p-Adic refinable functions and MRA-based wavelets

p -ADIC REFINABLE FUNCTIONS AND MRA-BASED W A VELETS A. YU. KHRENNIKO V, V. M. SHELK OVICH, AND M. SK OPINA Abstract. W e d escrib ed a wide class of p - adic refina ble equations generating p -adic multiresolution analysis. A m ethod for the const ruction of p -adic or- thogonal wa v elet bases within the fr amew ork of the M RA theory is suggested. A realization of this method is illustrated by an example, which gives a new 3-adic wa v elet basis. Another realization leads to the p -adic H aar bases which we re known before. 1. Intr oduction A sensation happ ened in the ea rly nineties beca use a general scheme for the construction of wa v elets w as developed. This sch eme is based o n the notion of m ultiresolution analysis (MRA in the sequel) in tro duced b y Y. Meyer and S. Mallat. Immediately sp ecialists started to implement new wa velet systems. No wada ys it is difficult to find an engineering a rea where wa velets a re not applied. In the p -a dic setting, situation is the following. In 2002 S. V. K ozyrev [9 ] found a co mpactly suppo rted p -adic w av elet basis for L 2 ( Q p ) which is an analog of the Haar basis. Another p -adic w av elet-type system generalizing Kozyrev’s basis was constructed in [6], [7]. It turned out that the men tioned a b ov e p -a dic wa velets are eigenfunctions of p -adic pseudo-differ e n tial op erators [1], [2], [5]– [7], [9], [1 0]. This fact implies that study of w av elets is imp ortant and gives a new powerful technique for solving p -adic problems (area s of applica tions can be found in [4], [8], [13]). Nevertheless, in the cited pap e rs, a the ory describing c ommon pr op e rties of p -adic wavelet b ases and giving gener al metho ds for their c onstruction was not develop e d . Moreov er, it was as sumed in [3] that there are obstacles for creating a p -a dic analog of the classica l MRA theory . T o construct a p - adic analog of a classical MRA we need a prop er p -adic r efinement e quation . In [6] the following co njecture was prop osed: the e qualit y (1.1) φ ( x ) = p − 1 X r =0 φ  1 p x − r p  , x ∈ Q p , may be consider ed as a r efinement e quation . A solution φ to this equation ( a r efinable fun ction ) is the character is tic function Ω  | x | p  of the unit disc. The equation (1.1) reflects a natur al “self-similar it y” of the space Q p : the unit disc 2000 Mathematics Subje ct Classific ation. Primary 11S80, 42C40; Secondary 11E95. Key wor ds and phr ases. p -adic multiresolution analysis; refinable equations, wa v elets. This paper was supp orted in part b y the grant of The Swedish Roy al Academ y of Sciences on collaboration with scien tists of former Soviet Union. The secon d author (V. S.) w as supported in part b y DFG Pr o ject 436 RUS 113/809 and Grant 05-01-04002-NNIOa of RFBR. The third author (M.S.) w as suppor ted in part b y Grant 06-01-00457 of RFBR. 1 2 A. YU. KHRENNIKO V, V. M. SHELKO VICH, AND M. SKOPINA B 0 (0) = { x : | x | p ≤ 1 } is repres en ted as a sum o f p mutually disjoint discs B 0 (0) = B − 1 (0) ∪  ∪ p − 1 r =1 B − 1 ( r )  , where B − 1 ( r ) =  x : | x − r | p ≤ p − 1  (see [13, I.3,Examples 1,2.]). The equa tion (1.1 ) is a n a nalog o f the r efinement e quation generating the Haar MRA in the rea l analysis. Using this idea, the notion of p - adic MRA was introduced and a general scheme for its c o nstruction was descr ibed in [12]. In [12], this scheme was rea lized for construction 2- adic Haar MRA with using (1.1) a s the gener ating refinement equation. In contrast to the real setting, the r efin able fun ction φ gener ating the Haar MRA is p erio dic , which never holds for rea l refinable functions. Do e to this fact, there exist infin ity many differ ent orthonor mal wa velet bases in the s ame Haar MRA. One of them co incide s with Kozyrev ’s wa velet basis. The present paper is dev oted to study o f p -adic r efine- ment e quations generating MRAs and a method for the construction of MRA-ba sed wa velets. Here a nd in wha t follows, we shall use the notations and the results from [13]. Let N , Z , C b e the sets of p ositive in tegers, integers, complex n um b ers, respe c tiv ely . The field Q p of p -adic n umbers is defined as the completion of the field of ratio nal nu mbers Q with resp ect to the non- Arc himedean p -adic norm | · | p . This p - adic norm is defined as follows: | 0 | p = 0; if x 6 = 0, x = p γ m n , where γ = γ ( x ) ∈ Z and the integers m , n are not div isible by p , then | x | p = p − γ . The norm | · | p satisfies the strong triangle inequality | x + y | p ≤ ma x( | x | p , | y | p ). The cano nical for m o f any p -adic num b er x 6 = 0 is (1.2) x = p γ ( x 0 + x 1 p + x 2 p 2 + · · · ) , where γ = γ ( x ) ∈ Z , x j = 0 , 1 , . . . , p − 1, x 0 6 = 0, j = 0 , 1 , . . . . W e sha ll w r ite the p -adic num b ers k = k 0 + k 1 p + · · · + k s − 1 p s − 1 , k 0 , . . . , k s − 1 = 0 , 1 , 2 , . . . , p − 1, in the usual (for the real analysis) form: k = 0 , 1 , . . . , p s − 1. Denote by B γ ( a ) = { x ∈ Q p : | x − a | p ≤ p γ } the disc of ra dius p γ with the center at a point a ∈ Q p , γ ∈ Z . An y tw o balls in Q p either ar e disjoint o r one c on tains the other. There exists the Haa r measure dx o n Q p ,which is p ositive, inv ar ian t under the shifts, i.e., d ( x + a ) = dx , and normalized by R | ξ | p ≤ 1 dx = 1. A complex-v a lued function f defined on Q p is called lo c al ly-c onstant if for any x ∈ Q p there exists an integer l ( x ) ∈ Z such that f ( x + y ) = f ( x ), y ∈ B l ( x ) (0). Denote b y D ( Q p ) the linear spaces of lo cally-constant compa ctly supported functions (so -called test functions) [13, VI.1.,2.]. The F ourier transform of ϕ ∈ D ( Q p ) is defined b y b φ ( ξ ) = F [ ϕ ]( ξ ) = R Q p χ p ( ξ · x ) ϕ ( x ) dx , ξ ∈ Q p , where χ p ( ξ · x ) = e 2 π i { ξ x } p is the additive character the field Q p , {·} p is a fractional par t of a num b er x ∈ Q p . The F o urier transform is extended to L 2 ( Q p ) in a sta ndard way . If f ∈ L 2 ( Q p ), 0 6 = a ∈ Q p , b ∈ Q p , then [13, VI I,(3.3)]: (1.3) F [ f ( ax + b )]( ξ ) = | a | − 1 p χ p  − b a ξ  F [ f ( x )]  ξ a  . According to [13, IV,(3.1)], (1.4) F [Ω( p − k | · | p )]( x ) = p k Ω( p k | x | p ) , k ∈ Z , x ∈ Q p , where Ω( t ) = 1 for t ∈ [0 , 1]; Ω( t ) = 0 for t 6∈ [0 , 1]. p -ADIC REFINABLE FUNCTIONS AND MRA-BASED W A VELETS 3 2. Mul tiresolution anal ysis Let I p = { a = p − γ  a 0 + a 1 p + · · · + a γ − 1 p γ − 1  : γ ∈ N ; a j = 0 , 1 , . . . , p − 1; j = 0 , 1 , . . . , γ − 1 } . It is well known that Q p = B 0 (0) ∪ ∪ ∞ γ =1 S γ , where S γ = { x ∈ Q p : | x | p = p γ } . Due to (1.2), x ∈ S γ , γ ≥ 1, if and o nly if x = x − γ p − γ + x − γ +1 p − γ +1 + · · · + x − 1 p − 1 + ξ , where x − γ 6 = 0, ξ ∈ B 0 (0). Since x − γ p − γ + x − γ +1 p − γ +1 + · · · + x − 1 p − 1 ∈ I p , we hav e a “natural” decomp osition o f Q p to a union of mutually disjoint discs : Q p = ∪ a ∈ I p B 0 ( a ). So, I p is a “natur al” set of shifts fo r Q p . Definition 2.1. ( [12]) A collection of clo sed spaces V j ⊂ L 2 ( Q p ), j ∈ Z , is called a mult ir esolution analysis ( MRA ) in L 2 ( Q p ) if the following axio ms ho ld (a) V j ⊂ V j +1 for all j ∈ Z ; (b) S j ∈ Z V j is de ns e in L 2 ( Q p ); (c) T j ∈ Z V j = { 0 } ; (d) f ( · ) ∈ V j ⇐ ⇒ f ( p − 1 · ) ∈ V j +1 for all j ∈ Z ; (e) there e x ists a function φ ∈ V 0 such tha t the system { φ ( · − a ) , a ∈ I p } is an orthonor mal basis for V 0 . The function φ from axiom (e) is called sc aling . It follows immediately from axioms (d) and (e) that the functions p j / 2 φ ( p − j · − a ), a ∈ I p , form a n orthono rmal basis for V j , j ∈ Z . According to the standa rd sch eme (see, e.g., [11, § 1.3]) for construction o f MRA-based wa velets, for each j , we define a space W j ( wavelet sp ac e ) a s the orthogonal complement of V j in V j +1 , i.e., V j +1 = V j ⊕ W j , j ∈ Z , where W j ⊥ V j , j ∈ Z . It is not difficult to see that (2.1) f ∈ W j ⇐ ⇒ f ( p − 1 · ) ∈ W j +1 , for all j ∈ Z and W j ⊥ W k , j 6 = k . T aking into account a xioms (b) and (c), w e obtain (2.2) M j ∈ Z W j = L 2 ( Q p ) (orthogonal direct sum) . If now w e find a finite num b er of functions ψ ( ν ) ∈ W 0 , ν ∈ A, such that the system { ψ ( ν ) ( x − a ) , a ∈ I p , ν ∈ A } is an orthono rmal basis for W 0 , then, due to (2 .1) and (2.2), the system { p j / 2 ψ ( ν ) ( p − j · − a ) , a ∈ I p , j ∈ Z , ν ∈ A } , is a n orthonor mal basis for L 2 ( Q p ). Such functions ψ ( ν ) are called wavelet functions and the bas is is a wavelet b asis . Let φ be a sc aling function for a MRA. As w as men tioned ab ov e, the system { p 1 / 2 φ ( p − 1 · − a ) , a ∈ I p } is a basis fo r V 1 . It follows from axio m (a) that (2.3) φ = X a ∈ I p β a φ ( p − 1 · − a ) , β a ∈ C . W e see that the function φ is a solution of a s p ecial kind of functional equation. Such equations a re called re finement e quations , and their solutions are calle d r efinable functions 1 . A na tural way for the construction of a MRA (see, e.g ., [1 1, § 1.2]) is the following. W e star t with an appr opriate function φ w ho se I p -shifts form an orthonormal system and set (2.4) V j = span  φ  p − j · − a  : a ∈ I p  , j ∈ Z . 1 Usually the terms “scaling function” and “refinable function” are synon yms in the l iterature, and they are used for both the senses: as a solution to a refinemen t equation and as a function generating MRA. W e separate the meanings of these terms. 4 A. YU. KHRENNIKO V, V. M. SHELKO VICH, AND M. SKOPINA It is clear that axioms (d) and (e) of Definition 2.1 are fulfilled. Of cours e, no t any s uc h a function φ provides axio m ( a ). In the r e al sett ing , the r elation V 0 ⊂ V 1 holds if and o nly if the scaling function sa tisfies a refinement equation. Situatio n is different in p -a dics. Ge ner ally sp eaking, a r efinemen t equation (2.3) do es not imply the including pr op erty V 0 ⊂ V 1 bec ause the set of the shifts I p do es not form a gr ou p . Indeed, we need a ll the functions φ ( · − b ), b ∈ I p , to b elong to the space V 1 , i.e., the identities φ ( x − b ) = P a ∈ I p α a,b φ ( p − 1 x − a ) should b e fulfilled for all b ∈ I p . Since p − 1 b + a is not in I p in ge ne r al, we c an not state that φ ( x − b ) = P a ∈ I p α a,b φ ( p − 1 x − p − 1 b − a ) ∈ V 1 for a ll b ∈ I p . Nev ertheless, some refinement equations imply including pro per t y , which ma y happen b ecause of different causes. The re finement e quation (1.1) is a particular case of (2.3). 3. Construction of refinable functions Now we are g oing to study p -adic r efinemen t e q uations and their s olutions. W e restrict ourselves b y the refinement equatio ns (2.3) with a finite num b er of the terms in the rig h t-hand side: (3.1) φ ( x ) = p s − 1 X k =0 β k φ  1 p x − k p s  . If φ ∈ L 2 ( Q p ), taking the F ourier transform a nd using (1.3), one can rewrite (3.1) as (3.2) b φ ( ξ ) = m 0  ξ p s − 1  b φ ( pξ ) , where (3.3) m 0 ( ξ ) = 1 p p s − 1 X k =0 β k χ p ( k ξ ) is a trigonometric p olynomial. It is clear tha t m 0 (0) = 1 whenever b φ (0) 6 = 0. Theorem 3.1. If φ is a re finable function such that supp b φ ⊂ B 0 (0) and the system { φ ( x − a ) : a ∈ I p } is orthonormal, then axiom ( a ) fr om Definition 2 .1 holds for the sp ac es (2 .4) . Pr o of. Since χ p ( − ξ ) = 1 for ξ ∈ B 0 (0), we ha ve χ p ( − ξ ) b φ ( ξ ) = b φ ( ξ ). Applying the F our ier transfo rm, w e obtain φ ( x + 1) = φ ( x ) . Thus φ is a 1 -p erio dic function. Since for a ll a ∈ I p and all k = 0 , 1 , . . . , p s − 1, either a p + k p s ∈ I p , or a p + k p s − 1 ∈ I p . Due to the 1 -p erio dicity of φ , it follo ws from (3.1) that φ ( x − b ) ∈ V 1 for all b ∈ I p . This implies V 0 ⊂ V 1 , simila rly V j ⊂ V j +1 for any j ∈ Z .  Theorem 3 .2. L et φ ∈ L 2 ( Q p ) b e a r efinable function, the system { φ ( x − a ) : a ∈ I p } b e orthonormal and supp b φ ⊂ B 0 (0) . A x iom ( b ) of Definition 2.1 holds for the sp ac es (2 .4) ( i.e., ∪ j ∈ Z V j = L 2 ( Q p )) if and only if (3.4) [ j ∈ Z supp b φ ( p j · ) = Q p . p -ADIC REFINABLE FUNCTIONS AND MRA-BASED W A VELETS 5 Pr o of. Firs t of all we note that, due to axio ms ( d ) a nd ( e ), eac h space V j is inv a riant with r e s pect to the shifts t = p j a , a ∈ I p . Show tha t the space ∪ j ∈ Z V j is inv ariant with resp ect to any shift t ∈ Q p . E v ery t ∈ Q p may b e approximated by a vector p j a , a ∈ I p , with arbitra ry larg e j ∈ Z . If f ∈ ∪ j ∈ Z V j , by axiom ( a ) (which holds due to Theor e m 3 .1), then f ∈ V j for all j ≥ j 1 . It follows from the contin uity of the function k f ( · + t ) k 2 that f ( · + t ) ∈ ∪ j ∈ Z V j . No w let t ∈ Q p . If g ∈ ∪ j ∈ Z V j , then approximating g by the functions f ∈ ∪ j ∈ Z V j , aga in using the contin uity of the shift op erator and the inv ariance of L 2 norm with resp ect to the shifts, we derive g ( · + t ) ∈ ∪ j ∈ Z V j . F or X ⊂ L 2 ( Q p ), set b X = { b f : f ∈ X } . By the Wiener theore m for L 2 (see, e.j., [11], all the arguments of the pro of g iv en ther e may b e rep eated word for word with replacing R by Q p ), a c losed subspace X of the spa ce L 2 ( Q p ) is inv ar ian t with resp ect to the shifts if and only if b X = L 2 (Ω) for so me set Ω ⊂ Q p . Let X = ∪ j ∈ Z V j , then b X = L 2 (Ω). Thus X = L 2 ( Q p ) if and only if Ω = Q p . Set φ j = φ ( p − j · ) , Ω 0 = ∪ j ∈ Z supp b φ j and prov e that Ω = Ω 0 . Since φ j ∈ V j , j ∈ Z , we hav e supp b φ j ⊂ Ω, and hence Ω 0 ⊂ Ω . Now a ssume that Ω \ Ω 0 contains a set of po sitiv e measure Ω 1 . If f ∈ V j , taking the F ourier transfor m from the expansio n f = P a ∈ I p h a φ ( p − j · − a ) we see that b f = 0 almost everywhere on Ω 1 . Hence the same is true for any f ∈ ∪ j ∈ Z V j . Passing to the limit we de duce that that the F ourier transform of a n y f ∈ X is equa l to zero a lmost everywhere on Ω 1 , i.e., L 2 (Ω) = L 2 (Ω 0 ). It remains to note that supp b φ j = supp b φ ( p j · )  Theorem 3.3. If φ ∈ L 2 ( Q p ) and t he system { φ ( x − a ) : a ∈ I p } is orthonormal, then axiom ( c ) of Definition 2.1 holds, i.e., ∩ j ∈ Z V j = { 0 } . Pr o of. Firs t, using the standard scheme (see, e.g., [11, Lemma 1.2.8.]), we pr o ve that for any f ∈ L 2 ( Q p ) (3.5) lim j →−∞ X a ∈ I p    f , p j / 2 φ ( p − j · − a )    2 = 0 , where ( · , · ) is the s calar pro duct in L 2 ( Q p ). Since the space D ( Q p ) is dense in L 2 ( Q p ) [13, VI.2], it suffices to prov e (3.5) for any ϕ ∈ D ( Q p ). If ϕ ∈ D ( Q p ), then there exists N s uc h that ϕ ( x ) = 0 for all | ξ | p > p N . Since | ϕ ( x ) | ≤ M for a ll | ξ | p ≤ p N , we hav e X a ∈ I p    ϕ ( · ) , p j / 2 φ ( p − j · − a )    2 ≤ p j X a ∈ I p  Z | x | p ≤ p N | ϕ ( x ) || φ ( p − j x − a ) | dx  2 ≤ p j + N M 2 X a ∈ I p Z | x | p ≤ p N | φ ( p − j x − a ) | 2 dx. By the c hange of v ariables η = p − j x − a , w e obtain X a ∈ I p    ϕ, p j / 2 φ ( p − j · − a )    2 ≤ M 2 p N Z A N j | φ ( η ) | 2 dη = M 2 p N Z Q p θ N j ( η ) | φ ( η ) | 2 dη , where θ N j is the characteristic function of the set A N j = ∪ a ∈ I p { η : | η + a | p ≤ 2 N + j } . Since lim j →−∞ θ N j ( η ) = 0 for an y η 6 = − a , using the Leb esgue dominated conv e rgence theo rem [13, IV.4], we obtain lim j →−∞ R Q p θ N j ( η ) | φ ( η ) | 2 dη = 0. 6 A. YU. KHRENNIKO V, V. M. SHELKO VICH, AND M. SKOPINA If now we assume that f ∈ ∩ j ∈ Z V j , then f ∈ V j for all j ∈ Z and, due to (3.5), k f k =  X a ∈ I p    f , p j / 2 φ ( p − j · − a )    2  1 / 2 → 0 , j → −∞ , i.e., k f k = 0 whic h was to be prov ed.  Theorem 3.4. L et φ b e a r efinable function such that supp b φ ⊂ B 0 (0) . If | b φ ( ξ ) | = 1 for al l ξ ∈ B 0 (0) then the system { φ ( x − a ) : a ∈ I p } is orthonormal. Pr o of. T aking into account formula (1.4), using the inc lus ion supp b φ ⊂ B 0 (0) and the P lanc herel formula, we have for any a ∈ I p  φ ( · ) , φ ( · − a )  = Z Q p φ ( x ) φ ( x − a ) dx = Z B 0 (0) | b φ ( ξ ) | 2 χ p ( aξ ) dξ = Z B 0 (0) χ p ( aξ ) dξ = Z Q p Ω( | ξ | p ) χ p ( aξ ) dξ = Ω( | a | p ) = δ a 0 .  So, to construc t a MRA we can take a function φ for which the hypotheses of Theorems 3.4 and (3.4) a re fulfilled. Next we are going to describ e all such functions. Prop osition 3.5. If φ ∈ L 2 ( Q p ) is a solution of r efinable e quation ( 3.2), b φ ( ξ ) is c ontinuous at the p oint 0 and b φ (0) 6 = 0 , then (3.6) b φ ( ξ ) = b φ (0) ∞ Y j =1 m 0  ξ p s − j  . Pr o of. Itera ting (3.2) N times, N ≥ 1, w e hav e (3.7) b φ ( ξ ) = N Y j =1 m 0  ξ p s − j  b φ ( p N ξ ) . T aking in to account that b φ ( ξ ) is contin uo us at the p oint 0 and the fact that | p N ξ | p = p − N | ξ | p → 0 as N → + ∞ for any ξ ∈ Q p , we obtain (3.6).  Prop osition 3.6. If b φ is define d by (3.6) , wher e m 0 is a trigonometric p olynomial (3.3) , m 0 (0) = 1 , then (3.2) holds. F urthermor e, if ξ ∈ Q p such that | ξ | p = p − n , then b φ ( ξ ) = b φ (0) for n ≥ s − 1 , and (3.8) b φ ( ξ ) = b φ (0) s − n − 1 Y j =1 m 0  ξ p s − j  = b φ (0) s − n − 1 Y j =1 m 0  ˜ ξ p s − j  = b φ ( ˜ ξ ) wher e ˜ ξ = ξ n p n + ξ n +1 p n +1 + · · · + ξ s − 2 p s − 2 , ξ n 6 = 0 , for n ≤ s − 2 . Pr o of. Relatio n (3.6) implies b φ ( pξ ) = b φ (0) Q ∞ j =1 m 0  ξ p s − j − 1  and, consequently , (3.2). Let n ≤ s − 2, | ξ | p = p − n , i.e., ξ = ξ n p n + ξ 1 p n +1 + · · · , ξ n 6 = 0. Since χ p  kξ ′ p s − j  = 1, whenever ξ ′ ∈ B − s +1 (0), j ∈ N , k = 0 , 1 , . . . , p s − 1, a nd χ p  kξ p s − j  = 1, whenever j ≥ s − n , we hav e (3.8). It is clear that b φ ( ξ ) = b φ (0) fo r n ≥ s − 1.  p -ADIC REFINABLE FUNCTIONS AND MRA-BASED W A VELETS 7 Corollary 3.7 . The fun ction b φ fr om Pr op osition 3.6 is lo c al ly-c ons t ant. Mor e over, if M ≥ − s + 2 , supp b φ ⊂ B M (0) , then for any k = 0 , 1 , . . . , p M + s − 1 − 1 and for al l x ∈ B s − 1  k p M  we have b φ ( x ) = b φ  k p M  . Prop osition 3.8. L et b φ b e define d by (3.6) , wher e m 0 is a t rigonometric p oly- nomial (3.3) . If m 0 (0) = 1 , m 0  k p s  = 0 for al l k = 1 , . . . , p s − 1 which ar e not divisible by p , then supp b φ ⊂ B 0 (0) , b φ ∈ L 2 ( Q p ) . If, furthermor e,   m 0  k p s    = 1 for al l k = 1 , . . . , p s − 1 which ar e divisible by p , then | b φ ( x ) | = | b φ (0) | for any x ∈ B 0 (0) . Pr o of. By Pr opo sition 3 .6, b φ satisfyes (3 .2) and (3.8). Let us chec k that b φ ( ξ ) = 0 for all ξ such that | ξ | p = p M , M ≥ 1. It fo llows from (3.2) that it suffices to consider only M = 1. Let | ξ | p = p , i.e., ξ = 1 p ξ − 1 + ξ 0 + ξ 1 p + · · · + ξ s − 2 p s − 2 + ξ ′ , where ξ − 1 6 = 0, ξ ′ ∈ B − s +1 (0). In view of (3.8), we hav e b φ ( ξ ) = b φ ( ˜ ξ ) = b φ  k p  , wher e k = ξ − 1 + ξ 0 p + ξ 1 p 2 + · · · + ξ s − 2 p s − 1 , ξ − 1 6 = 0. Note that the first fac tor of the pro duct in (3.8) is m 0  k p s  = 0. Th us b φ ( ξ ) = 0 for all ξ such that | ξ | p = p . The rest sta tements follow from P r opo sitions 3.5, 3 .6.  Due to Theorems 3 .1-3.4, the refinable functions with masks satisfying the hy- po theses of Pr o pos ition 3.8 generate MRAs. Next, we will see that all pro perties of a mask m 0 describ ed in Pro p ositio ns 3.8 a re necessary for the c o rresp onding refin- able function φ to be such that supp b φ ⊂ B 0 (0) and the system { φ ( x − a ) : a ∈ I p } is or tho normal. Theorem 3.9. L et b φ b e define d by (3.6) , wher e m 0 is a trigonometric p olyno- mial (3.3) . If supp b φ ⊂ B 0 (0) and the system { φ ( x − a ) : a ∈ I p } is orthonormal, then   m 0  k p s    = 0 whenever k is not divisible by p , and   m 0  k p s    = 1 whenever k is divisible by p , k = 1 , 2 , . . . , p s − 1 . Pr o of. Let a ∈ I p . Due to the orthono rmality of { φ ( x − a ) : a ∈ I p } , using the Plancherel formula and Corollar y 3.7, w e hav e δ a 0 =  φ ( · ) , φ ( · − a )  = Z Q p φ ( x ) φ ( x − a ) dx = Z B 0 (0) | b φ ( ξ ) | 2 χ p ( aξ ) dξ = p s − 1 − 1 X k =0 Z | ξ − k | p ≤ p − s +1 | b φ ( ξ ) | 2 χ p ( aξ ) dξ = p s − 1 − 1 X k =0 | b φ ( k ) | 2 Z | ξ − k | p ≤ p − s +1 χ p ( aξ ) dξ = p s − 1 − 1 X k =0 | b φ ( k ) | 2 χ p ( ak ) Z | ξ | p ≤ p − s +1 χ p ( aξ ) dξ = 1 p s − 1 Ω( | p s − 1 a | p ) p s − 1 − 1 X k =0 | b φ ( k ) | 2 χ p ( ak ) . Since Ω( | p s − 1 a | p ) 6 = 0 for a ll a ∈ B − s +1 (0), this yields 1 p s − 1 p s − 1 − 1 X k =0 | b φ ( k ) | 2 χ p ( ak ) = δ a 0 , a = 0 , 1 p s − 1 , . . . , p s − 1 − 1 p s − 1 . Consider these eq ualities a s a linear system w ith resp ect to the unknowns z k = | b φ ( k ) | 2 . It is well-known that the system has a unique solution z k = 1, k = 8 A. YU. KHRENNIKO V, V. M. SHELKO VICH, AND M. SKOPINA 0 , 1 , . . . , p s − 1 − 1, i.e., | b φ ( k ) | = 1 for all k = 0 , 1 , . . . , p s − 1 − 1. In particular, it follows that | b φ (0) | 2 = 1, which reduces (3 .6 ) to (3.9) b φ ( ξ ) = m 0  ξ p s − 1  m 0  ξ p s − 2  · · · m 0  ξ  . Let us c heck that   m 0  k p s    = 1 fo r all k divisible by p , k = 1 , 2 , . . . , p s − 1 − 1. This is equiv alent to   m 0  k p N    = 1 whenever N = 1 , 2 , . . . , p s − 1 , k = 1 , 2 , . . . , p N − 1, k is not divis ible by p . W e will prov e this statement by induction on N . F or the inductive base with N = 1, note that 1 = | b φ ( p s − 2 ) | = | m 0 ( 1 p ) | . F or the inductive step, assume tha t   m 0  k p n    = 1 for all n = 1 , 2 , . . . , N , N ≤ s − 2, k = 1 , 2 , . . . , p n − 1, k is not divisible by p . Using (3.9) and 1- perio dicity of m 0 , w e hav e 1 =    b φ  l p s − N − 2     =    m 0  l p N +1  m 0  l p N  · · · m 0  l p N − s +2     =    m 0  l p N +1     , for all l = 1 , 2 , . . . , p N +1 − 1, l is not divisible by p . Now as sume that m 0  k p s  6 = 0 for some k = 1 , 2 , . . . , p s − 1 not divis ible b y p . Since b φ ( k p ) = 0, due to (3.9 ), there exis ts n = 1 , 2 , . . . , s such that m 0  k p s − n  = 0 which contradicts to   m 0  k p s − n    = 1 .  W e have in vestigated refinable functions whose F o urier tr ansform is s uppor ted in the unit disk B 0 (0). Such functions provide a xiom ( a ) of Definition2.1 b ecause of a trivial arg umen t given in Theo rem 3.1. If supp b φ 6⊂ B 0 (0), g enerally sp eaking, the rela tion φ ( · − a ) ∈ V 1 do es not follow from the refinability of φ for all a ∈ I p bec ause I p is not a g r oup. Nevertheless, we obser v ed tha t some such refina ble functions also provide a xiom ( a ). Let p = 2, s = 3, φ b e defined by (3.6), where m 0 is given b y (3.3 ), m 0 (1 / 4) = m 0 (3 / 8) = m 0 (7 / 16) = m 0 (15 / 16 ) = 0 . It is not difficult to see that supp b φ ⊂ B 1 (0), supp b φ 6⊂ B 0 (0). Evidently , axio m ( a ) will b e fulfilled whenever φ  x − k 4  = P 7 r =0 γ kr φ  1 2 x − r 8  , k = 1 , 2 , 3, x ∈ Q 2 , which is equiv alent to b φ ( ξ ) χ 2  kξ 4  = m k  ξ 4  b φ (2 ξ ), k = 1 , 2 , 3 , ξ ∈ Q 2 , where m k ( ξ ) = 1 2 7 P r =0 γ k,r χ 2 ( rξ ). C o m bining this with (3.2) we hav e b φ (8 ξ )( m 0 ( ξ ) χ 2 ( k ξ )) − m k ( ξ )) = 0, k = 1 , 2 , 3, ξ ∈ Q 2 . These equa lities will b e fulfilled fo r any ξ ∈ Q 2 whenever they are fulfilled fo r ξ = l/ 16, l = 0 , 1 , . . . , 15. Desirable p olynomials m k , k = 1 , 2 , 3, exist b ecause we have b φ  1 2  = b φ  3 2  = b φ  5 2  = b φ  9 2  = b φ  11 2  = b φ  13 2  = b φ (1) = b φ (5) = 0. So, w e succeeded with providing axiom ( a ) of Definition2.1, but, unfortunately , such a φ is not a scaling function genera ting MRA b ecause axiom ( e ) is no t v a lid. Moreover, it is p ossible to s how that for an y refinable function whose F ourier transform is in B 1 (0) but not in B 0 (0) the shift system { φ ( x − a ) : a ∈ I p } is not ortho gonal. W e suggest the following c o njecture: it do es not exist c omp actly supp orte d r efinable functions with mut ual ly ortho gonal shifts { φ ( x − a ) : a ∈ I p } whose F ourier tr ansform is not supp orte d in B 0 (0). 4. W a velet bases Now we discuss ho w to find w av e let functions if we have already a p -adic MRA generating by scaling function. Let the refinement equation for φ b e (3.1). W e lo ok for wa velet functions ψ ( ν ) , ν = 1 , . . . , p − 1, in the form ψ ( ν ) ( x ) = P p s − 1 k =0 γ ν k φ  1 p x − p -ADIC REFINABLE FUNCTIONS AND MRA-BASED W A VELETS 9 k p s  , wher e the coefficients γ ν k are chosen such that (4.1) ( ψ ( ν ) , φ ( · − a )) = 0 , ( ψ ( ν ) , ψ ( µ ) ( · − a )) = δ ν µ δ 0 a , ν, µ = 1 , . . . , p − 1 , for any a ∈ I p . It is clear that (4.1) are fulfilled for all a 6 = 0 , 1 p s − 1 , . . . , p s − 1 − 1 p s − 1 . Set B = 1 √ p ( β 0 , . . . , β p s − 1 ) T , G ν = 1 √ p ( γ ν 0 , . . . , γ ν,p s − 1 ) T , ν = 1 , . . . , p − 1, S =       0 0 . . . 0 1 1 0 . . . 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . . 0 0 . . . 1 0       . T o provide (4.1) for a = 0 , 1 p s − 1 , . . . , p s − 1 − 1 p s − 1 we should find vectors G 1 , . . . , G p − 1 so that the matrix U = ( S 0 B , . . . , S p s − 1 − 1 B , S 0 G 1 , . . . , S p s − 1 − 1 G 1 , . . . , S 0 G p − 1 , . . . , S p s − 1 − 1 G p − 1 ) is unitary . Example 4.1 . Let s = 1. According to Pr opo sition 3.8, we set m 0 (0) = 1, m 0  k p  = 0 , k = 1 , . . . , p − 1 and find the mask m 0 ( ξ ) = 1 p P p − 1 k =0 χ p ( k ξ ) (here β k = 1 in (3.3) and B = 1 √ p (1 , . . . , 1) T ). The corres p onding refinement equa- tion (3.1) co inc ide s with the “na tur al” refinement eq uation (1.1), and its solution Ω  | · | p  is a refinable function generating a MRA b ecause o f Theor e ms 3 .1-3.4. T o find wav elets we obser v e that the unitary matrix { 1 √ p e 2 π ik l } k,l =0 ,...,p − 1 may be taken as U T . Computing the wa velet functions cor r espo nding to this matrix U , we derive the formulas which were found in [9]: ψ ( ν ) ( x ) = χ p  ν p x  Ω  | x | p  , ν = 1 , . . . , p − 1 . Example 4.2. Let s = 2, p = 3. According to P r opo sition 3.8, w e set m 0  k 9  = 0 if k is not div isible by 3, and m 0 (0) = 1, m 0  1 3  = m 0  2 3  = − 1. In this case m 0 ( z ) = 3 − 2  − 1 + 2 z + 2 z 2 − z 3 + 2 z 4 + 2 z 5 − z 6 + 2 z 7 + 2 z 8  , z = e 2 π iξ and b φ ( ξ ) =        1 , | ξ | p ≤ 1 3 , − 1 , | ξ − 1 | p ≤ 1 3 , − 1 , | ξ − 2 | p ≤ 1 3 , 0 , | ξ | p ≥ 3 . Thu s, the co rresp onding refinement equation (3 .1) is φ ( x ) = P 8 k =0 β k φ  x 3 − k 9  , where β 0 = − 1 3 , β 1 = β 2 = 2 3 , β 3 = − 1 3 , β 4 = β 5 = 2 3 , β 6 = − 1 3 , β 7 = β 8 = 2 3 , a nd (4.2) φ ( x ) =          − 1 3 , | x | p ≤ 1 3 2 3 , | x − 1 3 | p ≤ 1 2 3 , | x − 2 3 | p ≤ 1 0 , | x | p ≥ 9 = 1 3 Ω( | 3 x | p )  1 − e 2 π i { x } 3 − e 4 π i { x } 3  . So, we have B = 1 3 √ 3 ( − 1 , 2 , 2 , − 1 , 2 , 2 , − 1 , 2 , 2) T , and the vectors B , S B , S 2 B are orthonor mal. W e need to extend these three co lumns to a unitary matr ix U = ( B , S B , S 2 B , G 1 , S G 1 , S 2 G 1 , G 2 , S G 2 , S 2 G 2 ). It is no t difficult to s ee that the vectors G 1 = 1 √ 3 (1 , 0 , 0 , − 1 , 0 , 0 , 0 , 0 , 0 ) T , G 2 = 1 √ 3 (1 , 0 , 0 , 1 , 0 , 0 , − 2 , 0 , 0) T are 10 A. YU. KHRENNIKO V, V. M. SHELKO VICH, AND M. SKOPINA appropria te. The cor resp onding wa velet functions are ψ (1) = q 3 2  φ  x 3  − φ  x 3 − 1 3  , ψ (2) = 1 √ 2  φ  x 3  + φ  x 3 − 1 3  − 2 φ  x 3 − 2 3  . Substituting (4.2), w e obtain ψ (1) ( x ) =              − q 3 2 , | x | p ≤ 1 3 , q 3 2 , | x − 1 | p ≤ 1 3 , 0 , | x − 2 | p ≤ 1 3 , 0 , | x | p ≥ 3; ψ (2) ( x ) =          − 1 √ 2 , | x | p ≤ 1 3 , − 1 √ 2 , | x − 1 | p ≤ 1 3 , √ 2 , | x − 2 | p ≤ 1 3 , 0 , | x | p ≥ 3 . References 1. S. Alb everio, A.Y u. Khrenniko v, V.M. Shelko vic h, Harmonic analysis in t he p -adic L i zorkin sp ac e s: f r actional op er ators, pseudo-differ ential e quations, p -adic wavelets, T aub erian the o- r ems , Journal of F ourier Analysis and Applications, V ol. 12, Issue 4 (2006), 393–425. 2. S. Albeverio, A.Y u. Khrenniko v, V.M. Shelko vich, p -A dic semi-line ar evolutionary pseudo- differ enti al e quations in the Lizorkin sp ac e , Dokl. Ross. Ak ad. Nauk, 415 , no. 3, (2007), 295–299. English transl. in Russian Doklady Mathematics, 76 , no. 1, (2007), 539–543. 3. J.J. Benedetto, and R.L. Benedett o, A wavelet t he ory for lo c al fields and r elate d gr oups , The Journal of Geometric Analysis 3 (2004), 423–456. 4. A. Khrenniko v, Non-ar chime de an analysis: quantum p ar adoxes, dynamic al syste ms and bio- lo gic al mo dels . Kluw er Academic Publ., Dordrech t, 1997. 5. A.Y u. Khrenniko v, and S.V. Kozyrev, Pseudo differ e ntial op e r ators on ultr ametric sp ac es and ultr ametric wavelets , Izve stia Ak ademii Nauk, Seria Math. 69 no. 5 (2005), 133–148. 6. A.Y u. Khrennik ov , V.M. Shelko vich, p -A dic multidimensional wavelets and their applic ation to p -adic pseudo-differ ential op er ators . http://arxiv.org/abs/mat h-ph/0612049 7. A.Y u. Khrennik ov , V.M. Shelk o vich, Non-Haar p -adic wavelets and pseudo-differ ential op- er ators , Dokl. Ross. Ak ad. Nauk, 418 , no. 2, (2008). Engli sh transl. in Russian Doklady Mathematics, (2008) . 8. A.N. Koch ubei, Pseudo-differ enti al e quations and sto chastics over non-ar chime de an fields , Marcel Dekk er. Inc. New Y ork, Basel, 2001. 9. S.V. K ozyrev, Wavelet analysis as a p -adic sp e ctr al analysis , Izv estia Ak ademii Nauk, Seri a Math. 66 no. 2 (2002), 149–158. 10. S.V. Kozyrev, p -Ad ic pseudo differ ential op e r ators and p - adic wavelets , Theor. M ath. Physics 138 , no. 3 (2004), 1–42. 11. I. No viko v, V. Protasso v, and M. Sk opina, Wavelet The ory . Mosco w: Fizmatlit, 2005. 12. V. M. Shelk ovic h, M. Sk opina p -Ad ic Haar multir esolution analysis and pseudo -differ ential op er ators . htt 13. V.S. Vladimirov, I.V. V olovic h and E. I. Zelenov, p -Ad ic analysis and mathematic al physics . W orl d Scien tific, Singapore, 1994. Interna tional Center for Ma them a tical Modelling in Physics and Cogn itive Sciences MSI, V ¨ axj ¨ o University, SE-351 9 5, V ¨ axj ¨ o, Sweden. E-mail addr e ss : andre i.khrenniko v@msi.vxu.se Dep ar tment of Ma them a tics, S t.-Petersburg St a te Architecture and Civil Engineer- ing University, 2 Krasnoarmeiska y a 4, 190005 , S t. Peters burg, Russia. E-mail addr e ss : shelk v@vs1567.sp b.edu Dep ar tment of Applied Ma thema tics and Contr ol Processes, St. Petersburg St a te University, Universitetskii pr.-35, 198504 St. Petersburg, R ussia. E-mail addr e ss : skopi na@MS1167.s pb.edu