 (n−1) n−1 −1 = dx (y) dy c ((M ) (x − y))b ik kj Rs | det M |n−1 Rs k=1 ∫ ∫ m ∑ ∫

≤

k=1

Rs

|cik (x)| dx

(n−1)

Rs

|bkj

(x)| dx,

so the non-negative matrices Bn and C Bn−1 are connected by the relation Bn ≤ C Bn−1 . Therefore, by Corollary 2.1.5 from [3], ||Bn || ≤ ||C Bn−1 ||,

(24)

which implies the estimate ||Bn || ≤ ||C ||n , ⊔ ⊓

and the results follows from inequality (21).

(Rs ) Corollary 3 Let M be an expansive matrix. If all entries of the matrix c ∈ Lm×m 1 are non-negative almost everywhere, then ρp (WcM )

(∫

1

=

| det M |1/p



cij (x) dx

. Rs i,j=1 )m

(25)

Remark 2 If m = 1, a similar statement for the operator TcM was proved in [11]. It is interesting to note that for m = 1 Corollaries 2 and 3 produce the same result. However, for m > 1 the entries of the symbol matrix c must not necessarily be non-negative (or non-positive) in order to satisfy condition (23). Remark 3 In the proof of Corollary 2 we used a nice property (24) of non-negative matrices. However, if all entries of the matrix c are non-negative, the proof of relation (25) can be carried straightforward, without using Corollary 2.1.5 of [3].

12

Example 1 Let us compute the spectral radius of the operator TcM defined by (1), in the case where c is a matrix function from Lm×m (Rs ) satisfying condition (23). As 1 was already mentioned, TcM = WceM (25) one obtains ρp (TcM )

−1

where e c(x) = c(M x), x ∈ Rs . Therefore, from

(∫



cij (M x) dx

Rs i,j=1

(∫ )m



cij (x) dx

.

Rs i,j=1 )m

1

=

| det (M )−1 |1/p 1 | det M |1−1/p

=

Recall that for scalar operators TcM with non-negative compactly supported kernels this result was established in [11]. In the case of general symbols c ∈ Lm×m (Rs ), the evaluation of the norms of the 1 matrices Bn requires much more effort. Note that for m = 1 similar expressions have been previously studied by Goodman et al (see Lemmas 2.1-2.3 and Theorems 2.1-2.2 in [11]). Using their arguments, for compactly supported kernels c ∈ L∞ (Rs ) one can show that

(∫

lim

n→∞

Rs

)1/n

(n) |Bc,M (x)| dx

(n)

1/n

= lim ||Aa,M ||∞ . n→∞

This limit relation leads to the following result. Corollary 4 Let m = 1, c ∈ L∞ (Rs ) be a compactly supported function, and let M be an expansive matrix. Then

∫

ρp (WcM )

1 = | det M |1/p

c(x)dx . s

R

Acknowledgements The authors would like to thank an anonymous referee for pointing out an error in the first version of this paper.

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