Feb 4, 2017 - In the articles [14], [5] the mask m(0) and correspondent refinable function Ï were constructed using graph which is obtained from N-va...

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Abstract: We present an algorithm for construction step wavelets on local ﬁelds of positive characteristic. Key words: Local ﬁeld, scaling function, wavelets, multiresolution analysis.

arXiv:1702.01246v1 [math.FA] 4 Feb 2017

Introduction In 2004 H.Jiang, D.Li, and N.Jin [10] introduced the notion of multiresolution analysis (MRA) on local ﬁelds F (s) of positive characteristic p, proved some properties and constructed "Haar MRA" and corresponding "Haar wavelets". The wavelet theory developed in [1, 2, 3, 4, 11]. Construction of non-Haar wavelets is the a basic problem in this theory. The problem of constructing orthogonal MRA on the ﬁeld F (1) is studied in detail in the works [6, 7, 8, 12, 16, 17]. S.F.Lukomskii, A.M.Vodolazov [15, 18] considered local ﬁeld F (s) as a vector space over the ﬁnite ﬁeld GF (ps ) and constructed non-Haar wavelets. In [15] the authors construct the mask m(0) and correspondent reﬁnable function ϕ using some tree with zero as a root. In this case wavelets Ψ = (ψ (l) )l∈GF (ps ) may be found from the equality −1 ψˆ(l) = m(l) (χ)ϕ(χA ˆ ) l where A is a dilation operator, m(l) (χ) = m(0) (χr−l 0 ), and rk are Rademacher functions. In the article [13], the concept of N -valid tree was introduced and an algorithm for constructing the mask m(0) and correspondent reﬁnable function ϕ was indicated in the ﬁeld F (1) . In the articles [14], [5] the mask m(0) and correspondent reﬁnable function ϕ were constructed using graph which is obtained from N -valid tree by adding new arcs. But in this case we cannot deﬁne "masks" m(l) (χ) by the equation m(l) (χ) = m(0) (χr−l 0 ). In this article we give an algorithm for construction of "masks" m(l) (χ) in general case.

1

Basic concepts

Let p be a prime number, s ∈ N, GF (ps ) – ﬁnite ﬁeld. Local ﬁeld F (s) of positive characteristic p is isomorphic (Kovalski-Pontryagin theorem [9]) to the 1

Correspondence: [email protected], [email protected], [email protected] 2010 AMS Mathematics Subject Classification: 42C40, 43A25

1

set of formal power series a=

∞ X

ai ti , k ∈ Z, ai ∈ GF (ps ).

i=k

Addition and multiplication in the ﬁeld F (s) are deﬁned as sum and product of such series. Therefore we will consider local ﬁeld F (s) of positive characteristic p as the ﬁeld of sequences inﬁnite in both directions a = (. . . , 0n−1, an , an+1, . . .), aj ∈ GF (ps ) which have only ﬁnite number of elements aj with negative j nonequal to zero, and the operations of addition and multiplication are deﬁned by equalities ˙ = ((ai +b ˙ i))i∈Z , a+b X (ai bj ))l∈Z, ab = (

(1)

i,j:i+j=l

˙ and ” · ” are respectively addition and multiplication in GF (ps ). where ”+” The norm of the element a ∈ F (s) is deﬁned by the equality n 1 kak = k(. . . , 0n−1, an , an+1, . . .)k = , если an 6= 0. ps Therefore Fn(s) = {a = (aj )j∈Z : aj ∈ GF (ps ); aj = 0, ∀j < n} is a ball of radius p−ns. (s) Neighborhoods Fn are compact subgroups of the group F (s)+ . We will (s)+ denote them as Fn . They have the following properties: (s)+ (s)+ (s)+ 1). . . ⊂ F1 ⊂ F0 ⊂ F−1 ⊂ . . . (s)+ (s)+ (s)+ (s)+ 2)Fn /Fn+1 ∼ = GF (ps )+ и ♯(Fn /Fn+1 ) = ps . It is noted in [15] that the ﬁeld F (s) can be described as a linear space over GF (ps ). Using this description one may deﬁne the multiplication of element a ∈ F (s) on element λ ∈ GF (ps ) coordinatewise, i.e. λa = (. . . 0n−1, λan , λan+1, . . .), and the modulus λ ∈ GF (ps) can be deﬁned as 1, λ 6= 0, |λ| = 0, λ = 0. (s)

(s)

It is also proved there, that the system gk ∈ Fk \ Fk+1 is a basis in F (s) , i.e. any element a ∈ F (s) can be represented as: 2

λk gk , λk ∈ GF (ps).

P

a=

k∈Z

From now on we will consider gk = (..., 0k−1, (1(0), 0(1) , ..., 0(s−1))k , 0k+1, ...). In this case λk = ak . Let us deﬁne the sets (s)

˙ −2g−2 + ˙ . . . +a ˙ −s g−s }, s ∈ N. H0 = {h ∈ G : h = a−1g−1 +a ˙ −2 g−2+ ˙ . . . +a ˙ −s g−s , s ∈ N}. H0 = {h ∈ G : h = a−1 g−1+a The set H0 is the set of shifts in F (s) . It is an analogue of the set of nonnegative integers. We will denote the collection of all characters of F (s)+ as X. The set X generates a commutative group with respect to the multiplication of characters: (χ ∗ φ)(a) = χ(a) · φ(a). Inverse element is deﬁned as χ−1 (a) = χ(a), and the neutral element is e(a) ≡ 1. Following [15] we deﬁne characters rn of the group F (s)+ in the following (0) (1) (s−1) way. Let x = (. . . , 0k−1, xk , xk+1, . . .), xj = (xj , xj , . . . , xj ) ∈ GF (ps ). The element xj can be written in the form xj = (xjs+0, xjs+1, . . . , xjs+(s−1)). In this case x = (..., 0, xks+0, xks+1, . . . , xks+s−1, x(k+1)s+0, x(k+1)s+1, . . . , x(k+1)s+s−1, . . .) and the collection of all such sequences x is Vilenkin group. Thus the equality 2πi rn (x) = rks+l (x) = e p (xks+l ) deﬁnes Rademacher function of F (s)+ and every character χ ∈ X can be described in the following way: Y rnan , an = 0, p − 1. χ= (2) n∈Z

The equality (2) can be rewritten as Y a(0) a(1) (s−1) ak k k rks+0 rks+1 . . . rks+s−1 χ=

(3)

k∈Z

and let us deﬁne a

(0)

a

(1)

a

(s−1)

k k k rks+0 rks+1 . . . rks+s−1 = rakk

(0)

(1)

(s−1)

where ak = (ak , ak , . . . , ak

) ∈ GF (ps ). Then (3) takes the form Y rakk . χ=

(4)

k∈Z

(1,0,...,0)

We will refer to rk we set

= rk as the Rademacher functions. By deﬁnition 3

(rakk )bk

=

rakk bk ,

b

χ =( (0)

Y

rakk )b

(1)

=

(s−1)

It follows that if x = ((xk , xk , . . . xk GF (ps ) then (ruk , x)

=

Y

rakk b ,

ak , bk , b ∈ GF (ps ).

))k∈Z and u = (u(0), u(1) , . . . , u(s−1)) ∈

s−1 Y

e

2πi (l) (l) p u xk

.

l=0

In [15] the following properties of characters are proved ˙ 1) rku+v = ruk rvk , u, v ∈ GF (ps ). 2) (rvk , ugj ) = 1, ∀k 6= j, u, v ∈ GF (ps). s 3) The set of characters of the ﬁeld F (s) is a linear space (X, ∗, ·GF (p ) ) over the ﬁnite ﬁeld GF (ps) with multiplication being an inner operation and the power u ∈ GF (ps )being an outer operation. s 4) The set of Rademacher functions (rk ) is a basis in the spaceP (X, ∗, ·GF (p ) ). The dilation operator A in local ﬁeld F (s) is deﬁned as Ax := +∞ n=−∞ an gn−1 , P+∞ (s) where x = n=−∞ an gn ∈ F . In the group of characters it is deﬁned as (χA, x) = (χ, Ax).

2

Step Wavelets

We will consider a case of scaling function ϕ, which generates an orthogonal MRA, being a step function. The set of step functions constant on cosets of a (s) (s) (s) subgroup FM with the support supp(ϕ) ⊂ F−N will be denoted as DM (F−N ), (s) ⊥

M, N ∈ N. Similarly, D−N (FM ) is a set of step functions, constant on the (s) ⊥

(s) ⊥

cosets of a subgroup F−N with the support supp(ϕ) ⊂ FM . (s) an orthogonal MRA {Vn }, satisﬁes the reLet ϕ ∈ DM (F−N ) generate P ˙ [15], which we rewrite in a ﬁnement equation ϕ(x) = h∈H (N +1) βh ϕ(Ax−h) 0 frequency from −1 ϕ(χ) ˆ = m(0) (χ)ϕ(χA ˆ ),

where m(0) (χ) =

1 ps

X

(5)

βh (χA−1 , h)

(N +1)

h∈H0

is the mask of equation (5). There exist methods for constructing m(0) (χ) and ϕ(χ) ˆ (see e.g.[5]). We want to construct wavelets ψ (l) , l ∈ GF (ps ), l 6= 0 from 4

reﬁnable function ϕ. We will ﬁnd these wavelets ψ (l) from the equations −1 ψˆ(l) (χ) = m(l) (χ)ϕ(χA ˆ ),

and will call the functions m(l) (χ) masks, too. It is evident that ψˆ(0) (χ) = ϕ(χ). ˆ Theorem 2.1 Let m(k) (χ) (k ∈ GF (ps )) be a masks that are constant on the (s) ⊥

cosets of a subgroup F−N and periodic with any period ra1 1 ra2 2 . . . raν ν , aj ∈ GF (ps ), ν ∈ N. Define wavelets ψ (l) by the equations −1 ψˆ(l) (χ) = m(l) (χ)ϕ(χA ˆ ), (s) ˙ (l) )), where ϕ ∈ DM (F−N ) is a refinable function. The shifts system (ψ (l) (x−h l ∈ GF (ps), h(l) ∈ H0 will be orthonormal iff for any a−N . . . a−1 ∈ GF (ps ) X (s)⊥ a−N (s)⊥ a−N . . . ra0 0 )m(l) (F−N r−N m(k) (F−N r−N . . . ra0 0 ) = δk,l . (6) a0 ∈GF (ps )

(s) ⊥

Proof. The sufficiency. Let ϕ(χ) ˆ ∈ D−N (FM ) Consider scalar product l ˙ ˙ (ϕ(x−g), ψ (x−h)), where g, h ∈ H0. Z (l) (l) (x−h)d ˙ ˙ ˙ ˙ (ϕ(x−g), ψ (x−h)) = ϕ(x−g)ψ µ(x) = F (s)

=

Z

ˆ(l) (χ) = ϕˆ·−g ˙ (χ)ψ·−h ˙

Z

−1)(χ, g)(χ, h)m(l) (χ)d ν(χ) = ˆ ϕ(χ) ˆ ϕ(χA

X

X

=

Z

−1 2 (0) ˙ |ϕ(χA ˆ )| (χ, h−g)m (χ)m(l) (χ)d ν(χ) =

(s)⊥

FM

˜ = h−1g−1 +h ˙ =h ˙ −2g−2+ ˙ . . .| = = |h−g

=

Z

X

(s)⊥ a−N a −1 2 ˜ . . . rMM−1 |ϕ(F ˆ −N r−N −1 A )| (χ, h) dν(χ)·

a−N ...,a0 ,...,aM−1 (s)⊥ a−N aM−1 a F−N r−N ,...,r0 0 ,...,rM−1 (s)⊥ a

(s)⊥ a

−N −N . . . ra0 0 )m(l) (F−N r−N ·m(0) (F−N r−N . . . ra0 0 )d ν(χ) = X (s)⊥ a−N (s)⊥ a−N . . . ra0 0 ) . . . ra0 0 )m(l) (F−N r−N m(0) (F−N r−N =

a−N ,...,a0

5

(s)⊥ a−N +1 |ϕ(F ˆ −N r−N

X

a 2 . . . rMM−1 −2 )|

. . . ra0 1

a1 ,a2 ,...,aM−1

(s)⊥ a

Z

˜ (χ, h)dν(χ). a

a

−N M−1 ...r0 0 ...rM−1 F−N r−N

(7) ˙ By the orthonormality criteria for the system of shifts (ϕ(x−h)) of the res ﬁnable function ϕ ∀a−N , . . . , a0 ∈ GF (p ) the following equality holds: X (s)⊥ a−N +1 a 2 . . . ra0 1 . . . rMM−1 |ϕ(F ˆ −N r−N −2 )| = 1. a1 ,a2 ,...,aM−1

Consider integral from (7) Z 1 ˜ ˜ a−N (h) ˜ . . . ra−1 (h) ˜ = (χ, h)dν(χ) = sN 1F (s)⊥ (h)r −1 −N −N p (s)⊥ a

a

a

−N M−1 ...r0 0 ...rM−1 F−N r−N

=

−1 Y

1

p

˜ 1 (s)⊥ (h) sN F −N

(0) (0)

e

2πi p ((hj ,aj ))

,

j=−N

(s−1) (s−1)

aj is a scalar product. where (hj , aj ) = hj aj + . . . + hj Let us introduce the following notation: (s)⊥ a

(s)⊥ a

a0 a0 (l) (l) (0) −N −N m(0) a−N ...a0 = m (F−N r−N . . . r0 ), ma−N ...a0 = m (F−N r−N . . . r0 ).

Then we obtain ˙ ˙ (ϕ(·−g), ψ (l) (·−h)) = =

=

0 1

psN

For (ψ

1

psN

(k)

1 p

˜ 1 (s)⊥ (h) sN F −N

X

a−N ,...,a0

a−N ,...,a0

−1 Y

−1 Q

e

2πi p ((hj ,aj ))

a−N ,...,a−1 j=−N

2πi p ((hj ,aj ))

=

(s)⊥ ˜∈ if h / F−N ; ˜ = 0; if h

ma−N ...a0 ma−N ...a0

P

e

j=−N

(l)

(0)

P

(l) m(0) a−N ...a0 ma−N ...a0

P a0

(0) (l) ˜= ˜ ∈ F (s)⊥ . ma−N ...a0 ma−N ...a0 if h 6 0, h −N

(8)

˙ ˙ (x−g), ψ (x−h)) we can derive similar equality: (l)

˙ ˙ (ψ (k) (·−g), ψ (l) (·−h)) = 1

˜ = sN 1F (s)⊥ (h) −N p a

X

(l) m(k) a−N ...a0 ma−N ...a0

−N ,...,a0

−1 Y

j=−N

6

e

2πi p ((hj ,aj ))

=

0 1 sN p

1

psN

P

a−N ,...,a0

P

(s)⊥ ˜∈ if h / F−N ; ˜ = 0; if h

(l)

(k)

ma−N ...a0 ma−N ...a0 −1 Q

e

2πi p ((hj ,aj ))

a−N ,...,a−1 j=−N (j)

Thus, if masks m

P a0

(k) (l) ˜= ˜ ∈ F (s)⊥ . ma−N ...a0 ma−N ...a0 if h 6 0, h −N

(9) s

for all a−N . . . a−1 ∈ GF (p ) satisfy the condition X (l) m(k) a−N ...a0 ma−N ...a0 = δk,l , a0

˙ (l) )), l ∈ GF (ps ) is an orthonormal system. then the system of shifts (ψ (l) (x−h The necessity. Let us ﬁx k, l ∈ F G(ps) and consider equalities (8),(9) as a P (k) (l) ma−N ...a0 ma−N ...a0 and system of linear equation with unknowns xk,l a−N ...a−1 = a0

consider the matrix A of this system. It is obvious that A is a square matrix psN × psN . Let determinant is nonequal to zero. Let us start with N = 1, s = 1. In this case 1 1 1 ... 1 2πi 2πi 2πi 1 ep e p ·2 ... e p ·(p−1) 2πi 2πi 2πi 1 e p ·2·2 . . . e p ·2·(p−1) A = 1 e p ·2 p .. .. .. .. .. . . . . . 2πi 2πi 2πi 1 e p ·(p−1) e p ·(p−1)·2 . . . e p ·(p−1)·(p−1)

us prove that its

= V,

(10)

where V is Vandermonde matrix, which is known to have nonzero determinant. For the sake of clarity let us consider a case N = 2, s = 1. In this case the matrix A may be represented as block matrix V V V ... V 2πi 2πi 2πi V p V p ·2 V p ·(p−1) V e e . . . e 2πi 2πi 2πi 1 ·2 ·2·2 ·2·(p−1) p p p V e V ... e V = V ⊗ V, (11) e A= V p .. .. .. .. .. . . . . . 2πi 2πi 2πi V e p ·(p−1) V e p ·(p−1)·2 V . . . e p ·(p−1)·(p−1)V

where ⊗ symbol corresponds to Kronecker product. By the properties of Kronecker product det V ⊗ V = (det V )p(det V )p = (det V )2p 6= 0. Thus, again matrix A is nonsingular.

7

For the case of arbitrary N , s = 1 matrix A can be represented as A = V ⊗ V ⊗ . . . ⊗ V N times and will again have nonzero determinant by the properties of Kronecker product. Similarly, when N and s are both arbitrary A = V ⊗ V ⊗ . . . ⊗ V sN times. Thus, the system is nonsingular and has a unique solution, which proves the necessity. Theorem 2.1 can be reformulated in the following way: m(k) (χ) are the masks of corresponding step compactly supported orthonormal wavelets ψ (l) (χ) if and only if for each a−N . . . a−1 ∈ GF (ps) matrix M(a−N . . . a−1 ) with elements (s)⊥ a

−N . . . ra0 0 ) Ml,a0 (a−N , . . . , a−1) = m(l) (F−N r−N

is unitary. The suﬃciency of this theorem was proved in [10] (theorem 3). For step reﬁnable functions the condition (6) is necessary and suﬃcient. If the −1 condition (6) is fulﬁlled then the functions ψˆ(l) (χ) = m(l) (χ)ϕ(χA ˆ ) form a wavelet system [10]. For a step reﬁnable function we can describe an algorithm for constructing masks m(l) and wavelets ψ (l) , l ∈ GF (ps ). Let us assume we have all the values of m(0) (χ). We may obtain them using an algorithm presented in [5]. Recall the notation: (s)⊥ a

(s)⊥ a

a0 a0 (l) (l) (0) −N −N m(0) a−N ...a0 = m (F−N r−N . . . r0 ), ma−N ...a0 = m (F−N r−N . . . r0 ).

1) For each a−N . . . a−1 we construct a matrix M(a−N . . . a−1 ) ∈ Matps ×ps (C) with elements Ml,a0 (a−N . . . a−1 ) the following way. The ﬁrst row consists of all the values (0)

(0)

(0)

ma−N ...a−1 ,0 , ma−N ...a−1 ,1 , . . . , ma−N ...a−1 ,ps −1 (0)

(1)

(s−1)

where a−N . . . a−1 are ﬁxed and j = a0 +a0 p+. . .+a0 ps−1 calculated from (0) (1) (s−1) a0 = (a0 , a0 , . . . a0 ). Supplement this matrix to unitary in the following way. (0) If ma−N ...a−1 ,0 6= 0 then we make Ml,l = 1 for l 6= 0 and Ml,a0 = 0 for l 6= 0, l 6= a0 . (0) If ma−N ...a−1 ,0 = 0 then there exists number (0)

(1)

(s−1) s−1

j = j(a0 ) = a0 + a0 p + . . . + a0 (0)

p

for which ma−N ...a−1 ,j 6= 0. This nonzero value exists by the property of m(0) (see e.g.[10] ) In this case we make Mj,0 = 1, Ml,l = 1 for l 6= 0, l 6= j, and Ml,a0 = 0 in another case. 2) Run the Gram-Schmidt process on each matrix in order to make them unitary. 8

3) Now for each l ∈ GF (ps), l 6= 0 we ﬁnd the values of the mask m(l) from the equalities (s)⊥ a

−N . . . ra0 0 ) = Ml,a0 (a−N . . . a−1 ). m(l) (F−N r−N

. 4) The wavelets ψ (l) can be obtained using the formula −1 ψˆ(l) (χ) = m(l) (χ)ϕ(χA ˆ )

and performing inverse Fourier transform. First and second authors have performed the work of the state task of Russian Ministry of Education and Science (project 1.1520.2014K). The third author was supported RFBR, grant 16-01-00152.

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[9] Gelfand I, Graev M, Piatetski-Shapiro I. Theory of representations and authomorphic functions. M.:Nauka, 1966, 512p. (in russian.) (english translate: I.Gelfand, M.Graev, I.Piatetski-Shapiro. Theory of authomorphic functions. W.B.Saunders Company, Philadelphia, London, Toronto. 1969.) [10] Jiang H, Li D, Jin N. Multiresolution analysis on local ﬁelds. J Math Anal Appl 2004; 294: 523–532. [11] Li D, Jiang H. The necessary condition and suﬃcient conditions for wavelet frame on local ﬁelds. J Math Anal Appl 2008; 345: 500–510. [12] Lukomskii S. Step reﬁnable functions and orthogonal MRA on Vilenkin groups. J Fourier Anal Appl 2014; 20: 42–65. [13] Lukomskii S., Berdnikov G. N-Valid trees in wavelet theory on Vilenkin groups. International Journal of Wavelets, Multiresolution and Information Processing, Vol. 13, No. 5 (2015). [14] Lukomskii S., Berdnikov G., Kruss Iu. On the orthogonality of a system of shifts of the scaling function on Vilenkin groups. Mathematical Notes, 2015, 98:2, 339–342 [15] Lukomskii S., Vodolazov A. Non-Haar MRA on local Fields of positive characteristic. J Math Anal Appl 2016; 433: 1415–1440 [16] Protasov V. Approximation by dyadic wavelets. Mat Sb 2007; 198: 135– 152. (article in Russian with an abstract in English). [17] Protasov V., Farkov Yu. Dyadic wavelets and reﬁnable functions on a halfline. Mat Sb 2006; 197: 129–160. (article in Russian with an abstract in English). [18] Vodolasov A., Lukomskii S. MRA on Local Fields of Positive Characteristic. Izv Saratov Univ Mat Mekh Inform, 2014; 14: 511–518. (article in Russian with an abstract in English).

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