A Characterization of Wavelet Convergence in Sobolev Spaces

Transcription

1 A Charaterization of Wavelet Convergene in Sobolev Spaes Mark A. Kon 1 oston University Louise Arakelian Raphael Howard University Dediated to Prof. Robert Carroll on the oasion of his 70th birthday. Abstrat We haraterize uniform onvergene rates in Sobolev and loal Sobolev spaes for multiresolution analyses. 1. Introdution and definitions In [KR1] it is shown that onvergene rates of wavelet and multiresolution expansions depend on smoothness of the expanded funtion. Speifially, if is not larger than a ² ³ fixed parameter and / then the error of approximation is 6² ³, with dimension and the number of sales used in the expansion. This result is expeted (see [Wa]) and omparable to Fourier approximation orders. In this paper we study a very different phenomenon whih ours for funtion spaes beyond a ertain degree of smoothness. In these ases the rate of onvergene freezes and fails to improve, no matter what the smoothness of. Suh behaviors have been studied in the ontext of approximation theory. We show here that the smoothness level at whih suh freezing ours depends on the wavelet or saling funtion in a well-defined way, and that it more generally depends on the reproduing kernel of the multiresolution analysis (MRA). This ompletes a haraterization of pointwise onvergene rates in Sobolev spaes for general MRA's begun in [KR1], to inlude Sobolev spaes / with large. In addition (Theorem 8), we extend these results to loal Sobolev spaes whih are related to spaes with uniformly bounded derivatives. See [Ma, Me, Wa] for some results on 3 and 3 onvergene rates of -regular multiresolution expansions. 1 Researh partially supported by the National Siene Foundation, Air Fore Offie of Sientifi Researh, and U.S. Fulbright Commission Researh partially supported by the Air Fore Offie of Sientifi Researh and the National Seurity Agneny 1991 AMS Mathematis Subjet Classifiations: primary 4C15; seondary 40A30 1

2 In [KR1] it is assumed wavelets or saling funtions have suffiient regularity that the regularity of is the limiting fator in onvergene rates. Here we assume has suffiient regularity, and show that limitations on approximation rates then depend on regularity properties of or. We indiate more generally how the interation of the regularities of and of limits onvergene. This omplements results (see [D, Ma, Me]) whih relate onvergene rates for funtions to exat haraterizations of funtion spaes. These results rely on suffiient regularity for, and so do not give information when wavelets have lower regularity relative to. Our onditions on behavior of wavelets near the origin are more preise versions of Strang-Fix type onditions SF]. They an be translated into moment onditions on wavelets in the ase that the moment powers are integers; see also [KR]. For detailed definitions and theory of an MRA we refer to [D]. An MRA is defined as an inreasing sequene of subspaes = ¹ of 3 ² l ³ ( ³ suh that ²%³ = iff ²%³ = b, the intersetion of the spaes is ¹, the losure of their union is all of 3, and = is invariant under translations of integers. It is also generally assumed (though we do not require it here) that there exists a funtion ²%³ (the saling funtion) whose integer translates form an orthogonal basis for =. Let > be the orthogonal omplement of = in = b, i.e., > = = +1 m=, so that = +1 = = l>. From existene of it follows (see, e.g., [D], [Me], [Wo]) that there is a set of basi wavelets (%) ¹ $ (with $ a finite index set) suh that ( %) d ² %³ ( t, t ) form an orthonormal basis for > for fixed, and form an orthonormal basis for 3 ( l d ) as ÁÁ vary. Our results will hold for any wavelet set ¹ related to > whose translations and dilations form an orthonormal basis for 3 ( l d ), regardless of how they are onstruted (see [D], Ch. 10; [Me]; [HW]). It follows from the above definitions that there exist numbers ¹ t suh that the saling equation holds. We define ²%³ ²% ³ (1À1) ²³ (1.) to be the symbol of the MRA À Note it satisfies V² ³ ² ³ V ² ³, where V denotes Fourier transform. Our onvention for the Fourier transform is where h% %À h% V² ³ < ² ³² ³ ² ³ ²%³ % Definitions 1.1: We define 7 and 8 to be the 3 orthogonal projetions onto = and >, respetively, with kernels (when they exist) 7 ( %&, ) and 8 ( %&, ). We define 7 7. l

3 Given 3, ( i) the multiresolution approximation of is the sequene 7¹ ; ( ii) the wavelet expansion of is  ²%³, (1.3a) with the 3 expansion oeffiients of, and denoting onvergene in 3 ; ( iii) the saling expansion of is ²%³b ²%³, (1.3b)  where the, are 3 expansion oeffiients, and ²%³ ²%³À We say suh sums onverge in any given sense (e.g., pointwise, in 3, et.) if the sums are alulated in suh a way that at any stage in the summation there is a uniform bound on the range (largest minus smallest) of values for whih we have only a partial sum over,. Definitions 1.: A multiresolution analysis (MRA) or family of wavelets yields pointwise order of approximation (or pointwise order of onvergene) r in H if for any f H th, the order approximation 7 satisfies ² ³Á ² 14 À ³ as tends to infinity, if (if the left side of (1À4) is in fat infinite for some ). It yields best pointwise order of approximation (or onvergene) in / if s is the largest positive number suh that (1À4) holds for all f H À If the supremum of the numbers for whih (1À4) holds is not attained, then we denote the best pointwise order of onvergene by À The MRA yields optimal pointwise order of approximation (or onvergene) if is the best pointwise order of approximation for suffiiently smooth, i.e. for / for suffiiently large. Thus this order of onvergene is the best possible order in any Sobolev spae. We say if the best order of approximation in / beomes arbitrarily large for large. y onvention best order of approximation means that the supremum in (1À 4) fails to go to 0; thus by our definitions. We remark that our use of the term best approximation order differs from the term best approximation as used, e.g., by Singer [Si]. In addition the word best is used for tehnial reasons assoiated with the formulations of our statements. Speifially, in this paper an expansion has order of approximation if the optimal exponent in (1À 4) is or better, while it has best order of approximation if the optimal exponent is and no larger than. Definitions 1.3: The Sobolev spae / s is defined by 3

4 s s / F 3 ( l ): s O n ( ) O 1 +! < G. The homogeneous Sobolev spae is: s / F 3 ( l ): Á O ( ) O n < G. Note the spaes ontain the same funtions (by virtue of the fat that / is restrited to 3 ) À Only the norms differ, and the seond spae is inomplete as defined (its ompletion ontains non- 3 funtions whih grow at ³À Definitions 1.4: A funtion % ( ) on is radial if depends on % only. A real valued radial funtion is radial dereasing if O (%) O O (&) O whenever % &. A funtion ²%³ is in the radially bounded lass [R] (.f. [GK1,]) if it is absolutely bounded by a 1 positive 3 radial dereasing funtion ( %), i.e., ( %³ ²%³ when O%OO%OÁ with 1 d ( % 1) ( % ) whenever % 1 %, and ( %) 3 ( l ) (note we assume is defined and finite at the origin, so that must be bounded). l d Less general forms of the following two theorems were announed in [KKR, KR]; Theorems 1 through 4 were proved in [KR1] (see These theorems say that under mild assumptions on the MRA (i.e., the saling funtion or wavelets have a radially dereasing 3 majorant), for / ² l ³, the rate of onvergene ² ³ to of the error P 7P has sharp order À We emphasize that the onditions in Theorems 1 through 4 are equivalent. Theorem 1 [KR1]: Given a multiresolution analysis with either () i a saling funtion [R], ( ii) basi wavelets [R] or ( iii) a kernel 7 ²%Á &³ for the basi projetion P satisfying P ²%Á &³ H ²% &³ with H [R], then the following onditions ( a to b) are equivalent for, with the dimension: () a The multiresolution approximation yields pointwise order of approximation s in H s. ( a ) The multiresolution approximation yields best pointwise order of approximation s d/ in H s. ( a ) The multiresolution approximation yields best pointwise order of approximation r in H for all r À () b The projetion 0 P : H L is bounded, where 0is the identityà n h s Theorem is related to the vanishing moments property of the wavelets À Theorem [KR1]: Under the assumptions of Theorem 1, if there exists a family ¹ of basi wavelets orresponding to P n ¹ with (x) [R], then the following onditions are equivalent to those of Theorem 1: 4

5 () For every suh family of basi wavelets and eah, / ( ) For every suh family of basi wavelets and for eah, < s for some (or for all) > 0. ( ) For some suh family of basi wavelets, (1À5a) holds., the dual of / À ( ) d < 1 5 ² À ³ Definition 1.5: We define the spae </ to be the Fourier transforms of funtions in /, with the analogous definition for </ À Theorems 3 and 4 are related to the so-alled Strang-Fix onditions on the saling funtion and the low pass filter. Theorem 3 [KR1]: Under the assumptions of Theorem 1, if there exists a saling funtion orresponding to P n¹, the following onditions are equivalent to those of Theorems 1 and : / (d) For every suh saling funtion, 1 ( ) V < /. (d ) For every saling funtion [R] orresponding to P ¹ n < s ( 1 ( ) ( ) ) d < ² 1À5³ for some (or all) > 0. ( d ) For some saling funtion orresponding to P n ¹, (1À5 ) holds. ( d ) For every saling funtion [R] orresponding to P ¹ < M 0 ( + M) < ² 1À5³ d We define - = 0,1 ¹ to be the set of all -vetors with entries from the pair 0,1 ¹. Theorem 4 [KR1]: If m ² ³ is a symbol of a multiresolution expansion orresponding to a sequene of projetions 7 as in Theorem 1, the following onditions are equivalent to those in Theorems 1-3 for (e) For every symbol m ( ) orresponding to P n ¹, n < 0 (1 ( ) ) < ² 1À³ 5 for some (or all) > 0 (e ) For some symbol m0 () orresponding to P n¹, (1À5 ) holds. ( ) Every (or some) symbol m ( ) orresponding to P ¹ satisfies n 5

6 < for some (or all) > and for every -, where we define where - Á ¹ and denotes the zero vetor in ² 1.6³. For the remainder of the paper, we assume the following: m ²³ d < ² 15 À ³ - - ± ¹, ² 1À 6³ Assumptions: We assume in all of the following theorems that one of the following holds: ²³ The projetion 7 onto = satisfies O7 ²%Á &³O /²% &³ for some / [R] À If a saling funtion exists, ² 17 À ³ ²³ [R]. If a wavelet family exists, ²³ ²%³² ln² b O%O³ [R] for all. Remark: It is shown in [KKR1] that ²³ ²³ and ²³ ²³À This follows from the representations of 7²%Á&³ in terms of sums involving or when they exist. Note that the ondition on in ²³ is somewhat striter than that required for Theorems 1 through 4 above. It is required for existene of a kernel 7²%Á&³ for the projetion 7 satisfying O7 ²%Á &³O /²% &³, with /² h ³ a radial dereasing 3 funtion. This lass of wavelets inludes all r-regular wavelets (see [Me]) for any r 0. The assumptions are also needed for appropriate 3 and a.e. onvergene properties of wavelet expansions [KKR1]. Theorems 1-4 apply only to expansions of funtions in Sobolev spaes / for whih,07 / 3 is bounded (see (b) of Theorem 1). They say nothing about the ase of unbounded,. We show here that for larger (for whih, is unbounded), approximation rates are essentially the same as for the largest for whih, is bounded. Details of the approximation rates, however, depend somewhat deliately on the wavelets or saling funtion. efore giving an overview of our new results in Theorem 5, we refer the reader to formulas ² 1À 5a, b, d ³ as motivation for the following definition. Definitions 1.6: We define for Á 0 6

7 0 ( ) ² ³ -s 6 ( ) 7 d 1 O V -s () sup ² ³O d 1 3 ( ) (1 0( ) ) dà 1 -s In this paper an often-used onsequene of Theorems 1 through 4 is the existene of a least upper bound (best Sobolev parameter), depending only on the MRA, for whih ²³ of Theorem 1 holds. This motivates the following definition. Definition 1.7: The best Sobolev parameter of an MRA is = sup ( 0 7 ): / 3 is bounded ¹. y onvention if the set in the supremum is empty. Some bounds on follow from Theorem 1 above: Proposition 1.8: If the best Sobolev parameter, then, and the set satisfies ' ( 07 ): / 3 is bounded¹ ' = ², µ or ' = ², ). ² 1À8³ Proof: Assume Á so that (reall means there is no positive order of onvergene). Under any one of the assumptions (1.7), the kernel 7²%Á&³ of the projetion onto = is bounded by ²%&³, with [R]. Thus is bounded and in 3, and hene ²% &³ 3 &µ and is bounded in 3 &µ, uniformly in %. Thus for 3, 7 3 À ut for every nonnegative integer, there exist unbounded /. For suh,, ²0 7³ 7 is not in 3, and so, / 3 is unbounded. Similarly,, / 3 is unbounded for. Thus if, i.e., if, is bounded for some, then, must be bounded for an. Therefore, the supremum of for whih, is bounded, must satisfy. To prove statement (1.8) we need only show ' is onneted. This on the other hand follows by the equivalene of ² ³, ² ³, and ²³ of Theorem 1, showing ' ² Á ³ or ' ² Á µà From Theorems 1 and 4 we then have immediately Proposition 1.9: If the best Sobolev parameter, then 7

8 sup 0 ²³ ¹ sup ²³ ¹ sup 3 ²³ ¹. In Theorems 1-4, is important in that all statements hold only if. For approximation rates in /, we prove the following theorem. This summarizes onvergene rates in all / in terms of properties of the projetions 7 or of integrals involving the wavelets or saling funtions. Theorem 5: Given a multiresolution approximation P ¹, ²³ If, there is no positive order of approximation for the MRA 7 ¹ in any /, là If ²³ does not hold then and: () i For, the best pointwise order of approximation in / is ; ( ii) If s, the best pointwise order of approximation in H is s d/; ( iii) If s =, the best pointwise order of approximation in / is / if 0(0) H ( /) if 0 ² 0³ Â ( iv) If, the best pointwise order of approximation in / is 0 6 ³ H if +1/( ) = (1 ) ( Â ( /) otherwise () v In (iii) and (iv) above, 0 ²³ an be replaed by ²³ or by 3 ²³. Another way to say (iv) is that if, then there exists / ² ³ suh that ² ³ sup P7P À This says the onvergene rate annot be improved for funtions belonging even to very smooth Sobolev spaes. Moreover we note that the value b used above in (iv) is not ruial for its statement. Equivalent onditions to those in ( # ) exist in the form 0 b ²³ 6² ³ for any (or all). In terms of the Sobolev order of the expanded funtion and the best Sobolev parameter of the MRA, the following diagram gives rates for an MRA expansion in any Sobolev spae (or loal Sobolev spae; see below). The rates on the boundary region in ( iii) above are not indiated in the diagram. l 8

9 Figure: Approximation rate diagram; see Theorem 3 ²³ for rates on the boundary. The ²³ ²³ in ² ³ indiates that the supersript is present only in some ases. We will show that this diagram applies to expansions of funtions in / and in uniform loal spaes /", when the deay rate! of the saling funtion satisfies! (see Theorem 8³. In addition, on ompat subsets the rates in the diagram apply to funtions in loal Sobolev spaes /, when the wavelet has ompat support. See Definitions 1.10 below for definitions of the spaes /" and /. We now establish several equivalent onditions for failure of onvergene in all Sobolev spaes. Corollary 6: The following ((a) through (e)) are equivalent for the MRA 7 ¹ : (a) 07 / 3 is unbounded for all. (b) This operator is unbounded for ² Á b ³ for some. If there exists a family of wavelets ¹ ( ) For every Á O V ² ³O OO for some. If there exists a saling funtion ( ) For every Á ² ² ³ V ² ³³OO. ( ) In every Sobolev spae / of nonnegative order, the MRA fails to have any positive order of onvergene, i.e., the optimal order of onvergene is À We now state our results for optimal pointwise orders of onvergene in Sobolev spaes À Reall denotes the best Sobolev parameter of 7 ¹, and that optimal order of approximation denotes the highest order of approximation in suffiiently smooth Sobolev spaes. 9

10 Corollary 7: If the best Sobolev parameter, then the wavelet olletion [or saling funtion ] yield optimal pointwise order of approximation if 0 b ²³ 6² ³ [where 0 an be replaed by or L], and ² ³ otherwise. This optimal order is attained for all funtions with smoothness greater than, i.e., for / with Corollary 7 gives best possible pointwise onvergene rates, i.e., onvergene rates for the smoothest possible funtions. In fat this optimal rate in fat is largely independent of how smoothness is defined, i.e., whih partiular sale of spaes we are working with. Suh a statement is possible beause when the smoothness parameter is suffiiently large, the most used sales of smoothness spaes satisfy inlusion relations. For example for large the spae / is ontained in the sup-norm Sobolev spae 3 and in other 3 -type Sobolev spaes. Therefore the optimal rates of onvergene given here are upper bounds for onvergene rates in all 3 spaes, no matter how smooth. What is most important is that suh inlusions work in both diretions for the uniformly loal spaes in Definitions 1.10 below. For example, for suffiiently large, the uniformly loal 3 -Sobolev spae 3," is ontained in spaes in the sale 3 Á" ¹ for any fixed values of and (inluding ³. In addition, for suffiiently large the spae 3 ", ontains Sobolev spaes 3. This inludes 3 and its related smoothness spaes of funtions with bounded derivatives. This observation an then be used as follows. If ²%³ has deay rate! (Def. 1.10) with! (whih holds for many wavelets of interest), then by Theorem 8 below the optimal onvergene rate in all of the sales of uniformly loal spaes 3 Á" (inluding ), is either or ² ³, i.e., the same as in Corollary 7. Now the extension of Corollary 7 to the spaes 3 ", (Corollary 9) an be broadened, by the above argument, to more general sales of smoothness spaes, inluding smoothness spaes based on sup norms. (The aveat, however, is the saling funtion must have suffiiently rapid deay.) With this motivation, we now give the results for uniformly loal Sobolev spaes. Our results will also extend to loal Sobolev spaes with some aveats. / Definitions 1À10: The deay rate of a funtion is sup! O ²%³O O%O! for some ¹À We will assume here our deay rates! are positive unless otherwise speified. The loal Sobolev spae / is / D * ¹, where * is ompatly supported * " funtions. The uniform loal Sobolev spae /" is PP ¹Á where " the uniformly loal norm PhP is defined by ² here ) is the unit ball entered at %): Above, the loal norm is defined by " PP sup PP % Á). % l % 10

11 P P inf P P À ² 19 À ³ Á) % ie, i / )% Similarly, the spae 3 3 ² " ³ 3 ¹ has a loal version 3 Á" defined i analogously to the above with the norm P P in ² 1À 9 ³ replaed by the norm of the Sobolev spae 3À Thus /" onsists of funtions loally in / with loal / norms uniformly bounded. The following results for /" are effetively loal versions of our rates of onvergene results, modulo the spatial uniformity assumptions on funtions in / ". Suh uniformity assumptions also hold, e.g., for 3 Sobolev spaes. We require our working spaes /" to have uniformly bounded loal 3 Sobolev norms rather than 3 Sobolev norms, sine the latter would make our work more diffiult. As shown above, however, most other sales of smoothness spaes based on uniform ( 3 - type) bounds satisfy inlusion relations with the uniform Sobolev spaes /", extending optimal onvergene rate results to these spaes. Additionally, our results of ourse beome entirely loal (valid for loal Sobolev spaes) if wavelets involved have ompat support. Reall from the definitions that approximation order 0 in a spae? means the error, fails to have any positive rate of deay for some?à Theorem 8 ( Loalization) : The multiresolution or wavelet expansion orresponding to a saling funtion [R] has a best pointwise approximation order of at least min ²Á! ³ in /, with the rate of best approximation in / and! the deay rate of. " Corollary 9: If the best Sobolev parameter! (where! is the deay rate of ), then (a) The optimal approximation order in the sale of spaes /" is exatly if 0 b ²³ 6² ³ [where 0 an be replaed by or L], and ² ³ otherwise. (b) The same exat optimal approximation order holds in the sale of uniform loal spaes 3 for fixed, and in partiular also in the sale L and thus 3. Á" Indeed, note that when is an even integer (i.e. the operator ² ³ is loal) the spaes 3 and 3 Á" are idential, sine the first spae always is ontained in the seond, and if 3, then there is a sequene of unit balls ) for whih sup² " ³ is i Á" " unbounded, so that 3 Á"À Thus for eah, the sale 3 Á"¹ is eventually ontained in 3 for suffiiently large, and similarly 3 ¹ is eventually in 3 Á", so the two sales have idential optimal orders of onvergene. This type of inlusion also works for other sales of 3 Sobolev spaes, yielding idential optimal orders of approximation. % ) 11

12 Proposition 10: If is ompatly supported, the best pointwise approximation rate for the expansion of any H on any ompat l is the same as the rate for the global spae / À Examples: To illustrate these results we give appliations to some well-known wavelet approximations. 1. Haar wavelets We alulate the exat approximation order for Haar wavelets. The saling funtion is the harateristi funtion of the unit interval, whose Fourier transform is % V²³ % l ² ³ ² ³ l sin In this ase O V ² ³O l b 6²OO ³ so by Proposition, Á and 19 À 0 ²³ À In addition 0 ²³ ²² ³ O V ² ³O³OO b OO sin 8 9OO OO ² b 6² ³³ 6² ³. OO Thus by Theorem 5, in / Haar expansions have best order of onvergene J Á Á Á Á, with the same orders in the uniform loal Sobolev spaes /" by Theorem 8. y the same theorem, sine is ompatly supported, these orders of onvergene to ²%³ hold uniformly for % in a ompat set, for any ²%³ loally in /. Finally by Corollary 7, the optimal approximation order for suh expansions (i.e., for arbitrarily smooth funtions) is 1. y Corollary 9 this optimal order also holds, for example, in the sale 3 of 3 Sobolev spaes.. Meyer wavelets We now onsider standard Meyer wavelet expansions. The Fourier transform of the saling funtion is [D, page 137] 1

13 ² ³ Á OO V²³ os J ² ³ ² OO³µÁ OO otherwise, where is an appropriately hosen smooth funtion for whih V *. In this ase, so we have order of onvergene in eah Sobolev spae /, Á and onvergene order for À Note this implies that for funtions in the intersetion q/ of all Sobolev spaes, we have onvergene faster than any finite order. The same holds in the uniform loal spaes /" by Theorem 8. Thus the optimal order of onvergene in both these ases is, iàe., onvergene rates have no intrinsi limitations based on the wavelet for very smooth. 3. attle Lemarié wavelets Consider now attle-lemarié wavelets, whih effetively yield spline expansions of a given order. For splines of order 1 the -spline is O%OÁ O%O ²%³ H otherwise. The Fourier transform is V²³²³ sin 8 9. Here is not a saling funtion, sine it does not have orthonormal translates. The orthogonalization trik ([D], setion 5.4) yields a saling funtion with orthogonal translates, whose Fourier transform is Ṽ ²³ l ²³ sin À b os µ The orresponding wavelet has Fourier transform V² ³ l b sin sin ² ³ A 6² ³À b os b os µ C From this it follows from Proposition 1À 9 that À Further, ²³, while ²³ ²³ O V ² ³O OO 6² ³À b OO y Theorem 5, attle-lemarié expansions (and of ourse order one spline expansions, sine the saling spaes are the same) have order of onvergene = 13

14 Á 1 J Á 1 5 Á Á in /À In the uniform loal spaes /" the same approximation rates hold by Theorem 8. Analogous results hold for the higher order versions of these spline wavelets, and the orresponding spline expansions. 4. Daubehies wavelets For standard Daubehies wavelets of order, we onsider the symbol ²³ (see ² 1. ³ ; note the definition of the oeffiients in equation ² 1À1)): Here 0( ) = 1 [(1 + l3) + (3 + l 3) + (3 - l 3) + (1 l 3)e ] 8 b' b' b' µ ', and blá blá lál. Note ²³, while O ² ³O > ² b b b ³b²bb³ os b²b³ os bos? À Therefore O ² ³O e. Sine b, bb, and Á O ² ³O ² b b ³ b > os os? " os os #, and so O ² ³O f À In addition, O ² ³O f but O ² ³O > 36 os os?, so O ² ³O f À Therefore O ² ³O b 6²OO ³ ( ³, so Theorem 4 implies. Thus by Theorem 5, in / the best order of approximation for these Daubehies wavelets is 14

15 Á 1 J Á 1 5 Á 5 Á 5. Similar analyses an of ourse be done for higher order Daubehies expansions. As before, by Theorem 8, the global spae / an be replaed by the uniform loal spae /". We see the optimal order of onvergene for Daubehies wavelets of order is. For the ompatly supported Daubehies order wavelets, these are entirely loal results. Thus for any / o, the above exat approximation rates hold uniformly on any ompat l À Remark: Our results imply that for one dimensional [R] saling funtions with * Fourier transforms (e.g., for ompatly supported ones), optimal orders of onvergene are always integers. The reason is lear from Theorem 5, sine for suh is always a halfinteger (see Def. 1.6 and Prop. 1.9), given the Fourier transform V is always infinitely differentiable at the origin. Speifially OVO m VV is also infinitely differentiable at 0, and so O V ² ³O b 6²OO ³ with an integer and b (note here) À l However must be even sine O V ² ³O always has a maximum of at À In suh ases the Strang-Fix onditions, whih indiate integer onvergene orders and are related to moment and polynomial representation onditions, are entirely equivalent to those above. However, for non-ompatly supported saling funtions supported ases the two theories an diverge, in partiular our results allow for non-integer optimal onvergene rates (see [KR1]). The proofs for the new results 5-9 above are given in setion 7. These hold for multiresolution, saling, and wavelet expansions when they are defined. l. Preliminaries for proofs Let 7 and 8 be the kernels of the 3 projetions onto the spaes = and >, respetively. We inverse Fourier transform and obtain with 7 V ²%Á ³ 7 (%Á &); 8 V < ²%Á ³ < 8 (%Á &) ² 1 À ³ & & V & 7 ²%Á ³ ² ³ 7 ²%Á &³ & ² À ³ and 8²%Á³ V defined similarly. The transforms onverge everywhere and are ontinuous in if O7 ²%Á &³O /²% &³ with / [R] (see Def. 1À4). As usual, we have defined 77 here. The same onlusions hold if ²%³ ln ( + % ) [R] [KKR1]. The error, 0 7 is bounded in 3 À In Fourier spae its kernel is [KR1] 15

16 (, )(%) =, <, where, n has the kernel (in the variable ) / %, ²%Á ³ ( ) 7 V (%, ), ² 3 À ³ We denote,,à Reall the saling property 7 ( %Á&) = 70 ( %, &), ² 4 À ³ whih implies, ²%Á ³, ² %Á ³À ² 5 À ³ Also under our assumptions on the saling funtion, the Fourier kernel,²%á ³ of the remainder operator, is %, ²%Á ³ ² ³ A²%Á ³ V ² ³Á ² 6 À ³ where A²%Á ³ ( %) = ( %b) (À7) is the ak transform of. For later referene, it follows from [KR1], equation (3.1) and its sequel, along with properties of the ak transform, the Poisson summation formula, and the saling funtion, that the ak transform an be written % A²%Á ³ / % ( ) ( ) (.8) t % % ² ³ ( /) ( ), 0 - b t where - Á ¹. In addition, as alulated in KR1], we have from (.6) / %,²%Á ³ ( ) A (%, ) ( ) ² À 9³ / % % ( ) 1 ( ) ( b ) : ( ) ; À t The seond fator an be written in the form 16

17 1 ( ) % ( b ) ( ) ² À 10³ t 1 ( ) ( ) ( ) ( ) 8 9 % ( b). 0 For ompleteness we state a proposition relating approximation orders and operator norms, and two propositions relating operator norms and kernels. Proposition.1 [KR1]: Assume a anah spae (, a normed linear spae ), and a sequene of bounded operators Q ( ). Then the sequene 8 has order of approximation ²³, i.e., + ²0 8 ³+ ) * ²³ for all (, if and only if the operator norm * ²³, where * Á * are onstants (the latter depending on ). s Proposition.: An operator 9: / 3 with kernel 9 ²%Á ³ defined by 9²%³ 9 ²%Á ³² V ³ ² 11 À ³ has operator norm P9P sup O9 ²%Á ³O ² b OO ³ À / 3 % Proposition.3 [KR1]: For l, the operator R: / 3 defined by equation ² À 11 ³ is bounded if and only if the kernel 9 ²%Á ³ satisfies O9²%Á ³OOO *. Replaing the operator 9 by, 0 7 we get: Corollary.4: For the MRA P n ¹ has best pointwise order of approximation s in H if and only if, (%Á ) < / in the variable, uniformly in x, i.e., iff,² %Á ) O is essentially bounded in x. Proof: This follows from equivalene of ² ³ and ²³ of Theorem 1 and Proposition Growth rates of funtions The following results on growth of funtions are required in our proofs of sharpness of the best Sobolev parameter, and our main result, Theorem 5. The proofs are available for referene in an appendix to this paper on the Internet at with the same title as this paper. 17

18 Definitions 3.1: A funtion ²%³ on an open set E is loally bounded if it is bounded on ompat sets. We denote by ) the unit ball of l À Lemma 3.: Given a loally bounded positive funtion (²%³ on ) ¹ and Á (a) We have (²%³ 6²O%O ³ if and only if O(²%³ (²% ³O 6²O%O ³ where if we assume (²³ lim(²%³ À % ² 31 À ³ ( ) If and (²³ lim(²%³ exists, but we do not assume (²³, then in statement % (a), (²%³ 6²O%O ³ is replaed by O(²%³ (²³O 6²O%O ³. ( ) In (a), 6²h³ may be replaed by ²h³ (as % ). Definitions 3.3: Two funtions ( h) and ( h) are equivalent, ( h ) ( h), if there exist positive onstants and suh that for every in their domain, 1 ( ) ( ) ( ). 1 For * Á we define * * ² l ³ to be the lass of positive radial funtions ²O%O³ on l satisfying ² ³ ² ³ for and, i.e., suh that for all, We also define the norm ² ³ * *À ² ³ PP % % ²O%O³ Heneforth we assume all statements involving the order parameter. hold for Theorem 3.4: The following statements are equivalent for any fixed and a positive funtion ²%³ on the unit ball ) of l, with (all integrals are restrited to the unit ball): (a) The integral ( ) For some (or all), O%O and if, ²%³O%O % µà ( ') For some (or all) l, O%O ²%³% 6² ³À ²%³O%O % 6² ³ 18

19 O%O ²%³O%O % 6² ³À ( '') For some (or all), and some (or all) with b, + ²%³O%O ² b O%O³ % 6² ³ and if and b Á then lim ²%³O%O ² b O%O³ % exists and is finiteµà ( ) For any funtion ²O%O³ C suh that it follows that for some (or all) * À % ²O%O³%, ²O%O³ O%O ²%³ % Á Statements in brakets hµ may be inluded or exluded without hanging the equivalenes. In addition, 6²h³ may be replaed by ²h³ simultaneously in all of the above statements exluding (), and the equivalenes of (a)-( ) (i.e. all statements exluding ( )) ontinue to hold. Remark: For ompleteness (though this will not be used in the paper) we remark that the onditions in the above Theorem are also equivalent to the following onditions, listed below: O%O < l /< O%O< (b''') For some (or all), ²%³ O%O % 6² ³À (b'''') For some (or all) Á, ²%³ O%O ² b O%O³ % 6² ³À (d) O%O ²%³ % 6² ³À (e) O%O % ²%³ % 6²³À We now state a orollary whih gives uniformity for Theorem 3.4. Corollary 3.5 : Let q¹ q Q be a family of positive funtions from ) to là The following statements (with all inequalities uniform in ) are equivalent for fixed (note all integrals below are restrited to ) ) and : ²³ The integral ²³ For some (or all), O%O ²%³%. b 19

20 O%O ²%³O%O% q and for, ²%³O%O % for some independent of µà ² ³ For some (or all) l, O%O ²%³O%O% q. ² ³ For some (or all) hoies of with and with b, q²%³o%o ² b O%O³ % b gand if and satisfy b Á then ²%³O%O ² b O%O³ % for some independent of µ. ²³ For any funtion ²O%O³ suh that PP % ²O%O³%, it follows that * ²O%O³ O%O q ²%³% PPÁ for some (or all) * À ² ³ For any funtion ²O%O³ suh that PP, it follows that * ²O%O³ O%O q ²%³ % ²³Á for some (or all) *, where ²³ depends on but not on. The above onstants are all equivalent, i.e., there is a onstant suh that Á, ÃÁ, for any fixed hoie of Á Á and. The braketed statements in ²³ and ² ³ an be inluded or exluded without hanging the equivalenes. Remark: For ompleteness (though this will not be used in this paper), we remark that the onditions of Corollary 3.5 are also equivalent to the following: O%O < q l /< O%O < q (b''') For some (or all), ²%³O%O %. (b'''') For some (or all) Á, ²%³ O%O ² b O%O³ % À O%O (d) ²%³% q. (e) O%O % ²%³ %. q with the onstants through equivalent to through. The next Corollary relates divergene rates of two integrals as b 0

21 i i Corollary 3.6: Let ¹ 8 denote a family of positive funtions ) là The following statements (with all inequalities uniform in ) are equivalent for given ÁÁ lá with, and Á positive. (a) ) i ²%³²bO%O³ %, for some and all À (b) ) i b b ²%³²bO%O³ %, for some and all. Furthermore, if the above assumptions hold exept that is negative, then (a) implies that the left side of (b) is bounded uniformly in À Proof of Corollary 3.6: We first prove ( ) ( ) under the initial assumptions. Note that for fixed, statement ² ³ of Corollary 3.5 is equivalent to itself if "for some" is replaed by "for all". y the symmetry of ²³ and ²³, it suffies to prove ²³ ²³. Thus assume ²³ holds. We define onstants Á Á whih satisfy and b. ² 3À ³ Note that sine it follows. Defining the funtion by ²%³ ²%³O%O i we see that ² ³ of Corollary 3.5 is satisfied for our hoie of Á Á. Let, and now replae by b, and replae by. With these new values of and, we keep and unhanged, so that (3.) is still satisfied. y the equivalene of the "for some" and "for all" versions of statement ² ³ in Corollary 3.5, it follows that for this new value of, ² ³ still holds. However, ² ³ of Corollary 3.5 with the new value of is the same as ²³ of this Corollary, proving ²³ as desired. Now onsider the ase where is negative. We will show that ( ) implies that the left side of ( ) is uniformly bounded. We maintain all of the original assumptions of this Corollary, with the only hange that now is assumed negative instead of positive. Again define ÁÁ so that (3.) holds. Then with these values of ÁÁÁ ²³ above is again equivalent to the unbraketed part of ² ³ of Corollary 3.5. Sine the unbraketed part of ² ³ for one value of implies the braketed part for all values of suh that b, it follows that the braketed part of Corollary 3.5 holds for the new value of. Thus for some. This ompletes the proof. ) b, q²%³o%o ² b O%O³ % 4. Convergene rates and the best Sobolev parameter The main result of this setion, Theorem 4.3, shows that for any wavelet expansion the best pointwise rate of onvergene in / is independent of for, where is the best Sobolev parameter. We reall that, 0 7Á and 8,, b; see ² À 1 ³ and ².3 ³ for definitions of 8 V and,. V 1

22 The next proposition will be used later to establish that the funtion 0²³ (see Def. 1.6) in the statements of our theorems. ²³ an replae Proposition 4.1: If,, and, then for if and only if s 4²³ j, n²%, ³ dj 6² ³ 5²³ V s j Q ²%, ³ d j 6² ³, n where the norms are in xà Proof: Assume 4²³ 6² ³À Using saling properties of the kernels ² 5, À ³ 5²³ 8 V s (%Á) > P O, ( %, ³, ( %Á ) O OO > 6P O, ( %Á ³O OO + O, ( %Á ) OOO 7 OO b = ² 4( ) + 4( ) ³, proving 5²³ 6² ³À Conversely assume 5²³ 6² ³À Then we have -s 5²³ O, ( %Á ), (%Á ) O -s -s : O, ( %Á ) O P P, ( %Á ) ; 6 > > OO 7 s s O, ( %Á ) O P P O, ( %Á) O > OO ² 4² ³ 4²³ where all 3 norms are in %. To show this implies 4²³ 6² ³, define :²³ 4²³ À Then 5, 5²³ 6 :² ³² ³ :²³ 7 ²:² ³ :²³³ À Sine 5²³ 6² ³, b O:² ³ :²³O 6² ³ ² 41 À ³

23 b y ²³ of Lemma 3., if Á we have :²³ 6² ³, and so 4²³ 6² ³ as desired. If on the other hand, in order to apply Lemma 3. (a) we show :²³ lim :²³. To this end, we first bound 4²³ as follows. Define ² ³ P, ²%Á ³P P² ³ A²%Á ³ V² ³P Á where the above norms are in %. It is not diffiult to show (see [KR1] and the remarks for -² ³ before equation (.3.16 ³ there) ³. Thus Note also that -s -s P, ²%, ³P ² ³ d Á% 4²³ > n sup ³ ² ³ d P, ²%, ³P d ² ³, using the equivalene of (b) and (b') (in the ase where 6 is replaed by ) in Theorem 3.4, sine by our assumptions À Thus Á% lim :²³ lim 4²³ ²³ ² ³Á so that :²³ À Now applying Lemma 3. (a), we have by ² 4À 1 ³ in the ase b :²³ 6² ³, so 4²³ 6² ³ in this ase as well, ompleting the proof. that Theorem 4.: For, the MRA P n¹ has pointwise order of onvergene l in H if and only if ² ³ +, (%Á) ( + ) ² 4 À ³ for, uniformly in %. Proof: Assume first ² 4 À ³ holds. Then for / (letting = ) 3

24 ,% ( ) =,(%Á ) e ( ) e (, ) (1 + ), % ( ) (1 + ) =, (% Á ) ( + ) ( ) (1 + ) =, bb independently of %. In the third line we used the saling property ² À 5 ³ for kernels. Conversely assume the MRA has approximation order in H. Then for f / we have, =. Thus by Proposition.1 P, P / 3 6² ³, implying by Proposition. (and the equivalene of the fators ² b ³ and ² b ³ ³ that ess sup, n( %Á ) ²b ³ d % b ess sup,(%á ) ²bO ³ Á % implying ess sup, (%Á ) ² bo ³ % ² ³b as desired. Theorem 4.3: If the best Sobolev parameter, then the best pointwise order of onvergene of the MRA 7 ¹ in / is independent of for À Proof: Assume we have approximation order in /. Then uniformly a.e. in %, referring to the definition of, (%Á) in (.3) and Theorem 4.,, (%Á) ( + ) ++. ²À 4 3) Assume initially that b À We apply Corollary 3.6 with % i ² ³, (%Á ) and,, and. We show the hypotheses and (a) of the orollary are satisfied as follows. First, b. Seond, % i b ² ³² b OO³ O,²%Á ³O ² b OO³. ² 44 À ³ Note that sine b, it follows that ²0 7 ³ / 3 is unbounded. The integral on the left side of (4.4) diverges as, sine by the equivalene of ²³ and ²³ 4

25 in Theorem 1 and Corollary.4, s,²%á) O is unbounded in % for. Thus À Now, for an (to be determined later), we laim that (using ³, uniformly a.e. in %,,%Á ( ) ( + ) ² ( + ) ³ ² ( + ) ³ + + b. ²À 4 5) We first define more preisely. We will require that be suffiiently small that ² b ³ À (4.6) Further, we require that be hosen so that the exponent b in (4.5) is nonzero. efore ontinuing we will show that the part of the integral in (4.5) whih is outside the unit ball remains uniformly bounded a.e. in % and À To this end, note that the exponent in the integral satisfies ² ( + ) ³ Á sine. Thus the integral over the outside of the ball remains uniformly bounded, sine,²%á ³ is uniformly bounded (this follows from the definition of, and from the fat that 7²%Á&³ is in all ases bounded by an 3 onvolution kernel /²%&³ with / radially bounded; see [KR1]). Now we will show that under the above assumptions in fat b. Indeed, Corollary 3.6 implies that if b, then the left side of (4.5) is uniformly bounded a.e. in % as varies in ²Á³ (this inludes the portion of the integral outside the ball by the above remark). However, by (4.6), the left side of (4.5) diverges as sine (as above) by Theorem 1 and Corollary.4, s,²%á) O is unbounded in % for. Hene b. Thus by Corollary 3.6, (4.5) holds, and by Theorem 4. we onlude that we have order ² b ³ of onvergene in / À Thus sine ² b ³, we also have order of onvergene in / (when b ³. Thus if + 1/, and we have order of approximation in H, then we also have 1/ order of approximation in /. This means that if : is the set of orders of onvergene in /, then : Œ : for b À We know also that any order of onvergene in / also applies to / for >, and so : as a set is nondereasing with. Thus : : À Combining the above inlusions, : :, i.e., as a funtion of the set : is periodi with period 1/. Combining this with the fat that : is nondereasing, we onlude the set : must be onstant as a funtion of for >. Thus the valid orders of onvergene are the same in / for >. We remark that above any positive onstant ould have been used in plae of in the term b. Proposition 4.4: If E: / 3 is bounded, then for all, the MRA P n¹ has best order of onvergene / in /. Proof: Our assumption on the boundedness of, implies by Proposition 1.8 that À 5

26 y Theorem 1 if, the MRA has approximation order in /, and hene in / / for. Thus the best approximation order is at least / in /, and so : = Á /], where : denotes the set of approximation orders in / À Note we have used the equivalene of ( ) and ( ') in Theorem 1, whih implies that if we have approximation order - d/ in / this order is the largest possible in /. y Theorem 4.3 : is independent of. We laim for, : = Á/]. First note sine : is inreasing with and : =, /], we have : Œ Á/] for >. We show :, /] as follows. If it were true that and : then Theorem 4.3 would imply that : b, sine b À This would give by the b equivalene of ² ³ and ²³ in Theorem 1 that 0 7 / 3 is bounded. y definition of (Definitions 1.7), this would imply b, giving the desired ontradition. Thus it is impossible that : if > / and. Therefore : = Á /] for > as laimed, ompleting the proof. 5. Conditions for onvergene rates With Corollary.4 as motivation, we define (for 1³ 1( ) = sup, (%, ) d, % where the sup as usual is a.e. The following theorem is the analog of Theorem 5, using the 1 instead of the 0 integrals as riteria for approximation orders. Theorem 5.1: Given an MRA P ¹ with : n ( ) If s the best order of approximation of P ¹ in H is - d/à if 1(0) F ( ) if 1 (0) ( ) If s, the best order of approximation in / is n s À ( ) F d 1 6 ² ³ ( ) otherwise s if +1/( ) = ( ) If s, the best order of approximation in H is r = À Proof: Sine we have by Proposition 1.8. Statement ( ) follows from the definition of and from the equivalene in Theorem 1 of ( ) and ( ). If 1²³, statement (ii) follows from Corollary.4. On the other hand if 1²³, then by Corollary.4, approximation order fails. However, by Proposition 1.8 the set ' satisfies ' ² Á ³À Thus for any with, the operator, / 3 is bounded, and so the MRA has approximation order in / and hene also in / À Therefore, in / we have all orders of approximation less than, whih means the order of approximation is ² ³ À It remains to prove (iii). Sine by Theorem 4.3 we have best order of approximation independent of s for s >, we need only onsider a speifi value of, say = + /, and 6

27 find the best approximation order in / s. y Theorem 4. this is the supremum of values of for whih, (%, ) ( + ) d + b. ² 5À 1³ for. For any where ² 5À 1 ³ holds, we have order of onvergene in / +/. We first show (letting = ) -- +( d/) b, (%Á) ( + ) d ² 5À ³ if 1 () = 6( ), ² 53 À ³ +/ ² reall all order statements in hold for only). Then we show that ² 5À ³ is true for replaed by (i.e. it holds for all < ) if ² 5À 3 ³ fails. We also need to show these hoies of and are best possible (largest possible) in ² 5À ³. To prove the first statement, i.e., equation ² 5À ³, assume ² 5À 3 ³ holds, i.e., sup, (%Á) 1. % With the goal of applying Corollary 3.5, let ²³, % (%Á) OO and. Let b and, so that b and. Applying the equivalene of ²³ and ² ³ in Corollary 3.5, we onlude sup, (%Á ) ² bo ³ 1. % This proves ² 5À ³ and shows that if 1+1/( ) = 6( ) we have approximation order +1/ / in / by Theorem 4.. Note that this order is in fat best by Theorem 1. Indeed, sine this order is the same in all / for, it holds for b for À However by Theorem 1 the approximation order in / in this ase annot be better than b, and so the (onstant) order of approximation in / for annot be better than this for all, and hene annot be better than. Thus for b (and so for all ) the best approximation order in / is as desired. Now we onsider when 1+1/( ) = 6( ) fails to hold, and show we have best b approximation order ( /) in /. y (ii) above the best approximation order is at least ( /) and we must show it annot be better. However sine ² 5À 3 ³ fails by our assumption, it is easy to show by the same arguments as above that ² 5À ³ fails. y Theorem 4. therefore, we fail to have order of onvergene ( /) in / +1/, so the best approximation order must be ( /). Thus by Theorem 4.3 this is the best approximation order in / for all. 6. Preliminaries for the proof of Theorem 5 7

28 We present some tehnial lemmas required in the proofs of the main results, parts ( iv) and ( v) of Theorem 5. Reall that the lass * and the norm PhP are given in Def. 3.3, and that the integral / 1 0 ²³ 6( ) ( ) 7. (6.1) b 1 Lemma 6.1: Let and *. Assume 0 b ²³6² ³ (for ). Then for ²OO³ * with PP, we have (defining ² ³ ²OO³OO ) ²³ < (1 O0( /) ) ² ³ < ²³ O O 0( b/) ² ³ < for - Á ¹ À Proof: Using the assumption 0 b ²³ 6² ³ and the equivalene of ( ) and ( ) in Corollary 3.5, letting ² ³ 1 ( ) V 6 ( ) 7OO and, we have In addition, so OO Thus sine we have 61 ( ) V( ) 7² ³. OO 61 ( ) V ( ) 7²³ 61 ( ) V ( ) ( ) V( ) 7²³, OO 61 ( ) V( ) 7 ² ³ <. ( ) 1 = ( ) V ( ) 1 ( ) ( ) V ( ) 1 7, ² 6À) 0 0 < 0 (1 O ( ) ) ² ³ d < Á (6.3) proving ²³À Additionally, it is known that (e.g., [KR1, equation (.3.9) and Lemma.3.1]), If 0, by ² 64 À ³Á ( + ) = 1  d ² + M³ = ² ³ À (6.4) t d 0 - M 8

29 ( b/) = 1 ( b ) 1 ( ), so by ² 6À 3³ OO yielding ²³ and ompleting the proof. ( b/) ² ³ < Á (6.5) 0 The Fourier kernel,²%á ³ is given in ².6 ³. We also have: Lemma 6.: Let and *. Assume 0 b ²³6² ³ ( À ) Then the kernel,²%á ³ satisfies OO OP,²%Á ³P P,²%Á ³P O² ³ for any ²OO³ * with PP. Here ²³²³OO, and the PhP norm is taken with respet to %. Proof: We have from (.8): A%Á ( ) ( ) ( ) ( ) A (%, ) b ( ) ( ) ( b ) A (%, ), and so A%Á ( ) ( ) A (%Á ) ( /) ²À³ 6 6 ( ( ) 1) ( ) A (%, ) b ( ) ( ) ( b/) A (%, / ) À We now use (6.6) and ²³ and ²³ of Lemma 6.1 (with and as in the Lemma), noting P, ²%Á ³P P, ²%Á ³P, to obtain 9

30 8 O P,²%Á ³P P,²%Á ³PO ² ³ 9 OO % % 8 f A (%Á ) ( ) ² ³ A (%Á ) ( /) ² ³ f ² ³9 < A (%, ) ( ) A (%, /) 8 ( ) ² ³ d9 < = : ( ( ) 1) 0 ( ) A (%, /) < + ( ) ( ) 0 0( b /) A (%, / ) ² ³ ; 0 ( ( ) 1) 8 O 0 ( ) A (%Á ) O ² ³ 9 < r + m ( ) u O 0 ( ) O0( b ) O A (%Á ) O ² ³ s < v Á 0 where we have used that A%Á ( ), V ²³, and ²³ are uniformly bounded in % and À Lemma 6.3: Assume that. There exists a number * suh that for any ²O O³ * ² l ³, the following holds (defining ² ³² ³OO ): For any ³ ³, ( /) ² ³, then O ² ³ ² ³. <1 ²OO³ Proof: Define if OO ²OO³ H, with *, and * to be determined. Let otherwise ² ³ ²OO³OO, so that from the assumption of the Lemma 1/ 1/ 1/ 1/ ( /) ² ³ d. l d ² 67 À ³ Now hoose * so * Á whih is possible sine À Then l O@ ² ² ³O ² ³ O@ ² ³² ² ³² ³ ² ³O l where (reall * ; see Definitions 3.3) 30

31 ²OO³ ² ³ ² ³ ² ³ * ²O O³ We have defined above. ( ) be a positive sequene in 3 whih onverges pointwise ) We may assume onvergene ours suh that, defining.² ³@² ³² ³ and. ² ³@ ² ³² ³ Á. ( ) ² ³. ( ).( ) ² ³. ( ). ² 68 À ³ For example, ² ³, we ould ² ³ ²@ ² ³ ³, with + denoting the greater of the argument and 0. Then O@ ² ³@ ² ³O O@² ³@² ³Ofor all Á l,and O. ² ³ ² ³. ² ² ³² ² ³² ² ³² ² ³² ³.² ³ ² ³.² ³O. b as desired. Now. () ² ³. ( ) P. () P ² ³. ( ) P ² * ³. () P ² 69 À ³ y our hoie of * we have *. y ² 6À 7³ l.² ³ ² ³.² ( ( ) ² ³ d. ld Thus by dominated onvergene and ² 68, À ³ the left side of ² 69 À ³ onverges, so the right side is bounded in. Thus the sequene P. ( ) P is bounded, and sine. ( ) onverges to.( ) pointwise from below, P. ( ) P is finite, proving the lemma. Reall for an inner produt spae = (with inner produt h³, a family of vetors A ¹ = forms a frame if there exist onstants ( and )suh that for all =, (PP O h A O )PP À Lemma 6.4: In a finite dimensional spae, the optimal frame bound is a ontinuous funtion of the frame. Speifially, if the vetors A m ² ³¹ form a frame in l for eah, and if the A ²³ vary ontinuously in, then O ha² ³O O ha² ³O sup Á inf OO l OO l are ontinuous funtions of. (6.10) 31

32 Proof: We write sup O ha² ³O OO l ÁO O sup O ha² ³O. (6.11) l eause suprema of equiontinuous funtion families are ontinuous, it is only neessary that we hek that the family O ha² ³O ¹ S is equiontinuous, where : OO¹. ut for this it suffies to show that for eah, O ha ² ³O ¹ S forms an equiontinuous family of funtions, whih is lear by the Shwartz inequality. Lemma 6.5: Let f( ) and A²%Á ³ be vetor funtions on l, let, and assume ²³ The vetors A²%, ³¹ span l, i.e., there is no nonvanishing vetor suh that A²%Á ³ h a.e. %µ. ²³ A²%Á ³ is ontinuous in. Then if ²³ is a vetor funtion in 3² l ³ suh that it follows that OO O ²³h A²%Á ³O 6² ³ a.e. %µ, (6.1) OO O ² ³O 6² ³À Proof: Assume (6.1). Let % ¹ be suh that A²% Á ³¹ is a basis for l, and suh that the equality in (6.1) holds for %% D. Then by the previous Lemma, for OOsuffiiently small, say OOÁ O² ³O sine A( % Á 0) ¹ forms a basis and thus a frame. Now write (for small ) O ² ³h A²% Á ³O Á O²³O O²³O b O²³O O O O O O O ² ³h A²% Á ³O b O² ³O O O O 6² ³b 6² ³À 7. Proof of Theorem 5 The next theorem establishes the equivalene of the funtions 0 and 3. 3

33 Theorem 7.1: For Á,! b,! 6² ³ ² À ³ ( 1 ( ) ( ) ) 7 1 if and only if 0 -t (1 ( ) ) 6² ³ 7À ² ³ Proof: Let the left side of ² 7.1 ³ be ;²³ and that of ² 7.) be <²³À Reall V V V! ² ³ ² ³ ² ³À Let :²³ ² ² ³ O ² ³O ³OO À Note by fatoring OO the integrand that :²³ 6² ³ iff ; ²³ 6² ³. Assume ² 7.1 ³. Then (6.) gives ²7. ³ (after fatoring the differenes of squares on the right of ² 6À ³ ). Conversely assume ²7. ³ holds, i.e., that <²³ 6² ³À Then note 61 ( ) V ( ) 761 ( ) V ( ) 7 = ( ) (1 ( ) ) V ( ). ² 7À 3³ The fators in the integrand of ² 7. ³ are positive sine they symbol ²³ assumes its maximum of 1 at. Thus, without loss redefining ² ³ O V ² ³O for O O,! 6² ³ ( 1 ( ) ( ) ) ( 1 ( ) ( /) ) OO!! f ( 1 ( ) ( ) ) OO (1 ( ) () )! f OO! O:²³ :² ³OÀ Defining 7²³ so! :²³ we have! OO 6² ³ O ²7 ²³ 7 ² ³³OÁ b! O7 ²³ 7 ² ³O 6² ³. ² 74 À ³! Hene if!, by Lemma 3., 7 ²³ 6² ³, so ; ²³ 6² ³Á as desired. On the other hand, if! Áthen we an again apply Lemma 3. (a) if we an / show 7 ²³ lim7 ²³ À To this end, note that sine (1 - ( ) ( ) ) Á /! (1 ( )! ( ) ) ² ³ O)²³O ² ² ³ O V! sup ² ³O³ ² ³Á OO 0 33

34 where O)²³O denotes the volume in dimensions of the ball of radius. Thus by the seond paragraph of the statement of Theorem 3.4 (note sine! we have! ) relating to replaement of 6by,! ;²³ (1 ( )! ( ) ) ² ³À 1 So by fatoring the integrand below,! :²³ (1 ( ) ( ) ) ² ³ 1!!, and thus 7²³ :²³. Thus we an apply Lemma 3. to (7.4), to again obtain! 7 ²³ 6² ³Á and ; ²³ 6² ³Á ompleting the proof. Note for later referene that / < ( (%)) = ( ). Reall also formula (.6) for,²%á ³. We now give the omplete proof of Theorem 5, the main result of this paper. Proof of Theorem 5: ( o) If it suffies to show that we fail to have any positive order of onvergene in / for. Reall means 07 / 3 is unbounded for all À To begin we laim that in this ase the onvergene rate of the MRA for is independent of À The rest of the proof is similar to that of Theorem 4.3. Assume we have approximation order in /. Then uniformly (as usual a.e.) in %, we have by Theorem 4. bb, (%Á) ( b ) d. ² 7À5) Assume for the moment that b À We apply Corollary 3.6 with % i ² ³, (%Á ),,, and b, with to be determined below À Note that the supremum of the left side of ² 7À 5³ diverges as by Corollary.4, the equivalene of ² ³ and ²³ in Theorem 1 and the fat that, so that À Similarly to the proof of Theorem 4.3, it follows from Corollary 3.6 that for some we have uniformly a.e. in %: ² ² b ³³ ² ² b ³³ + +, (%Á) ( + ) b. ² 7À6) More preisely, we hoose so small that ² b ³, and also so that the exponent b 0. Then as in the previous proof, we onlude that in fat 34