Research Article Frame Multiresolution Analysis on Local ...downloads.hindawi.com/archive/2015/216060.pdf · Research Article Frame Multiresolution Analysis on Local Fields of Positive

Research ArticleFrame Multiresolution Analysis on Local Fieldsof Positive Characteristic

Firdous A Shah

Department of Mathematics University of Kashmir South Campus Anantnag Jammu and Kashmir 192101 India

Correspondence should be addressed to Firdous A Shah fashahkugmailcom

Received 21 August 2014 Accepted 12 January 2015

Academic Editor Gerd Teschke

Copyright copy 2015 Firdous A Shah This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present a notion of framemultiresolution analysis on local fields of positive characteristic based on the theory of shift-invariantspaces In contrast to the standard setting the associated subspace 119881

0of 1198712(119870) has a frame a collection of translates of the scaling

function 120593 of the form 120593(sdot minus 119906(119896)) 119896 isin N0 where N

0is the set of nonnegative integers We investigate certain properties of

multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA)on local fields of positive characteristic Finally we provide a characterization of wavelet frames associated with FMRA on localfield119870 of positive characteristic using the shift-invariant space theory

1 Introduction

Multiresolution analysis is considered as the heart of wavelettheory The concept of multiresolution analysis providesa natural framework for understanding and constructingdiscrete wavelet systems A multiresolution analysis is anincreasing family of closed subspaces 119881

119895 119895 isin Z of 1198712(R)

such that⋂119895isinZ 119881119895= 0 and⋃

119895isinZ 119881119895is dense in 119871

2(R) which

satisfies 119891 isin 119881119895if and only if 119891(2sdot) isin 119881

119895+1 Furthermore

there exists an element 120593 isin 1198810such that the collection of

integer translates of function 120593 120593(sdot minus 119896) 119896 isin Z representsa complete orthonormal system for 119881

0 The function 120593 is

called the scaling function or the father wavelet The conceptofmultiresolution analysis has been extended in various waysin recent years These concepts are generalized to 119871

2(R119889)

to lattices different from Z119889 allowing the subspaces ofmultiresolution analysis to be generated byRiesz basis insteadof orthonormal basis admitting a finite number of scalingfunctions replacing the dilation factor 2 by an integer119872 ge 2

or by an expansive matrix 119860 isin 119866119871119889(R) as long as 119860 sub 119860Z119889

(see [1 2])On the other hand this elegant tool for the construc-

tion of wavelet bases has been extensively studied by sev-eral authors on the various spaces namely Cantor dyadicgroups [3] locally compact Abelian groups [4] 119901-adic fields

[5] zero-dimensional groups [6] and Vilenkin groups [7]Recently R L Benedetto and J J Benedetto [8] developed awavelet theory for local fields and related groupsTheydid notdevelop the multiresolution analysis (MRA) approach theirmethod is based on the theory of wavelet setsThe local fieldsare essentially of two types zero and positive characteristic(excluding the connected local fields R and C) Examples oflocal fields of characteristic zero include the 119901-adic field Q

119901

whereas local fields of positive characteristic are the Cantordyadic group and the Vilenkin 119901-groups The structures andmetrics of the local fields of zero and positive characteristicare similar but their wavelet and MRA theory are quitedifferent The concept of multiresolution analysis on a localfield119870 of positive characteristic was introduced by Jiang et al[9] They pointed out a method for constructing orthogonalwavelets on local field119870with a constant generating sequenceSubsequently tight wavelet frames on local fields of positivecharacteristic were constructed by Shah and Debnath [10]using extension principles As far as the characterization ofwavelets on local fields is concerned Behera and Jahan [11]have given the characterization of all wavelets associated withmultiresolution analysis on local field 119870 based on results onaffine and quasiaffine frames Recently Shah and Abdullah[12] have introduced the notion of nonuniform multiresolu-tion analysis on local field 119870 of positive characteristic and

Hindawi Publishing CorporationJournal of OperatorsVolume 2015 Article ID 216060 8 pageshttpdxdoiorg1011552015216060

2 Journal of Operators

obtained the necessary and sufficient condition for a function120593 to generate a nonuniformmultiresolution analysis on localfields More results in this direction can also be found in[13 14] and the references therein

Since the use of multiresolution analysis has proven to bea very efficient tool in wavelet theory mainly because of itssimplicity it is of interest to try to generalize this notion asmuch as possiblewhile preserving its connectionwithwaveletanalysis In this connection Benedetto and Li [15] consideredthe dyadic semiorthogonal frame multiresolution analysis of1198712(R) with a single scaling function and successfully applied

the theory in the analysis of narrow band signals Thecharacterization of the dyadic semiorthogonal frame mul-tiresolution analysis with a single scaling function admittinga single frame wavelet whose dyadic dilations of the integertranslates forma frame for1198712(R)was obtained independentlyby Benedetto and Treiber by a direct method [16] and by Kimand Lim by using the theory of shift-invariant spaces [17]Later on Yu [18] extended the results of Benedetto and Lirsquostheory of FMRA to higher dimensions with arbitrary integralexpansive matrix dilations and has established the necessaryand sufficient conditions to characterize semiorthogonalmultiresolution analysis frames for 1198712(R119899)

In this paper we introduce the notion of frame multire-solution analysis (FMRA) on local field 119870 of positive char-acteristic by extending the above describedmethodsWe firstinvestigate the properties ofmultiresolution subspaces whichwill provide the quantitative criteria for the constructionof FMRA on local fields of positive characteristic We alsoshow that the scaling property of an FMRA also holdsfor the wavelet subspaces and that the space 119871

2(119870) can

be decomposed into the orthogonal sum of these waveletsubspaces Finally we study the characterization of waveletframes associated with FMRA on local field 119870 of positivecharacteristic using the shift-invariant space theory

The paper is organized as follows In Section 2 wediscuss some preliminary facts about local fields of positivecharacteristic including the definition of a frame The notionof frame multiresolution analysis of 1198712(119870) is introduced inSection 3 and its quantitative criteria are given by means ofTheorem 12 In Section 4 we establish a complete charac-terization of wavelet frames generated by a finite number ofmother wavelets on local field119870 of positive characteristic

2 Preliminaries on Local Fields

Let 119870 be a field and a topological space Then 119870 is calleda local field if both 119870

+ and 119870lowast are locally compact Abelian

groups where119870+ and119870lowast denote the additive andmultiplica-

tive groups of119870 respectively If119870 is any field and is endowedwith the discrete topology then119870 is a local field Further if119870is connected then 119870 is either R or C If 119870 is not connectedthen it is totally disconnectedHence by a local field wemeana field 119870 which is locally compact nondiscrete and totallydisconnected The 119901-adic fields are examples of local fieldsMore details are referred to in [19 20] In the rest of thispaper we use NN

0 and Z to denote the sets of natural and

nonnegative integers and integers respectively

Let119870 be a fixed local fieldThen there is an integer 119902 = p119903where p is a fixed prime element of 119870 and 119903 is a positiveinteger and a norm | sdot | on 119870 such that for all 119909 isin 119870 we have|119909| ge 0 and for each 119909 isin 119870 0 we get |119909| = 119902

119896 for someinteger 119896 This norm is non-Archimedean that is |119909 + 119910| le

max|119909| |119910| for all 119909 119910 isin 119870 and |119909 + 119910| = max|119909| |119910|whenever |119909| = |119910| Let 119889119909 be the Haar measure on thelocally compact topological group (119870 +) This measure isnormalized so that int

D119889119909 = 1 whereD = 119909 isin 119870 |119909| le 1 is

the ring of integers in119870 DefineB = 119909 isin 119870 |119909| lt 1The setB is called the prime ideal in 119870 The prime ideal in 119870 is theunique maximal ideal in D and hence as a result B is bothprincipal and prime Therefore for such an idealB inD wehaveB = ⟨p⟩ = pD

Let Dlowast = D B = 119909 isin 119870 |119909| = 1 Then it is easy toverify thatDlowast is a group of units in 119870

lowast and if 119909 = 0 then wemay write 119909 = p1198961199091015840 1199091015840 isin Dlowast Moreover each B119896 = p119896D =

119909 isin 119870 |119909| lt 119902minus119896

is a compact subgroup of119870+ and is knownas the fractional ideals of119870+ (see [19]) LetU = 119886

119894119902minus1

119894=0be any

fixed full set of coset representatives of B in D then everyelement 119909 isin 119870 can be expressed uniquely as 119909 = sum

infin

ℓ=119896119888ℓpℓ

with 119888ℓisin U Let 120594 be a fixed character on119870

+ that is trivial onD but is nontrivial onBminus1 Therefore 120594 is constant on cosetsof D implying that if 119910 isin B119896 then 120594

119910(119909) = 120594(119910119909) for 119909 isin

119870 Suppose that 120594119906is any character on 119870

+ then clearly therestriction 120594

119906|D is also a character onD Therefore if 119906(119899)

119899 isin N0 is a complete list of distinct coset representatives of

D in119870+ then as it was proved in [20] the set 120594

119906(119899) 119899 isin N

0

of distinct characters onD is a complete orthonormal systemonD

The Fourier transform 119891 of a function 119891 isin 1198711(119870)cap119871

2(119870)

is defined by

119891 (120585) = int119870

119891 (119909) 120594120585(119909)119889119909 (1)

It is noted that

119891 (120585) = int119870

119891 (119909) 120594120585(119909)119889119909 = int

119870

119891 (119909) 120594 (minus120585119909) 119889119909 (2)

Furthermore the properties of Fourier transform on localfield are much similar to those on the real line In particularFourier transform is unitary on 119871

2(119870)

We now impose a natural order on the sequence 119906(119899)

119899 isin N0 SinceDB cong 119866119865(119902) where119866119865(119902) is a 119888-dimensional

vector space over the field 119866119865(119902) (see [20]) we choose a set1 = 120577

0 1205771 1205772 120577

119888minus1 sub Dlowast such that span 120577

119895119888minus1

119895=0cong 119866119865(119902)

For 119899 isin N0such that 0 le 119899 lt 119902 we have

119899 = 1198860+ 1198861119901 + sdot sdot sdot + 119886

119888minus1119901119888minus1

0 le 119886119896lt 119901

119896 = 0 1 119888 minus 1

(3)

Define

119906 (119899) = (1198860+ 11988611205771+ sdot sdot sdot + 119886

119888minus1120577119888minus1

) pminus1

(4)

For 119899 isin N0and 0 le 119887

119896lt 119902 119896 = 0 1 2 119904 we write

119899 = 1198870+ 1198871119902 + 11988721199022+ sdot sdot sdot + 119887

119904119902119904 (5)

Journal of Operators 3

such that

119906 (119899) = 119906 (1198870) + 119906 (119887

1) pminus1

+ sdot sdot sdot + 119906 (119887119904) pminus119904 (6)

Also for 119903 119896 isin N0and 0 le 119904 lt 119902

119896 we have

119906 (119903119902119896+ 119904) = 119906 (119903) p

minus119896+ 119906 (119904) (7)

Further it is easy to verify that 119906(119899) = 0 if and only if 119899 = 0

and 119906(ℓ) + 119906(119896) 119896 isin N0 = 119906(119896) 119896 isin N

0 for a fixed

ℓ isin N0 Hereafter we use the notation 120594

119899= 120594119906(119899)

119899 ge 0Let the local field 119870 be of characteristic 119901 gt 0 and let

1205770 1205771 1205772 120577

119888minus1be as above We define a character 120594 on 119870

as follows

120594 (120577120583pminus119895) =

exp(2120587119894

119901) 120583 = 0 119895 = 1

1 120583 = 1 119888 minus 1 or 119895 = 1

(8)

Definition 1 Let H be a separable Hilbert space A sequence119891119896

119896 isin N0 in H is called a 119891119903119886119898119890 for H if there exist

constants 119860 and 119861 with 0 lt 119860 le 119861 lt infin such that

1198601003817100381710038171003817119891

1003817100381710038171003817

2

2le sum

119896isinN0

1003816100381610038161003816⟨119891 119891119896⟩1003816100381610038161003816

2le 119861

10038171003817100381710038171198911003817100381710038171003817

2

2 forall119891 isin H (9)

The largest constant 119860 and the smallest constant 119861 satisfying(9) are called the upper and the lower frame bound respec-tively A frame is said to be tight if it is possible to choose119860 = 119861 and a frame is said to be exact if it ceases to be a framewhen any one of its elements is removed An exact frame isalso known as a Riesz basis

The following theorem gives us an elementary character-ization of frames

Theorem 2 (see [15]) A sequence 119891119896 119896 isin N

0 in a Hilbert

space H is a frame for H if and only if there exists a sequence119886 = 119886

119896 isin 1198972(N0) with 119886

1198972(N0)

le 119862119891 119862 gt 0 such that

119891 = sum

119896isinN0

119886119896119891119896 (10)

and sum119896isinN0

|⟨119891 119891119896⟩|2lt infin for every 119891 isin H

For 119895 isin Z and 119910 isin 119870 we define the dilation operator 120575119895

and the translation operator 120591119910as follows

120575119895119891 (119909) = 119902

1198952119891 (pminus119895119909) 120591

119910119891 (119909) = 119891 (119909 minus 119910)

119891 isin 1198712(119870)

(11)

Our study uses the theory of shift-invariant spaces developedin [21 22] and the references therein A closed subspace 119878 of1198712(119870) is said to be shift-invariant if 120591

119896119891 isin 119878 whenever 119891 isin 119878

and 119896 isin N0 A closed shift-invariant subspace 119878 of 1198712(119870) is

said to be generated by Φ sub 1198712(119870) if 119878 = span120591

119896120593(119909) =

120593(119909 minus 119906(119896)) 119896 isin N0 120593 isin Φ The cardinality of the

smallest generating set Φ for 119878 is called the length of 119878 whichis denoted by |119878| If |119878| = finite then 119878 is called a finite shift-invariant space (FSI) and if |119878| = 1 then 119878 is called a principal

shift-invariant space (PSI) Moreover the spectrum of a shift-invariant space is defined to be

120590 (119878) = 120585 isin D 119878 (120585) = 0 (12)

where 119878(120585) = 119891(120585 + 119906(119896)) isin 1198972(N0) 119891 isin 119878 119896 isin N

0

3 Frame Multiresolution Analysison Local Fields

We first introduce the notion of a frame multiresolution anal-ysis (FMRA) of 1198712(119870)

Definition 3 Let 119870 be a local field of positive characteristic119901 gt 0 and let p be a prime element of 119870 A frame multire-solution analysis of 1198712(119870) is a sequence of closed subspaces119881119895 119895 isin Z of 1198712(119870) satisfying the following properties

(a) 119881119895sub 119881119895+1

for all 119895 isin Z

(b) ⋃119895isinZ 119881119895is dense in 119871

2(119870) and⋂

119895isinZ 119881119895= 0

(c) 119891(sdot) isin 119881119895if and only if 119891(pminus1sdot) isin 119881

119895+1for all 119895 isin Z

(d) the function 119891 lying in 1198810implies that the collection

119891(sdot minus 119906(119896)) isin 1198810 for all 119896 isin N

0

(e) the sequence 120591119896120593 = 120593(sdot minus 119906(119896)) 119896 isin N

0 is a frame

for the subspace 1198810

The function 120593 is known as the scaling function while thesubspaces 119881

119895rsquos are known as approximation spaces or mul-

tiresolution subspaces A frame multiresolution analysis issaid to be nonexact and respectively exact if the frame forthe subspace 119881

0is nonexact and respectively exact In mul-

tiresolution analysis studied in [9] the frame condition isreplaced by that of an orthonormal basis or an exact frame

Next we establish several properties of multiresolutionsubspaces that will help in the construction of frame mul-tiresolution analysis on local field119870 of positive characteristicThe following proposition shows that for every 119895 isin Z thesequence 120593

119895119896 119896 isin N

0 where

120593119895119896 (119909) = 119902

1198952120593 (pminus119895119909 minus 119906 (119896)) (13)

is a frame for 119881119895

Proposition 4 Let 120591119896120593 119896 isin N

0 be a frame for 119881

0=

span120591119896120593 119896 isin N

0 and

119881119895= 119891 isin 119871

2(119870) 119891 (p

119895sdot) isin 119881

0 119895 isin Z (14)

Then the sequence 120593119895119896

119896 isin N0 defined in (13) is a frame for

119881119895with the same bounds as those for 119881

0

4 Journal of Operators

Proof For any 119891 isin 119881119895 we have

sum

119896isinN0

10038161003816100381610038161003816⟨120575minus119895119891 120591119896120593⟩

10038161003816100381610038161003816

2

= sum

119896isinN0

1003816100381610038161003816100381610038161003816int119870

119902minus1198952

119891(p119895119909)120593(119909 minus 119906(119896))119889119909

1003816100381610038161003816100381610038161003816

2

(15)

= sum

119896isinN0

1003816100381610038161003816100381610038161003816int119870

119891(119909)1199021198952

120593(pminus119895119909 minus 119906(119896))119889119909

1003816100381610038161003816100381610038161003816

2

(16)

= sum

119896isinN0

10038161003816100381610038161003816⟨119891 120593119895119896

⟩10038161003816100381610038161003816

2

(17)

Since 120591119896120593 119896 isin N

0 is a frame for 119881

0 therefore we have

1198601003817100381710038171003817119891

1003817100381710038171003817

2

2= 119860

10038171003817100381710038171003817120575minus11989511989110038171003817100381710038171003817

2

2le sum

119896isinN0

10038161003816100381610038161003816⟨119891 120593119895119896

⟩10038161003816100381610038161003816

2

le 11986110038171003817100381710038171003817120575minus11989511989110038171003817100381710038171003817

2

2

= 1198611003817100381710038171003817119891

1003817100381710038171003817

2

2

(18)

This completes the proof of the proposition

We now characterize all functions of FSI space by virtueof its Fourier transforms

Proposition 5 Let 120591119896120593 119896 isin N

0 120593 isin Φ be a frame for

its closed linear span119881 where Φ = 1205931 1205932 120593

119871 sub 1198712(119870)

Then 119891 isin 1198712(119870) lies in 119881 if and only if there exist integral

periodic functions ℎℓisin 1198712(D) ℓ = 1 119871 such that

119891 (120585) =

119871

sum

ℓ=1

ℎℓ (120585) 120593ℓ (120585) (19)

Proof Since the system 120591119896120593 119896 isin N

0 120593 isin Φ is a frame for

119881 then by Theorem 2 there exists a sequence 119886ℓ

119896 isin 1198972(N0)

for ℓ = 1 119871 such that

119891 (119909) =

119871

sum

ℓ=1

sum

119896isinN0

119886ℓ

119896120593ℓ(119909 minus 119906 (119896)) (20)

Taking Fourier transform on both sides of (20) we obtain

119891 (120585) =

119871

sum

ℓ=1

ℎℓ (120585) 120593ℓ (120585) (21)

where ℎℓ(120585) = sum

119896isinN0119886ℓ

119896120594119896(120585) are the integral periodic

functions in 1198712(D) The converse is established by taking ℎ

ℓ

as above and applying the inverse Fourier transform on bothsides of (19)

We now study some properties of the multiresolutionsubspaces 119881

119895of the form (14) by means of the Fourier

transform

Proposition 6 Let 120591119896120593 119896 isin N

0 be a frame for 119881

0=

span 120591119896120593 119896 isin N

0 and for 119895 isin Z define119881

119895by (14)Then for

any function 120595 isin 1198811 there exists periodic function 119866 isin 119871

2(D)

such that

(pminus1

120585) = 11990212

119866 (120585) 120593 (120585) (22)

Proof By the definition of 119881119895 it follows that 120595(psdot) isin 119881

0 By

Proposition 5 there exists a periodic function 119866 isin 1198712(D)

such that (120595(psdot))and

= (pminus1120585) = 11990212

119866(120585)120593(120585) lies in 1198712(119870)

The following theorem establishes a sufficient conditionto ensure that the nesting property holds for the subspaces119881119895rsquos

Theorem 7 Let 120591119896120593 119896 isin N

0 be a frame for 119881

0=

span 120591119896120593 119896 isin N

0 and for 119895 isin Z define 119881

119895by (14) Assume

that there exists a periodic function 119867 isin 119871infin(D) such that

120593 (120585) = 11990212

119867(p120585) 120593 (p120585) (23)

Then 119881119895sube 119881119895+1

for every 119895 isin Z

Proof Given any 119891 isin 119881119895 there exists a sequence 119886

119896119896isinN0

isin

1198972(N0) such that

119891 (119909) = 1199021198952

sum

119896isinN0

119886119896120593 (pminus119895119909 minus 119906 (119896)) (24)

Let 1198980(120585) = sum

119896isinN0119886119896120594119896(120585) isin 119871

2(D) and let 119898

1(p120585) =

1198980(120585)119867(p120585) Then clearly 119898

1lies in 119871

2(D) as 119867 lies in

119871infin(D) Therefore by Parsevalrsquos identity there exists a

sequence 119887119896119896isinN0

isin 1198972(N0) such that 119898

1(120585) = sum

119896isinN0119887119896120594119896(120585)

lies in 1198712(119870)

Taking Fourier transform of (24) and using assumption(23) we obtain

119891 (120585) = 1199021198952

1198980(p119895120585) 120593 (p

119895120585)

= 119902(119895+1)2

1198980(p119895120585)119867 (p

119895+1120585) 120593 (p

119895+1120585)

= 119902(119895+1)2

1198981(p119895+1

120585) 120593 (p119895+1

120585)

(25)

By implementing inverse Fourier transform to (25) we have

119891 (119909) = 119902(119895+1)2

sum

119896isinN0

119887119896120593 (pminus119895minus1

119909 minus 119906 (119896)) (26)

Using Proposition 4 we observe that 119891 isin 119881119895+1

Moreover itis easy to verify that the function119867 in (23) is not unique

The following theorem is the converse to Theorem 7

Theorem 8 Let 120591119896120593 119896 isin N

0 be a frame for 119881

0=

span 120591119896120593 119896 isin N

0 and for 119895 isin Z define 119881

119895by (14) Assume

that 1198810sube 1198811andΦ(120585) = 120593(120585 minus 119906(119896))

2

1198972(N0)

Then there existsperiodic function 119867 isin 119871

infin(D) such that (23) holds

Proof Since 120591119896120593 119896 isin N

0 is a frame for 119881

0 therefore there

exist positive constants 119860 and 119861 such that

119860 le Φ (120585) le 119861 ae on 120590 (1198810) (27)

Since 1198810sube 1198811 we have 120593 isin 119881

1 By Proposition 6 there exists

a periodic function1198670isin 1198712(D) such that

120593 (pminus1

120585) = 11990212

1198670 (120585) 120593 (120585) (28)

Journal of Operators 5

Therefore we have

1003816100381610038161003816120593 (120585)1003816100381610038161003816

2= 119902

10038161003816100381610038161198670(p120585)1003816100381610038161003816

2 1003816100381610038161003816120593(p120585)1003816100381610038161003816

2 ae (29)

Let S = B 120590(1198810) and 119867 isin 119871

2(D) be a periodic function

such that119867 = 1198670 ae on 120590(119881

0) and119867 is bounded onS by a

positive constant 119862 Then it follows from the above fact that119867 is not unique so that (29) also holds for119867 that is

1003816100381610038161003816120593 (120585)1003816100381610038161003816

2= 119902

1003816100381610038161003816119867 (p120585)1003816100381610038161003816

2 1003816100381610038161003816120593 (p120585)1003816100381610038161003816

2 ae (30)

Taking 119899 = 119896119901 + 119903 where 119896 isin N0and 119903 = 0 1 119902 minus 1 we

have1003816100381610038161003816120593(120585 + 119906(119899))

1003816100381610038161003816

2

= 1199021003816100381610038161003816119867(p120585 + p119906(119903))

1003816100381610038161003816

2|120593 (p120585 + p119906 (119903) + 119906 (119896)

1003816100381610038161003816

2 ae(31)

Summing up (31) for all 119896 isin N0and 119903 = 0 1 119902 minus 1 we

have

sum

119899isinN0

1003816100381610038161003816120593 (120585 + 119906 (119899))1003816100381610038161003816

2

=119902

119902minus1

sum

119903=0

1003816100381610038161003816119867 (p120585 + p119906 (119903))1003816100381610038161003816

2sum

119896isinN0

|120593 (p120585 + p119906 (119903) +119906 (119896)1003816100381610038161003816

2 ae

(32)

which is equivalent to

Φ (120585) = 119902

119902minus1

sum

119903=0

1003816100381610038161003816119867 (p120585 + p119906 (119903))1003816100381610038161003816

2Φ (p120585 + p119906 (119903)) ae (33)

or

Φ(pminus1

120585) = 119902

119902minus1

sum

119903=0

1003816100381610038161003816119867(120585 + p119906(119903))1003816100381610038161003816

2Φ (120585 + p119906 (119903)) ae (34)

Note that Φ(pminus1120585) le 119861 ae and hence (34) becomes

119902minus1

sum

119903=0

1003816100381610038161003816119867 (120585 + p119906 (119903))1003816100381610038161003816

2Φ (120585 + p119906 (119903)) le 119902119861 ae (35)

This implies that for almost every 120585 isin Bminus1 and 119903 = 0 1

119902 minus 1 we have

1003816100381610038161003816119867(120585 + p119906(119903))1003816100381610038161003816

2Φ (120585 + p119906 (119903)) le 119902119861 (36)

Also if Φ(120585 + p119906(119903)) = 0 then |119867(120585 + p119906(119903))| le 119862 and ifΦ(120585+p119906(119903)) gt 0 thenwemay assume that119860 le Φ(120585+p119906(119903)) le

119861 Thus for almost every 120585 isin Bminus1 and 119903 = 0 1 119902 minus 1 wehave

1003816100381610038161003816119867(120585 + p119906(119903))1003816100381610038161003816

2le max 119862

2 119902119861119860minus1

(37)

Hence 119867 is essentially bounded on D This proves thetheorem completely

The following two propositions are proved in [23]

Proposition 9 Suppose 1198810= span 120591

119896120593 119896 isin N

0 and for

each 119895 isin Z define 119881119895by (14) such that 119881

0sube 1198811 Assume that

|120593| gt 0 119886119890 on a neighborhood of zeroThen the union⋃119895isinZ119881119895

is dense in 1198712(119870)

Proposition 10 Let 120593 isin 1198712(119870) and define 119881

0= span 120591

119896120593

119896 isin N0 For each 119895 isin Z define 119881

119895by (14) Then one has

⋂119895isinZ 119881119895= 0

Lemma 11 Let 119881119895be the family of subspaces defined by (14)

with119881119895sube 119881119895+1

for each 119895 isin Z Suppose 120593 isin 1198712(119870) is a nonzero

function with 1198810= span 120591

119896120593 119896 isin N

0 Then for every 119895 isin

Z 119881119895is a proper subspace of 119881

119895+1

Proof Suppose that 119881ℓ= 119881ℓ+1

for some ℓ isin Z Let 119891 isin 119881119895+1

then for any given 119895 isin Z we have 119891(p119895+1minusℓminus1119909) isin 119881

119895+1 Since

119891(p119895minusℓ119909) isin 119881ℓ therefore 119891 lies in 119881

119895and 119881

119895= 119881119895+1

Hence⋂119895isinZ 119881119895

= 1198810 By Proposition 10 it follows that 119881

119895= 0

which is a contradiction

Combining all our results so far we have the followingtheorem

Theorem 12 Let 120593 isin 1198712(119870) and define 119881

0= span 120591

119896120593 119896 isin

N0 For each 119895 isin Z define 119881

119895by (14) and Φ(120585) = 120593(120585 minus

119906(119896))2

1198972(N0)

Suppose that the following hold

(i) 119860 le Φ(120585) le 119861 ae on 120590(1198810)

(ii) there exists a periodic function 119867 isin 119871infin(D) such that

120593 (120585) = 11990212

119867(p120585) 120593 (p120585) ae (38)

(iii) |120593| gt 0 ae on a neighborhood of zero

Then 119881119895 119895 isin Z defines a frame multiresolution analysis

of 1198712(119870)

Proof Since 1198810is a shift-invariant subspace of 1198712(119870) there-

fore the system 120591119896120593 119896 isin N

0 forms a frame for119881

0with frame

bounds 119860 and 119861 ByTheorem 7 and Lemma 11 it follows that119881119895sub 119881119895+1

for every 119895 isin Z Hence by the definition of 119881119895 119891

lies in 119881119895if and only if 119891(p119895) lies in 119881

0 while 119891(pminus1sdot) lies in

119881119895+1

if and only if 119891(p119895+1sdot) lies in 1198810 Thus 119891 lies in 119881

119895if and

only if 119891(pminus1sdot) lies in 119881119895+1

Moreover by assumption (iii) andProposition 10 it follows that ⋃

119895isinZ119881119895 is dense in 1198712(119870) and

⋂119895isinZ 119881119895

= 0 Thus the sequence 119881119895

119895 isin Z satisfiesall the conditions to be a frame multiresolution analysis of1198712(119870)

In order to constructwavelet frames associatedwith framemultiresolution analysis on local fields 119870 of positive charac-teristic we introduce the orthogonal complement subspaces119882119895 119895 isin Z of119881

119895in119881119895+1

It is easy to verify that the sequenceof subspaces 119882

119895 119895 isin Z also satisfies the scaling property

that is

119882119895= 119891 isin 119871

2(119870) 119891 (p

119895sdot) isin 119882

0 119895 isin Z (39)

6 Journal of Operators

Theorem 13 Let 119881119895

119895 isin Z be an increasing sequence ofclosed subspaces of 1198712(119870) such that ⋃

119895isinZ119881119895 is dense in 1198712(119870)

and ⋂119895isinZ 119881119895= 0 Let 119882

119895be the orthogonal complement of

119881119895in119881119895+1

for each 119895 isin Z Then the subspaces119882119895are pairwise

orthogonal and

1198712(119870) = ⨁

119895isinZ

119882119895 (40)

Proof Assume that 119894 lt 119895 then ⟨119891119894 119891119895⟩ = 0 for any119891

119894isin 119882119895as

119882119894sub 119881119894+1

sub 119881119895 Let 119875

119895be the orthogonal projection operators

from 1198712(119870) onto 119881

119895 then lim

119895rarrinfin119875119895119891 = 119891 lim

119895rarrminusinfin119875119895119891 =

0 and 119882119895

= 119891 minus 119875119895119891 119891 isin 119881

119895+1 Therefore for any 119891 isin

1198712(119870) we have

119891 = sum

119895isinZ

(119875119895+1

119891 minus 119875119895119891) (41)

Thus the result of the direct sum follows since 119875119895+1

minus119875119895is the

orthogonal projector from 1198712(119870) onto119882

119895

4 Characterization of Wavelet Frameson Local Fields

In this section we give the characterization of wavelet framesassociated with frame multiresolution analysis on local fieldsof positive characteristic First we will characterize theexistence of a function 120595 in 119882

0 where 119882

0is the orthogonal

complement of1198810in1198811 by virtue of the analysis filters 119866 and

119867 defined as in Section 3

Theorem 14 Let 119867 be a periodic function associated with theframe multiresolution analysis 119881

119895 119895 isin Z satisfying the

condition (23) Define 1198820as the orthogonal complement of 119881

0

in 1198811 Let 120595 isin 119881

1such that

(120585) = 11990212

119866 (p120585) 120593 (p120585) (42)

where 119866 is a periodic function in 1198712(D) Then 120595 lies in 119882

0if

and only if

119902minus1

sum

119903=0

119867(p120585 +p119906 (119903))Φ (p120585 +p119906 (119903)) 119866 (p120585+ p119906 (119903)) =0 119886119890 120585

(43)

Proof We note that 120595 lies in 1198820if and only if

⟨120595 120591119896120595⟩ = ⟨120595 120595 (sdot minus 119906 (119896))⟩ = 0 forall119896 isin N

0 (44)

Define

119865 (120585) = sum

119896isinN0

120593 (120585 + 119906 (119896)) (120585 + 119906(119896)) (45)

Then it is easy to verify that 119865 lies in 1198711(D) by using Mono-

tonic ConvergenceTheorem and the Plancherel Theorem as

intD

1003816100381610038161003816119865 (120585)1003816100381610038161003816 119889120585 le int

D

sum

119896isinN0

1003816100381610038161003816120593 (120585 + 119906 (119896)) (120585 + 119906 (119896))1003816100381610038161003816 119889120585

= sum

119896isinN0

intD

1003816100381610038161003816120593 (120585 + 119906 (119896)) (120585 + 119906 (119896))1003816100381610038161003816 119889120585

= int119870

1003816100381610038161003816120593 (120585) (120585)1003816100381610038161003816 119889120585

le1003817100381710038171003817120593

10038171003817100381710038172

100381710038171003817100381710038171003817100381710038172

=1003817100381710038171003817120593

10038171003817100381710038172

100381710038171003817100381712059510038171003817100381710038172

(46)

For a fixed 119899 isin N0 we define 119865

119872as

119865119872 (120585) =

119872

sum

119896=0

120593 (120585 + 119906 (119896)) (120585 + 119906(119896)) 120594119899 (120585) (47)

Then in view of (23) and (42) we have

119865119872

(120585) = 119902

119902minus1

sum

119903=0

sum

119902119896+119903le119872

119867(p120585 + p119906 (119903))1003816100381610038161003816120593(p120585 + p119906(119903) + 119906(119896))

1003816100381610038161003816

2

sdot 119866(p120585 + p119906(119903))120594119899(120585)

(48)

Using Monotonic Convergence Theorem and the Cauchy-Schwartz inequality we obtain

1003817100381710038171003817119865119872 minus 119865120594119899

10038171003817100381710038171198712(D)

le intD

sum

119896ge119872+1

1003816100381610038161003816120593 (120585 + 119906 (119896)) (120585 + 119906 (119896))1003816100381610038161003816 119889120585

= sum

119896ge119872+1

intD

1003816100381610038161003816120593 (120585 + 119906 (119896)) (120585 + 119906 (119896))1003816100381610038161003816 119889120585

= sum

119896ge119872+1

int119909+D

1003816100381610038161003816120593 (120585) (120585)1003816100381610038161003816 119889120585

le int|120585|gt119872

1003816100381610038161003816120593 (120585) (120585)1003816100381610038161003816 119889120585

le int|120585|gt119872

|120593(120585)|2119889120585

12

int|120585|gt119872

1003816100381610038161003816 (120585)1003816100381610038161003816

2119889120585

12

997888rarr 0 as119872 997888rarr infin

(49)

Hencelim119872rarrinfin

1003817100381710038171003817119865119872 minus 119865120594119899

10038171003817100381710038171198712(D)= 0 (50)

Therefore there exists a subsequence 119865119872119895

such that

lim119895rarrinfin

100381710038171003817100381710038171003817119865119872119895

minus 119865120594119899

1003817100381710038171003817100381710038171198712(D)= 0 ae (51)

Hence

119865 (120585) = 119902

119902minus1

sum

119903=0

119867(p120585 + p119906 (119903))

sdot Φ (p120585 + p119906 (119903)) 119866(p120585 + p119906(119903)) ae

(52)

Journal of Operators 7

Using (50) and the Dominated Convergence Theorem wehave for all 119899 isin N

0

⟨120595 120591minus119899

120593⟩ = int119870

(120585) 120593(120585)120594119899(120585) 119889120585

= sum

119896isinN0

int119909+D

(120585) 120593(120585)120594119899 (120585) 119889120585

= lim119872rarrinfin

119872

sum

119896=0

intD

(120585 + 119906 (119896))

sdot 120593(120585 + 119906(119896))120594119899(120585) 120594119896(120585) 119889120585

= lim119872rarrinfin

intD

119865119872 (120585) 119889120585

= intD

119865 (120585) 120594119899(120585) 119889120585

(53)

Consequently 119865 = 0 ae is the necessary and sufficientcondition for (44) to hold for all 119899 isin N

0

Lemma 15 Let 119882119895

119895 isin Z be a sequence of pairwiseorthogonal closed subspaces of 119871

2(119870) such that 119871

2(119870) =

⨁119895isinZ119882119895 Then for every 119891 isin 119871

2(119870) there exist 119891

119895isin 119882119895

119895 isin Z such that 119891(119909) = sum119895isinZ 119891119895(119909) Furthermore

10038171003817100381710038171198911003817100381710038171003817

2

2= sum

119895isinZ

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817

2

2 (54)

Proof For any arbitrary function 119891 isin 1198712(119870) we have

lim119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891 minus

119899

sum

119895=minus119899

119891119895

100381710038171003817100381710038171003817100381710038171003817100381710038172

= 0 (55)

where 119891119895isin 119882119895 for each 119895 isin Z Moreover for a fixed 119899 isin N

we have10038171003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119895=minus119899

119891119895

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

2

=

119899

sum

119895=minus119899

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817

2

2 (56)

Since the norm sdot2is continuous therefore the desired result

is obtained by taking 119899 rarr infin on both sides of the aboveequality

Theorem 16 Let 120593 be the scaling function for a framemultiresolution analysis 119881

119895 119895 isin Z and suppose that 119882

119895

is the orthogonal complement of 119881119895in 119881119895+1

Let Ψ = 1205951

1205952 120595

119871 sub 119882

0 Then the collection

FΨ

= 120595ℓ

119895119896(119909) = 119902

1198952120595ℓ(pminus119895119909 minus 119906 (119896))

119895 isin Z 119896 isin N0 ℓ = 1 119871

(57)

constitutes a wavelet frame for 1198712(119870)with frame bounds119860 and119861 if and only if

120591119896120595ℓ 119896 isin N

0 ℓ = 1 119871 (58)

forms a frame for 1198820with frame bounds 119860 and 119861

Proof Suppose that the systemFΨgiven by (57) is a wavelet

frame for 1198712(119870) with bounds 119860 and 119861 Then it follows from(39) that the family of functions 120595

ℓ

119895119896lies in 119882

119895 for ℓ =

1 119871 119895 isin Z and 119896 isin N0

By applyingTheorem 13 to an arbitrary function 119891 isin 1198820

we have

sum

119895isinZ

sum

119896isinN0

10038161003816100381610038161003816⟨119891 120595ℓ

119895119896⟩10038161003816100381610038161003816

2

= sum

119896isinN0

10038161003816100381610038161003816⟨119891 120591119896120595ℓ⟩10038161003816100381610038161003816

2

(59)

Using the frame property of the systemFΨ we have

1198601003817100381710038171003817119891

1003817100381710038171003817

2

2le

119871

sum

ℓ=1

sum

119896isinN0

10038161003816100381610038161003816⟨119891 120591119896120595ℓ⟩10038161003816100381610038161003816

2

le 1198611003817100381710038171003817119891

1003817100381710038171003817

2

2 (60)

and it follows that the collection 120591119896120595ℓ 119896 isin N

0 ℓ = 1 119871

is a frame for1198820

Conversely suppose that the collection 120591119896120595ℓ

119896 isin

N0 ℓ = 1 119871 is a frame for 119882

0with bounds 119860 and 119861

For any fixed 119895 isin Z and 119891 isin 119882119895 we have from (39) that

119891(p119895sdot) isin 1198820 Moreover by making use of the fact that

⟨119891 120595ℓ

119895119896⟩ = 1199021198952

int119870

119891 (119909) 120595ℓ (pminus119895119909 minus 119906 (119896))119889119909

10038171003817100381710038171003817119902minus1198952

119891(p119895sdot)10038171003817100381710038171003817

2

2= 119902minus119895

int119870

10038161003816100381610038161003816119891 (p119895119909)

10038161003816100381610038161003816

2

119889119909 =1003817100381710038171003817119891

1003817100381710038171003817

2

2

(61)

we have

11986010038171003817100381710038171003817119902minus1198952

119891(p119895sdot)10038171003817100381710038171003817

2

2le

119871

sum

ℓ=1

sum

119896isinN0

10038161003816100381610038161003816⟨119891 120595ℓ

119895119896⟩10038161003816100381610038161003816

2

le 11986110038171003817100381710038171003817119902minus1198952

119891(p119895sdot)10038171003817100381710038171003817

2

2

(62)

Thus for a given 119895 isin Z the collection 120595ℓ

119895119896 119896 isin N

0 ℓ =

1 119871 constitutes a frame for119882119895with frame bounds119860 and

119861Let 119891 be an arbitrary function in 119871

2(119870) then by Theo-

rem 13 and Lemma 15 there exist 119891119895isin 119882119895such that

119891 = sum

119895isinZ

119891119895 ⟨119891

119894 120595ℓ

119895119896⟩ = 0 119894 = 119895 (63)

Therefore we have119871

sum

ℓ=1

sum

119895isinZ

sum

119896isinN0

10038161003816100381610038161003816⟨119891 120595ℓ

119895119896⟩10038161003816100381610038161003816

2

=

119871

sum

ℓ=1

sum

119895isinZ

sum

119896isinN0

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119894isinZ

⟨119891119894 120595ℓ

119895119896⟩

1003816100381610038161003816100381610038161003816100381610038161003816

2

=

119871

sum

ℓ=1

sum

119895isinZ

sum

119896isinN0

10038161003816100381610038161003816⟨119891119895 120595ℓ

119895119896⟩10038161003816100381610038161003816

2

(64)

Using (62) we obtain

119860sum

119895isinZ

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817

2

2le

119871

sum

ℓ=1

sum

119895isinZ

sum

119896isinN0

10038161003816100381610038161003816⟨119891119895 120595ℓ

119895119896⟩10038161003816100381610038161003816

2

le 119861sum

119895isinZ

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817

2

2 (65)

Combining (64) (65) and Lemma 15 we have

11986010038171003817100381710038171003817119891119895

10038171003817100381710038171003817

2

2le

119871

sum

ℓ=1

sum

119895isinZ

sum

119896isinN0

10038161003816100381610038161003816⟨119891119895 120595ℓ

119895119896⟩10038161003816100381610038161003816

2

le 11986110038171003817100381710038171003817119891119895

10038171003817100381710038171003817

2

2 (66)

This completes the proof of the theorem

8 Journal of Operators

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] L Debnath and F A Shah Wavelet Transforms and TheirApplications Birkhauser New York NY USA 2015

[2] S G Mallat ldquoMultiresolution approximations and waveletorthonormal bases of 119871

2(R)rdquo Transactions of the American

Mathematical Society vol 315 no 1 pp 69ndash87 1989[3] W C Lang ldquoOrthogonal wavelets on the Cantor dyadic grouprdquo

SIAM Journal on Mathematical Analysis vol 27 no 1 pp 305ndash312 1996

[4] Y A Farkov ldquoOrthogonal wavelets with compact support onlocally compact Abelian groupsrdquo IzvestiyaMathematics vol 69no 3 article 623 2005

[5] A Y Khrennikov V M Shelkovich and M Skopina ldquo119901-adicrefinable functions and MRA-based waveletsrdquo Journal ofApproximation Theory vol 161 no 1 pp 226ndash238 2009

[6] S F Lukomskii ldquoMultiresolution analysis on product of zero-dimensional Abelian groupsrdquo Journal of Mathematical Analysisand Applications vol 385 no 2 pp 1162ndash1178 2012

[7] S F Lukomskii ldquoStep refinable functions and orthogonalMRA on Vilenkin groupsrdquo The Journal of Fourier Analysis andApplications vol 20 no 1 pp 42ndash65 2014

[8] J J Benedetto and R L Benedetto ldquoA wavelet theory for localfields and related groupsrdquoThe Journal of Geometric Analysis vol14 no 3 pp 423ndash456 2004

[9] H Jiang D Li and N Jin ldquoMultiresolution analysis on localfieldsrdquo Journal of Mathematical Analysis and Applications vol294 no 2 pp 523ndash532 2004

[10] FA Shah andLDebnath ldquoTightwavelet frames on local fieldsrdquoAnalysis vol 33 no 3 pp 293ndash307 2013

[11] B Behera andQ Jahan ldquoCharacterization of wavelets andMRAwavelets on local fields of positive characteristicrdquo CollectaneaMathematica vol 66 no 1 pp 33ndash53 2015

[12] F A Shah andAbdullah ldquoNonuniformmultiresolution analysison local fields of positive characteristicrdquo Complex Analysis andOperator Theory 2014

[13] F A Shah and Abdullah ldquoWave packet frames on local fields ofpositive characteristicrdquo Applied Mathematics and Computationvol 249 pp 133ndash141 2014

[14] F A Shah and Abdullah ldquoA characterization of tight waveletframes on local fields of positive characteristicrdquo Journal ofContemporaryMathematical Analysis vol 49 no 6 pp 251ndash2592014

[15] J J Benedetto and S Li ldquoThe theory of multiresolution analysisframes and applications to filter banksrdquo Applied and Com-putational Harmonic Analysis Time-Frequency and Time-ScaleAnalysis Wavelets Numerical Algorithms and Applications vol5 no 4 pp 389ndash427 1998

[16] J J Benedetto and O M Treiber ldquoWavelet frames multireso-lution analysis and extensionprinciplerdquo in Wavelet Transformsand Time-Frequency Signal Analysis L Debnath Ed pp 3ndash36Birkhauser Boston Mass USA 2000

[17] H O Kim and J K Lim ldquoOn frame wavelets associated withframe multiresolution analysisrdquo Applied and ComputationalHarmonic Analysis Time-Frequency and Time-Scale Analysis

Wavelets Numerical Algorithms and Applications vol 10 no1 pp 61ndash70 2001

[18] X Yu ldquoSemiorthogonal multiresolution analysis frames inhigher dimensionsrdquo Acta Applicandae Mathematicae vol 111no 3 pp 257ndash286 2010

[19] D Ramakrishnan and R J Valenza Fourier Analysis on NumberFields vol 186 ofGraduate Texts in Mathematics Springer NewYork NY USA 1999

[20] M H Taibleson Fourier Analysis on Local Fields PrincetonUniversity Press Princeton NJ USA 1975

[21] A Ron and Z Shen ldquoFrames and stable bases for shift-invariantsubspaces of 1198712(R119889)rdquo Canadian Journal of Mathematics vol 47no 5 pp 1051ndash1094 1995

[22] A Ron and Z Shen ldquoAffine systems in 1198712(R119889) the analysis of

the analysis operatorrdquo Journal of Functional Analysis vol 148no 2 pp 408ndash447 1997

[23] B Behera and Q Jahan ldquoMultiresolution analysis on local fieldsand characterization of scaling functionsrdquoAdvances in Pure andApplied Mathematics vol 3 no 2 pp 181ndash202 2012

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