Multivariate box spline wavelets in higher dimensional Sobolev spaces

R E S E A R C H

Open Access

Multivariate box spline wavelets in

higher-dimensional Sobolev spaces

Raj Kumar

_{and Manish Chauhan}

*_{Correspondence:}

[email protected] 2_{Department of Mathematics,}

University of Delhi, New Delhi, India Full list of author information is available at the end of the article

Abstract

We construct wavelets and derive a density condition of MRA in a higher-dimensional Sobolev space. We give necessary and sufficient conditions for orthonormality of wavelets inHs_(Rd_{). We construct nonseparable orthonormal wavelets in}

a higher-dimensional Sobolev space by using multivariate box spline.

Keywords: Wavelets; Box Splines; Multiresolution analysis; Sobolev space

1 Introduction

Box splines are refinable functions, and we can easily choose various directions to have a box spline function with a desired order of smoothness. Naturally, they have been used to construct various wavelet functions. Mathematically box splines offer an elegant toolbox for constructing a class of multidimensional elements with flexible shape and support. In multivariate setting, box splines are often considered as a generalization of B-splines [1]. Theoretically, the computational complexity of a box spline is lower than that of an equiva-lent B-spline, since its support is more compact and its total polynomial degree is lower. To investigate this potential in practice, several attempts were made. Recurrence relation [1,

2] is the most commonly used technique for evaluating box splines at an arbitrary position. There are many papers on multivariate spline wavelet theory, in particular, on orthogonal spline wavelets [3], compactly spline prewavelets [4–6], bivariate and trivariate compactly supported biorthogonal box spline wavelets [7,8], and multivariate compactly supported tight wavelet frames [9].

Wavelets in a Sobolev space and their properties were instigated by Bastin et al. [10,11], Dayong and Dengfeng [12], and Pathak [13]. Regular compactly supported wavelets and compactly supported wavelets of integer order in a Sobolev space by B-spline are given in [10,11]. Further, bivariate box splines in a Sobolev space were introduced in [14].

Inspired by the works mentioned, in this paper, we study nonseparable wavelets in a higher-dimensional Sobolev space by using a multivariate box spline. To the best of our knowledge, no previous studies of multivariate box spline wavelets exist in higher-dimensional Sobolev spaces. This paper is organized as follows. In Sect.2, we hereby present construction of wavelets and density conditions of MRA in a higher-dimensional Sobolev space. Also, we give necessary and sufficient conditions for the orthonormality of wavelets inHs_(Rd_{). In Sect.3, we construct nonseparable wavelets in a higher-dimensional}

Sobolev space by using a multivariate box spline.

1.1 Sobolev spaceHs_(Rd₎

For any real numbers, the Sobolev spaceHs(Rd) consists of tempered distributions in

S(Rd_{) such that} f2_s:= 1

(2π)d

Rd

1 +ξ2sfˆ(ξ)2dξ,

where · denotes the Euclidean norm inRd_{, and the corresponding inner product is}

f,gs:=

1 (2π)d

Rd

1 +ξ2sfˆ(ξ)gˆ(ξ)dξ.

The Fourier transformfˆoff ∈L1(Rd) is defined as

ˆ

f(ξ) :=

Rde

–ix,ξ

f(x)dx,

wherex,ξis the inner product of two vectorsxandξinRd_. 2 Multiresolution analysis

To adapt classical theory of MRA overHs(Rd), we first derive an orthonormality and den-sity condition. The main problem is thatHs_{-norm is not dilation invariant. We also don’t}

achieve orhtonormality at each level of dilation by a single scaling function. This lead us to a more general construction of MRA, where the scaling function depends on the level of dilation. Throughout this paper, the superscriptjof a functionϕ(j)_{represents levelj.}

Proposition 2.1 If s is a real number,ϕ(j)_∈Hs_(Rd_{),and j is an integer,then the}

distribu-tionsϕ(_j,j_k)(x) = 2jd/2_ϕ(j)₍₂j_x–k),k∈Zd_{,are orthonormal in H}s_(Rd_)iff

k∈Zd

1 + 22jξ+ 2kπ2sϕˆ(j)(ξ+ 2kπ)2= 1 (1)

almost everywhere.It follows that we have the bound

ϕ_ˆ(j)2–jξ≤1 +ξ2–s/2.

Proof Sinceϕ_j(_,j_k)(t)∈Hs(Rd), the series

M(ξ) =

r∈Zd

ϕ_ˆ(j)(ξ+ 2πr)21 + 22j(ξ+ 2πr)2s

converges almost everywhere, belongs toL1

loc(Rd), and is 2πZd-periodic, that is,M(ξ)∈

L1_(Td_{), where T}d_{= [0, 2π]}d_{is thed-dimensional torus. Moreover, for everyl∈Z}d_{, we}

have

TdM(ξ)e

–iξ,(k–l)_dξ

=

r∈Zd

Td

ϕ_ˆ(j)(ξ+ 2πr)21 + 22j(ξ+ 2πr)2se–iξ,(k–l)dξ

=

Rd

= 2–jd

Rd

ϕ_ˆ(j)2–ju21 +u2se–i2–ju,(k–l)du

=

Rd

1 +u2se–i2–ju,k2–jd/2ϕˆ(j)2–jue–i2–j_u,l

2–jd/2_ϕˆ(j)₂–j_udu

=

Rd

1 +u2sFϕ_j(_,j_k)(t) (u)Fϕ_j(_,j_l)(t)(u)du

= (2π)dϕ_j(_,j_k)(t),ϕ_j(_,j_l)(t)_s.

Since{1/(2π)d_e–iξ,(k–l)_:k,l∈Zd_{}is an orthonormal basis forL}2_(Td_{), we have}

1 (2π)d

Td

M(ξ)e–iξ,(k–l)_dξ=ϕ(j)

j,k(t),ϕ

(j)

j,l(t)

s=δk,l

ifM(ξ) = 1. From (1) we get

1 + 22jξ2sϕˆ(j)(ξ)2≤1,

which implies

ϕ_ˆ(j)(ξ)≤1 + 22jξ2–s/2.

Proposition 2.2 Letϕ(j)_{,j∈Z,be a sequence of elements of H}s_(Rd_{)such that,for every j,}

the distributionsϕ_j(_,j_k)(x) = 2jd/2_ϕ(j)₍₂j_x–k),k∈Zd_{,are orthonormal in H}s_(Rd_{).If P} jis the

orthogonal projection from Hs_(Rd_{)onto V}

j:=span{ϕ(j,jk):k∈Zd},then,for every h∈Hs(Rd),

we have

lim

j→+∞

Pjh2s–

1 (2π)d

Rd

1 +ξ22shˆ(ξ)2ϕˆ(j)2–j_ξ2_dξ

= 0.

Moreover,if there are A,α> 0such that

Rd

1 +ξαϕˆ(j)(ξ)2dξ≤A

for every j≤0,then_jj_=–=∞_∞Vj={0}d.

Proof Let us prove the first part withh∈C₀∞(Rd_{). By the definition ofP} jwe get

Pjh2s=

k∈Zd

h,ϕ_j(_,j_k)

s

2 = 2

–jd

(2π)2d

k∈Zd

_Rd

1 +ξ2shˆ(ξ)ϕˆ(j)₂–j_ξei2–jk,ξ

dξ 2

.

Moreover, sincehandϕ(j)_{belong toH}s_(Rd_),

Rd

1 +ξ2shˆ(ξ)ϕˆ(j)₂–j_ξei2–jk,ξ

dξ

=

]0,2j_2π[de

i2–j_k,ξ

p∈Zd

Hence, using the Parseval formula inL2_{(]0, 2}j_2π[d_{), we get} Pjh2s

= 1 (2π)d

]0,2j_2π[d

p∈Zd

1 +ξ+ 2j2πp2shˆξ+ 2j2πpϕˆ(j)₂–j_{ξ+ 2πp}

2

dξ

= 1 (2π)d

q∈Zd

Rd

1 +ξ2s1 +ξ+ 2j2πq2shˆ(ξ)ϕˆ(j)₂–j_ξ

× ˆhξ+ 2j_2πqϕˆ(j)₂–j_{ξ+ 2πqdξ}

= 1 (2π)d

Rd

1 +ξ22shˆ(ξ)2ϕˆ(j)2–jξ2

+ 1 (2π)d

q∈Zd_\{0}d

Rd

1 +ξ2s1 +ξ+ 2j2πq2shˆ(ξ)ϕˆ(j)₂–j_ξ

× ˆhξ+ 2j_2πqϕˆ(j)₂–j_{ξ+ 2πqdξ.}

The term associated withq={0}d_,{0}d_{= (0, 0, . . . , 0)∈Z}d_{is used as an approximation for} Pjh2s. Using Proposition2.1, the inequality|ϕ(j)(2–jξ)| ≤(1 +ξ2)–s/2, and the fact that

ˆ

hbelongs to the Schwartz spaceS(Rd_{) (i.e.,|ˆh(ξ)| ≤C(1 +ξ}2₎–α_{for anyα> 0), we obtain}

that the sum of the other ones is bounded by

q∈Zd_\{0}d

Rd

1 +ξ2s/21 +ξ+ 2j2πq2s/2hˆ(ξ)hˆξ+ 2j_2πqdξ

≤C

q∈Zd_\{0}d

1 (1 +2j_2πq2₎2

Rd

1 (1 +ξ2₎2dξ

≤C

q∈Zd_\{0}d

1 (2j_2πq2₎2

Rd

1 (1 +ξ2₎2dξ

≤C2–4(j+1)

q∈Zd_\{0}d

1

|q|2 _Rd 1

(1 +ξ2₎2dξ,

where|q|= (d_r=1|qr|2)1/2,q= (q1,q2, . . . ,qd)∈Zd. This expression converges to 0 asj→

+∞.

Now leth∈Hs_(Rd_{). Recall the inequality} f+g2≤(1 +ε)f2+

1 +1

ε

g2,

which is valid for everyε> 0 and any seminorm. For anyχ∈C₀∞(Rd_{), we have} Pjh2s–

1 (2π)d

Rd

1 +ξ22shˆ(ξ)2ϕˆ(j)2–jξ2dξ

≤(1 +ε)Pjχ2s+

1 +1

ε

Pj(h–χ)

2

– 1

(2π)d_{(1 +ε)}

Rd

+ 1 (2π)d_(ε)

Rd

1 +ξ22shˆ(ξ) –χ(ξˆ )2ϕˆ(j)2–jξ2dξ

≤(1 +ε)

Pjχ2s–

1 (2π)d

Rd

1 +ξ22sχ(ξˆ )2ϕˆ(j)2–jξ2dξ

+

1 +2 ε

h–χ2_s+

1 +ε– 1 (1 +ε)

χ2_s.

By the same way, we can obtain a similar lower bound. To prove that the left-hand side converges to 0 asjconverges to +∞, we first takeεsufficiently small. Then we chooseχ approximatinghand finallyjlarge.

For the second part, we have to prove that, for everyh∈C₀∞(Rd_),P

jhconverges to zero

inHs(Rd_{) asj→–∞. We use the last expression ofP}

jhsobtained previously. We first

es-timate the sum overqwithout the integral. By the Cauchy–Schwarz inequality and Propo-sition2.1we have

q∈Zd

1 +ξ+ 2j2πq2shˆξ+ 2j2πqϕˆ(j)₂–j_{ξ+ 2πq}

≤

q∈Zd

1 +ξ+ 2j2πq2shˆξ+ 2j2πq2

1/2 .

We know that

q∈Zd

1 +ξ+ 2j2πq2shˆξ+ 2j2πq22j+1πd→

Rd

1 +ξ2shˆ(ξ)2dξ

ifj≤–1. It follows that

Pjh2s≤

1 (2π)d

Rd

1 +ξ2shˆ(ξ)ϕˆ(j)2–jξ2–jdChsdξ

≤2–jdChs

(2π)d

Rd

1 + 2–j_ξα_ϕˆ(j)₂–j_ξ2_dξ

1/2

×

Rd

1 + 2–jξ–α1 +ξ22shˆ(ξ)2dξ 1/2

≤C √

Ahs

(2π)d

Rd

1 + 2–jξ–α1 +ξ22shˆ(ξ)2dξ 1/2

.

The last expression converges to zero asjconverges to –∞. Now we construct wavelets inHs_(Rd_{) with the help of previous propositions.}

By definition,Vjis the set of allf∈Hs(Rd) such that ˆ

f(ξ) =m2–jξϕˆ(j)2–jξ,

where m∈L2_loc(Rd_{) is 2πZ}d_{-periodic. This follows immediately from the fact that the}

We haveVj⊂Vj+1 for everyj∈Zdiff there are 2πZd-periodic functionsm(0j)∈L2loc(Rd) such that the following scale relation holds:

ˆ

ϕ(j)(2ξ) =m(₀j+1)(ξ)ϕˆ(j+1)(ξ); (2)

moreover,ϕ(j)_andϕ(j+1)_{satisfy the hypothesis of Proposition2.1. Now, using our theorems} and propositions, we develop the definition of MRA inHs(Rd).

Definition 2.3 Letsbe a real number. The MRA of Hs(Rd) is a sequence Vj,j∈Z, of

closed linear subspaces ofHs_(Rd_{) such that} (a) Vj⊂Vj+1,

(b) j_j=_=–∞_∞Vj=Hs(Rd), (c) j_j=_=–∞_∞Vj={0}d, and

(d) for everyj, there is a functionϕ(j)such that the distributions2jd/2_ϕ(j)₍₂j_x–k),

k∈Zd_{, form an orthonormal basis forVj.}

Before giving a necessary condition for the orthonormality, we defineEd:={0, 1}das the

unit cube in thed-dimensional Euclidean space.

Theorem 2.4 Ifϕ(j)andϕ(j+1)satisfy the hypothesis of Proposition2.1,then

2d–1

q=0

m(₀j+1)(ξ+γqπ)= 1, γq∈Ed,q= 1, 2, . . . , 2d– 1.

Proof We know from Proposition2.1that if the system is orthonormal, then

δk,l=

ϕ_j(_,j_k),ϕ_j(_,j_l)_s

= 2 –jd

(2π)d

Rd

ϕ_ˆ(j)2–jξ2e–i2–jξ,(k–l)1 +ξ2sdξ

= 1 (2π)d

Rd

ϕ_ˆ(j)(u)2e–iu,(k–l)1 + 22ju2sdu

= 1 (2π)d

Rd

m(₀j+1)(u/2)2ϕˆ(j+1)(u/2)2e–iu,(k–l)1 + 22ju2sdu

= 2

d

(2π)d

Rd

m(₀j+1)(ν)2ϕˆ(j+1)(ν)2e–i2ν,(k–l)1 + 22(j+1)ν2sdν

= 1 (π)d

Td

_m(j+1) 0 (ν)

2

r∈Zd

ϕ_ˆ(j+1)(ν+ 2πr)2

×1 + 22(j+1)ν+ 2πr2se–i2ν,(k–l)_dν

= 1 (π)d

Td

m(₀j+1)(ν)2e–i2ν,(k–l)dν

= 1 (π)d

[0,π)d

2d_–1

q=0

m(₀j+1)(ν+γqπ)

2

which implies that

2d–1

q=0 _m(j+1)

0 (ξ+γqπ)

2

= 1, γq∈Ed,

ifk=l.

With the help of (2) and Theorem2.4, we may defineϕ(j)by

ˆ

ϕ(j)(ξ) =m0(j+1)(ξ/2)ϕˆ(j+1)(ξ/2)

=

J

t=1

m₀(j+t)ξ/2tϕˆ(j+J)ξ/2J

=· · ·= 1 (1 +ξ2₎s/2

+∞

t=1

m(₀j+t)ξ/2t (3)

forj∈Z. ForVj, letWjbe the orthogonal complement ofVjinVj+1. We have

ψ_j(_,j_k)_,p:= 2jd/2ψ_p(j)2jx–k∈Vj+1 (4)

if there are 2πZd_{-periodic functionsm}(j)

1,m (j) 2 , . . . ,m

(j)

2d_–1∈L2loc(Rd) such that

ˆ

ψ_p(j)2–jξ=m(_pj+1)2–j–1ξϕˆ(j+1)2–j–1ξ, p= 1, 2, . . . , 2d– 1.

Theorem 2.5 The distributionsψ_j(_,j_k)_,p(x) = 2jd/2_ψ(j)

p (2jx–k)are orthonormal if

2d–1

q=0

m(_pj+1)(ξ+γqπ)= 1, γq∈Ed,∀p= 1, 2, . . . , 2d– 1,

and they are orthogonal to Vjif

2d_–1

q=0

m(_pj+1)(ξ+γqπ)m(0j+1)(ξ+γqπ) = 0, γq∈Ed,∀p= 1, 2, . . . , 2d– 1. (5)

Proof

ψ_j(_,j_k)_,p,ψ_j(_,j_l)_,p

s

= 2 –jd

(2π)d

Rd

ψˆ_p(j)2–jξ2e–i2–jξ,(k–l)1 +ξ2sdξ

= 1 (2π)d

Rd

ψˆ_p(j)(u)2e–iu,(k–l)1 + 22ju2sdu

= 1 (2π)d

Rd

m(_pj+1)(u/2)2ϕˆ(j+1)(u/2)2e–iu,(k–l)1 + 22ju2sdu

= 2

d

(2π)d

Rd

= 1 (π)d

Td

m(_pj+1)(ν)2

r∈Zd

ϕ_ˆ(j+1)(ν+ 2πr)2

×1 + 22(j+1)ν+ 2πr2se–i2ν,(k–l)dν = 1

(π)d

Td

_m(j+1)

p (ν)

2

e–i2ν,(k–l)dν

= 1 (π)d

[0,π)d

2d–1

q=1

m(_pj+1)(ν+γqπ)

2

e–i2ν,(k–l)dν

= 1 (π)d

[0,π)de

–i2ν,(k–l)_dν.

Therefore

ψ_j(_,j_k)_,p,ψ_j(_,j_l)_,ps= 1

if2_qd₌₀–1|m(pj+1)(ξ+γqπ)|= 1,γq∈Ed, andk=l.

Now we prove second part of the theorem:

0 =ψ_j(_,j_k)_,p,ϕ_j(_,j_l)_,p

s

= 2 –jd

(2π)d

Rd

1 +ξ2sψˆ_p(j)2–jξϕˆ(j)₂–j_ξe–i2–jξ,(k–l)_dξ

= 1 (2π)d

Rd

1 + 22jξ2sm_p(j+1)(ξ/2)m(₀j+1)(ξ/2)ϕˆ(j+1)(ξ/2)2e–iξ,(k–l)_dξ

= 2

d

(2π)d

Rd

1 + 22(j+1)ξ2sm_p(j+1)(ξ)m(₀j+1)(ξ)ϕˆ(j+1)(ξ)2e–i2ξ,(k–l)dξ

= 1 (π)d

Tdm

(j+1)

p (ξ)m

(j+1) 0 (ξ)

×

r∈Zd

1 + 22(j+1)ξ+ 2rπ2sϕˆ(j+1)(ξ+ 2rπ)2e–i2ξ,(k–l)dξ

= 1 (π)d

[0,π)d

2d–1

q=1

m(_pj+1)(ξ+γqπ)m(0j+1)(ξ+γqπ)e–i2ξ,(k–l)dξ,

which implies

2d_–1

q=0

m(_pj+1)(ξ+γqπ)m(0j+1)(ξ+γqπ) = 0, γq∈Ed,∀p= 1, 2, . . . , 2d– 1.

Now we define unitary matrix with the help of our theorems,

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

m(₀j)(ξ+γ0π) m(0j)(ξ+γ1π) · · · m(0j)(ξ+γ2d_–1π)

m(₁j)(ξ+γ0π) m(1j)(ξ+γ1π) · · · m(1j)(ξ+γ2d_–1π)

. .. . .. . .. . ..

m(j)

2d_–1(ξ+γ0π) m₂(jd)_–1(ξ+γ1π) · · · m₂(jd)_–1(ξ+γ2d_–1π)

⎤ ⎥ ⎥ ⎥ ⎥

Theorem 2.6 Suppose that the scaling functionϕ(j)_{,j∈Z,generate an MRA{V}

j}of Hs(Rd)

andϕ_j(_,j_k),k∈Zd_{,form an orthonormal basis for V}

j,j∈Z.Suppose that,for each j∈Z,m(pj)for

p= 1, 2, . . . , 2d_{–1are such that matrix(6)is unitary.Defineψ}(j)

j,k,pby(4)for p= 1, 2, . . . , 2d–1

and j∈Z.Then Wj=Wj,1⊕Wj,2⊕ · · · ⊕Wj,2d_–1with Wj,p=span{2jd/2ψp(j)(2jx–k) :k∈

Z},p= 1, 2, . . . , 2d_{– 1,is perpendicular to V}

jin Vj+1,and Vj+1=Vj⊕Wj.Therefore

2jd/2ψ_p(j)2jx–k, k∈Z,p= 1, 2, . . . , 2d– 1,

is an orthonormal basis for Hs(Rd).

Proof First,we show thatψ_j(_,j_k)_,p⊥Vj, for allk∈Zdandp= 1, 2, . . . , 2d– 1. Indeed,

(2π)dψ_j(_,j_k)_,p(x),ϕ_j(_,j_k)(x)_s = (2π)d2jd/2ψ_p(j)2jx–k1

, 2jd/2ϕ_j(_,j_k)2jx–k2

s

=

Rd

1 +ξ2sF2jd/2ψ_p(j)2jξ–k1

F2jd/2_ϕ(j)

j,k

2j_ξ–k

2

dξ

= 2–jd

Rd

1 +ξ2sψˆ_p(j)2–jξϕˆ(j)₂–j_ξe2–jξ,(k2–k1)_dξ

=

Rd

1 + 22jξ2sm_p(j+1)(ξ/2)ϕˆ(j+1)(ξ/2)

×m(₀j+1)(ξ/2)ϕˆ(j+1)_(ξ/2)e2–jξ,(k2–k1)_dξ

=

Td

l∈Zd

1 + 22jξ+ 2πl2sm_p(j+1)(ξ/2 +πl)ϕˆ(j+1)(ξ/2 +πl)

×m(₀j+1)(ξ/2 +πl)ϕˆ(j+1)_{(ξ/2 +πl)e}2–jξ,(k2–k1)_dξ

=

Td

₂d_–1

q=0

m(_pj+1)(ξ/2 +γqπ)m(0j+1)(ξ/2 +γqπ)

e2–jξ,(k2–k1)_dξ

by Proposition2.1. This expression is equal to zero because matrix (6) is unitary. Similarly, we can show thatWj,p1⊥Wj,p2 for allp1,p2∈ {1, 2, . . . , 2d– 1}.

We know show thatVj+1=Vj⊕Wj,1⊕Wj,2⊕ · · · ⊕Wj,2d_–1for anyf ∈Vj+1. We write

ˆ

f(ξ) =B2–j–1ξϕˆ(j+1)2–j–1ξ.

We will demonstrate that there exist 2πZd-periodic functionsG(2–jξ) andHp(2–jξ) such

that

ˆ

f(ξ) =G2–jξϕˆ(j)2–jξ+ 2d–1

p=1

Hp

2–jξψˆ_p(j)2–jξ.

Now, we have

B(ξ/2)ϕˆ(j+1)(ξ/2) =G(ξ)ϕˆ(j)(ξ) + 2d–1

p=1

=G(ξ)m(₀j+1)(ξ/2)ϕˆ(j+1)(ξ/2) + 2d_–1

p=1

Hp(ξ)m(pj+1)(ξ/2)ϕˆ(j+1)(ξ/2).

It follows that

B(ξ/2) =G(ξ)m(₀j+1)(ξ/2) + 2d–1

p=1

Hp(ξ)m(pj+1)(ξ/2).

By the periodicity (2πZd_{-periodic) ofGandH}

pwe have

B(ξ/2 +γqπ) =G(ξ)m0(j+1)(ξ/2 +γqπ) +

2d–1

p=1

Hp(ξ)m(pj+1)(ξ/2 +γqπ)

forq= 0, 1, . . . , 2d_{– 1. This completes proof.} 3 Multivariate box spline

Now we give an example of multivariate box splines in a Sobolev space. Using them, we construct a wavelet inHs_(Rd_).

Let D be the direction matrix of order d × d_i=1+1mi,mi ∈ N0,∀i, whose column

vectors consist of (m1,m2, . . . ,md+1) copies of the following d + 1 column vectors: (1, 0, . . . , 0)T_{, (0, 1, 0, . . . , 0)}T_{, . . . , (0, 0, . . . , 1)}T_{, and (1, 1, . . . , 1)}T_.

Fixs≥0 and the natural numbers (m1,m2, . . . ,md+1) such that

m[D] :=min{mi+mj:i=jfor alli,j= 1, 2, . . . ,d+ 1}

+1

2>s.

LetMm1,m2,...,md+1 be a multivariate box spline function defined in terms of the Fourier

transform by

Mm1,m2,...,md+1(ξ) =

d+1

j=1

1 –e–ikj,ξ

ikj,ξ

mj

, kj∈D,mj∈N0,∀j.

The multivariate box splineMm1,m2,...,md+1belongs toC

m[D]–1_{, wherem[D] + 1 is the} min-imum number of columns that can be discarded fromDto obtain a matrix ofrank<d

(see [15]). For

W_m(j)_{1,m2,...,md+1}(ξ) :=

l∈Zd

1 + 22jξ+ 2πl2sMm1,m2,...,md+1(ξ+ 2πl)

2 ,

it is known that there existc,C≥0 such that 0≤c≤

l∈Zd

Mm1,m2,...,md+1(ξ+ 2πl)

2

≤C<∞.

Consideringξ:= (ξ1,ξ2, . . . ,ξd) andl:= (l1,l2, . . . ,ld), we have

l∈Zd

1 + 22jξ+ 2πl2sMm1,m2,...,md+1(ξ+ 2πl)

= (l1,l2,...,ld)∈Zd

1 + 22j

d

i=1

|ξi+ 2πli|2

s

Mm1,m2,...,md+1(ξ+ 2πl)

2

. (7)

By mathematical induction we know that, for positive real numbersxi,i= 1, . . . ,d,

_d

‘i=1

xi

m ≤dm

_d

i=1 (xi)m

, xi∈R+. (8)

From (7) and (8) we have

(l1,l2,...,ld)∈Zd

1 + 22j

d

i=1

|ξi+ 2πli|2

s

Mm1,m2,...,md+1(ξ+ 2πl)

2

≤(d+ 1)s

c+ 22js (l1,l2,...,ld)∈Zd

_d

i=1

|ξi+ 2πli|2s

Mm1,m2,...,md+1(ξ+ 2πl)

2

≤(d+ 1)s

c+ 22jsC

(l1,l2,...,ld)∈Zd

_d

i=1

Mmi–s,md+1(ξ+ 2πl)

2

≤Cj< +∞,

whereC,Cj> 0, andmi–s,md+1is theith term subtracted bys. Hence we have the fol-lowing:

Lemma 3.1 There exists two constants cjand Cjsuch that

0 <cj≤Wm(j)1,m2,...,md+1(ξ)≤Cj< +∞.

Now, for everyj∈Z, we define

ˆ

ϕ(j)(ξ) =Mm1,m2,...,md+1(ξ)

Wm(j)1,m2,...,md+1(ξ)

. (9)

Now we find a 2πZd_{-periodic functionm}(j)

0 ∈L2(Zd) for which the scaling relation (5) holds:

ϕ(j)(2ξ) =m0(j+1)(ξ)ϕ(j+1)(ξ).

From (9) we get

m(₀j+1)(ξ) = Mm1,m2,...,md+1(2ξ)

Mm1,m2,...,md+1(ξ)

! !

"Wm(j+1)1,m2,...,md+1(ξ)

Wm(j)1,m2,...,md+1(2ξ)

=

d+1

i=1

1 +e–iki,ξ

2

mi!!

"Wm(j+1)1,m2,...,md+1(ξ)

Wm(j)1,m2,...,md+1(2ξ)

Finally, let us construct wavelets associated with the scaling functionϕ(j)_{,j∈Z. We define} the 2πZd_{-periodic functionsm}(j)

p,p= 1, 2, . . . , 2d–1, by

m(_pj)(ξ) =e–iγp,ξ_L(j)

p(2ξ)m

(j+1)

0 (ξ+γpπ),

where the trigonometric polynomialL(pj)is to be chosen such thatmp(j)satisfies (6) for allp. Proposition 3.2 Supposeϕ(j)is a scaling function for an MRA Vj,j∈Z,of Hs(Rd)and m(0j)

is the associated low pass filter.Then the distributions2j/2ψ(j)(2jx–k),j∈Z,k∈Zd,are an orthonormal basis for Hs_(Rd_{)if and only if}

ˆ

ψ_p(j)(2ξ) =e–iγp,ξ_L(j)

p(2ξ)m

(j+1)

0 (ξ+γpπ)ϕˆ(j+1)(ξ), ∀p= 1, 2, . . . , 2d–1,

a.e.onRd_{for some2πZ}d_{-periodic functionL}(j)

p such that

_L(j)

p(ξ)= 1, ∀p= 1, 2, . . . , 2d–1,a.e.ξ∈Td.

Proof From Proposition2.1we get

k∈Zd

1 + 22jξ+ 2kπ2sψˆ_p(j)(ξ+ 2kπ)2= 1, ∀p= 1, 2, . . . , 2d–1. (10)

Now, we have only to verify the density condition

lim

j→+∞ϕ

(j)₂–j_{ξ=1 +ξ}2–s/2_.

By definition,

W_m(j)_1,m2,...,m

d+1

2–jξ=

k∈Zd

1 +ξ+ 2j+1πk2sMm1,m2,...,md+1

2–jξ+ 2πk2.

The term associated withk= 0 converges to (1 +ξ2)s_{. Using the estimates}

Mm1,m2,...,md+1

2–jξ+ 2πk=

d+1

j=1

1 –e–ikj,2–jξ

ikj, 2–jξ+ 2πk

m_j

forξ= (ξ1,ξ2, . . . ,ξd) and

d+1

j=1

1 –e–ikj,2–jξ

ikj, 2–jξ+ 2πk

m_j

≤

_d+1

j=1

sin(2–(j+1)ξj)

2–j–1_ξ

j+πk

mj

sin(2–(j+1)d j=1ξj)

2–j–1d

j=1ξj+dπk

md+1

≤2

–(j+1)(d_j+1=1mj)

(#d_j=1|ξj|mj)(|

d

j=1ξj|md+1)

|k|

d+1

for 2–(j+1)₍#d

j=1|ξj|mj)(|

d

j=1ξj|md+1) < 1 andk= 0, we see that, asj→+∞, the sum of

the other terms converges to 0. The conclusion follows easily.

Ifψp(j)is an orthonormal wavelet, then the orthonormality of{2j/2ψp(j)(2j·–k) :j∈Z,k∈

Zd_{,p= 1, 2, . . . , 2}d–1_{}gives us}

1 =

k∈Zd

1 + 22jξ+ 2kπ2sψˆ_p(j)(ξ+ 2kπ)2

=

k∈Zd

1 + 22jξ+ 2kπ2sL_p(j)(ξ)2ϕˆ(j+1)(ξ/2 +kπ)2

×m(₀j+1)(ξ/2 +kπ+γpπ)2

=L(_pj)(ξ)2

l∈Zd

1 + 22(j+1)ξ/2 + 2lπ2sϕˆ(j+1)(ξ/2 + 2lπ)2

×

2d_–1

q=1

m(₀j+1)(ξ/2 +γqπ)

2 +

l∈Zd

1 + 22(j+1)ξ/2 + 2lπ+γqπ2

s

×ϕˆ(j+1)(ξ/2 + 2lπ+γqπ)

2

m(₀j+1)(ξ/2)2

=L(_pj)(ξ)2 ₂d_–1

q=0

m(₀j+1)(ξ/2 +γqπ)2

=L(_pj)(ξ)2, p= 1, 2, . . . , 2d–1,

for a.e.ξ∈Tdandγq,γp∈Ed, which finishes our proof. 4 Conclusion

In this paper, we have successfully generalized MRA over higher-dimensional Sobolev spaces by giving orthonormality and density conditions. Further, we constructed nonsep-arable orthonormal wavelets in a higher-dimensional Sobolev space by using multivari-ate box splines. The main obstacle in constructing wavelets is constructing low-pass and high-pass filters with the help of multivariate box splines, which satisfy the condition of orthonormality inHs_(Rd_{) for every scalej(because theH}s_{-norm is not dilation invariant).}

Acknowledgements

The authors would like to thank the anonymous referees for their insightful comments and suggestions.

Funding

The work of the corresponding author is supported by the Senior Research Fellowship of University Grants Commission (UGC), India (Grant no. SRF/AA/139/F-236/2013-14).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript. Both authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Kirori Mal College, University of Delhi, New Delhi, India.}2_{Department of Mathematics,}

University of Delhi, New Delhi, India.

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