Gibbs Phenomenon for Wavelets

Transcription

1 APPLIED AND COMPUTATIONAL HAMONIC ANALYSIS 3, (1996) ATICLE NO. 6 Gibbs Phenoenon for Wavelets Susan E. Kelly 1 Departent of Matheatics, University of Wisconsin La Crosse, La Crosse, Wisconsin 5461 Counicated by Wolfgang Dahen eceived March 4, 1994; revised May 25, 1995 When a Fourier series is used to approxiate a function with a jup discontinuity, an overshoot at the discontinuity occurs. This phenoenon was noticed by Michelson [6] and explained by Gibbs [3] in This phenoenon is known as the Gibbs effect. In this paper, possible Gibbs effects will be looked at for wavelet expansions of functions at points with jup discontinuities. Certain conditions on the size of the wavelet kernel will be exained to deterine if a Gibbs effect occurs and what agnitude it is. An if and only if condition for the existence of a Gibbs effect is presented, and this condition is used to prove existence of Gibbs effects for soe copactly supported wavelets. Since wavelets are not translation invariant, effects of a discontinuity will depend on its location. Also, coputer estiates on the sizes of the overshoots and undershoots were coputed for soe copactly supported wavelets with sall support. c 1996 Acadeic Press, Inc. 1. GIBBS PHENOMENON FO FOUIE SEIES To illustrate what is happening in the Gibbs effect, let us exaine the partial sus of a Fourier series. Let g(x) be a periodic, piecewise sooth function with a jup discontinuity at x. For any fixed x 1, not equal to x, the partial sus of g(x) at x 1 approach g(x 1 ). That is, if s n is the partial su of g, then s n(x 1 )g(x 1 ). n However, if x is allowed to approach the discontinuity as the partial sus are taken to the it, an overshoot, or undershoot, ay occur. That is, are possible. This overshoot, or undershoot, is called the Gibbs phenoenon. Proposition 1.1 Let f be a function of bounded variation, 2π-periodic function. At each jup discontinuity x of f, the Fourier series for f will overshoot (undershoot) f(x + ) and undershoot (overshoot) f(x ) if f(x + ) f(x ) is positive (negative). The overshoot and undershoot will be approxiately 9% of the agnitude of the jup f(x + ) f(x ). For further details for the Fourier series, see [8]. 2. GENEAL WAVELET STUCTUE AND COMPACTLY SUPPOTED WAVELETS A general structure, called a ultiresolution analysis, for wavelet bases in L 2 () was described by Mallat [5]. Let V 2 V 1 V V 1 V 2 be a faily of of closed subspaces in L 2 () V {}, V L 2 (), Z Z f V f(2 ) V +1, and there is a φ V such that {φ,n } is an orthonoral basis of V, φ,n (x) 2 /2 φ(2 x n). (1) and n xn x + s n (x n ) g(x + ) Define W such that V +1 V W. Thus, L 2 () W. Then there exists a ψ W such that {ψ,n } is an orthonoral basis of W, and {ψ,n }, is a wavelet basis of L 2 (), 1 E-ail: kelly@ath.uwlax.edu. n s n (x n ) g(x ) xn x ψ,n (x) 2 /2 ψ(2 x n). (2) The function φ is called the scaling function, and ψ is called the other function /96 $12. Copyright c 1996 by Acadeic Press, Inc. All rights of reproduction in any for reserved.

2 GIBBS PHENOMENON 73 Two special sus will be used in this paper. For f L 2 (), the projection ap of L 2 () onto V is : L 2 () V defined by f(x) 1 j ψ j,k ψ j,k (x) k f, f, φ,n φ,n (x), and Π f(x) will be called a dyadic su of f. Also, a general partial su of f will be defined by a projection of L 2 () into V +1, naely, is defined by S lσ f(x) 1 j k S lσ : L 2 () V +1 f, ψ j,k ψ j,k (x) + l f, ψ,σ(k) ψ,σ(k) (x), {σ(k)} l k is a set of l +1distinct integers. Copactly supported wavelets will be used to illustrate the Gibbs effect. The basic structure and properties needed for this paper are provided below. Further details can be found in Daubechies s paper [1]. Based on a decoposition and reconstruction algorith of Mallat which utilizes wavelet s ultiresolution analysis structure, Daubechies extracted the necessary conditions for constructing wavelets fro a sequence of nubers {h(n)} without reference to a ultiresolution analysis. Proposition 2.1. Define (ξ) (1/ 2) h(n)e inξ, the h(n) s satisfy h(n) n ɛ < for soe ɛ>, (3) and k h(n) 2, (4), k h(n)h(n +2k)δ,k. (5) 1,k Also, (ξ) can be written in the following for: (ξ) [ 1 (1 + N )] 2 e iξ F(ξ),N Z+, with F(ξ) f(n)e inξ, f(n) n ɛ < for soe ɛ>, (6) and sup F(ξ) < 2 N 1. (7) ξ Then define g(n) ( 1) n h( n +1), and let ˆφ(ξ) Π j1 (2 j ξ), η l (x) 2 h(n)η l 1 (2x n) φ(x) l η l(x), (8) and η (x) χ [ 1 2,1 2 ) (x). (9) Also, define ψ(x) 2 ( 1) n h( n+1)φ(2x n). Then, the set of φ,n (x) 2 /2 φ(2 x n) defines a ultiresolution analysis, and the {ψ,n (x)}, is the associated wavelet basis. eark. Condition (3) guarantees that ˆφ is well defined; (4) akes φ(x) dx 1; (5) gives the orthonorality to the {φ,n }; and (6) and (7) ensure that φ is continuous. Note that if N 1and F(ξ) 1in the above proposition, then φ(x) χ [,1) (x), and one gets the Haar syste. In this case, the continuity condition (7) fails. Daubechies showed that φ(x) and ψ(x) have copact support if and only if only a finite nuber of the h(n) s in Proposition (2.1) are nonzero. The following definition of Daubechies will be used in the following sections. Definition 2.2. Let N φ and N ψ be the function defined by {h(n)} satisfying the conditions of Theore (2.1), h(n) for n<and n>2n 1,h() and h(2n 1). With these assuptions, the support size of N φ can be deterined. Proposition 2.3. The sallest interval which contains the support of N φ is [, 2N 1]. eark. In this paper, the fact that this is the sallest such interval is needed. A proof of this was not provided in [1], so a proof of this will be provided here. Proof. Fro (8), it follows that the supp N φ [, 2N 1]. It is left to show that this interval is the sallest such interval. Clai: There does not exist an ɛ>such that Nφ [,ɛ]. Assue there does exist such an ɛ, and let ɛ be the largest such ɛ; a largest ust exist since N φ is continuous. By (8), Nφ(x) 2N 1 2 h(n) N φ(2x n), (1) and for x in(ɛ, 1/2+ɛ /2), N φ(x) 2h()φ(2x). (11)

3 74 SUSAN E. KELLY Since h(), (11) iplies that φ(x) for x in(2ɛ, 1+ɛ ). This violates the axiality of ɛ, hence there does not exist such an ɛ and the clai is true. To show that there does not exist an ɛ>such that Nφ [2N 1 ɛ,2n 1], the sae procedure can be carried out, using x 2N 1 in(ɛ, 1/2 +ɛ /2) in (1) and using the fact that h(2n 1). 3. GIBBS PHENOMENON AT THE OIGIN To study the Gibbs effect of functions of bounded variation with a jup discontinuity at zero, it suffices to look at wavelet expansions of the function 1 x, 1 x< f(x) 1 x, <x 1 (12), else since other functions with a jup discontinuity at zero can be written in ters of f plus a function which is continuous at the origin A General Forula for Dyadic Sus We first restrict attention to the study of dyadic su wavelet expansions of f(x) : L p () V f(x) f(y)k (x, y) dy (13) The absolute value of the arguent of the integral is bounded by K (a, u) + K (a, u), which is an integrable function because of the rate of decay of wavelets. Also, the it of this arguent as tends to infinity is χ [, ) (u){k (a, u) K (a, u)}. Thus, applying the Doinated Convergence Theore to (14), one has Π f(2 a) {K (a, u) K (a, u)} du 2 K (a, u) du 1, since integrating the kernel over all reals is one. eark. If instead of choosing a as a fixed nuber, we had a sequence 2 a, there would be two possibilities: If a, then we would end up with Eq. (15) with a, and we would have the sae expression as if we had chosen a.ifa, since 2 a ust tend to zero, a ust tend to infinity slower than 2. Thus, because of the decay conditions of φ, the expression of Eq. (14) would tend to zero, and there would be no overshoot. This explains our choice of x 2 a. The following theore has now been obtained. Theore 3.1. For f defined in (12), a and using the notation of (13) f(2 a)2 K (a, u) du 1. (15) Thus, studying a Gibbs phenoenon reduces to looking at the above integral of the wavelet kernel. Specifically, a Gibbs effect occurs near the origin if and only if K (x, y) φ,n (x)φ,n (y), and φ is the scaling function. Now, 1 f(x) ( 1 y)k (x, y) dy + (1 y)k (x, y) dy 1 χ [,2 ] (u)(1 2 u){k (2 x, u) and (or) K (a, u) du > 1, for soe a > K (a, u) du <, for soe a<. K (2 x, u)} du. Since what is of interest is the region about the origin as tends to infinity, x will be set to 2 a, a isafixed real nuber (see reark below). The above expression then becoes f(2 a) χ [,2 ] (u)(1 2 u){k (a, u) K (a, u)} du. (14) A siilar result was proved independently by Goes and Cortina [4] for a ore general class of expansions. So far, results pertain to all wavelets. We will now look at wavelets which have copact support. It is easy to see fro Theore (3.1) that for the Haar syste, φ(x) χ [,1) (x), there is no Gibbs effect at the origin. The existence of a Gibbs effect near the origin for wavelet expansions of f can be proved for certain copactly supported wavelets using the following result. Theore 3.2. A Gibbs phenoenon for a dyadic wavelet expansion of f(x) generated by the function N φ, defined in Definition (2.2), occurs at the right hand side of

4 GIBBS PHENOMENON 75 f(x) if and only if there exists an a>such that 1 2 Nφ(a +1) Nφ(t)dt + N φ(a +2) Nφ(t)dt + since Nφ(t) dt 1, Nφ(t)dt 1 (19) + N φ(a+()) Nφ(t) dt <. (16) Proof. Fro Theore (3.1), there exists a Gibbs effect at the right hand side of the origin if and only if there exists an a>such that ( ) K (a, u) du > 1 K (a, u) du. (17) Since the support of N φ is contained in [, 2N 1] and the integral of N φ over the reals is one, (17) reduces to finding an a>such that n Nφ(a + n) > n Nφ(t) dt Nφ(a + n) Nφ(t) dt. (18) Subtracting the appropriate ters of (18) yields (16). Thus, the theore is proved. To prove the existence of a Gibbs effect for soe copactly supported wavelets the following technical leas will be needed. Lea 3.3. For N φ, N > 2,h()+h(2N 1) 2. Proof. Assue the Lea is false and h(2n 2) + h(2n 1) 2. That would iply that h 2 (2N 2) + 2h(2N 2)h(2N 1) + h 2 (2N 1)2. Equation (5) iplies that the su of the first and last ter is less than or equal to 1. Thus, h(2n 2)h(2N 1) 1 2, which by the assuption of the proof can be rewritten as [ h(2n 1) 1 2 ] 2. This stateent is only true if h(2n 1)1/ 2. By the assuption, this iplies that h()1/ 2and then by Eq. (5), h(n) would have to be zero for N 2N 3. This last stateent is false since h(), fro the construction requireents for these wavelets. Hence, the assuption in the proof is false and the Lea is proved. Lea 3.4. For N φ, N > 2, there exists a positive integer n<2n 1such that n Nφ(t) dt. Proof. Assue that the lea is false; that is, assue that n Nφ(t) dt for all integers n<2n 1. Then, and k k 1 Nφ(t) dt for k,...,. (2) Using Eq. (19) and (2), and integrating Eq. (1) over [2N 2, 2N 1], one obtains 1 Nφ(t)dt { 2 h(2n 2) + h(2n 1) { 2 Nφ(2t (2N 2)) dt } Nφ(2t (2N 1)) dt h(2n 2) Nφ(t) dt 2 } 2N 1 + h(2n 1) Nφ(t) dt 2N 3 2 {h(2n 2) + h(2n 1)}. 2 Thus, h()+h(2n 1) 2, which is false by Lea (3.3). The assuption ade in the proof is incorrect, and the lea is true. The following result can now be proved. Theore 3.5. If h(2n 1) <, then there exists a Gibbs phenoenon on the right hand side of the origin for the dyadic su wavelet expansion of f(x) generated by N φ. Proof. Letting n be the sallest integer to satisfy Lea (3.4), equation (16) reduces to looking for an a> such that n n+1 Nφ(a + n) Nφ(t) dt + N φ(a +(n+1)) Nφ(t) dt + + N φ(a+()) Nφ(t) dt <. (21) To siplify the above expression, a can be chosen such that 2N 2 a+n<2n 1. Then, Eq. (21) reduces to Nφ(a + n) n Nφ(t) dt <. (22) By the assuption on n, n Nφ(t) dt, and (22) can be verified if two nubers x 1,x 2 [2N 2, 2N 1] can be found such that N φ(x 1 ) and N φ(x 2 ) have opposite signs.

5 76 SUSAN E. KELLY Choose x 1 2N 1.5 such that N φ(x 1 ). Then, Nφ(x 1 ) 2h(2N 1) N φ(2x 1 (2N 1)). Since h(2n 1) <, letting x 2 2x 1 (2N 1), the needed nubers x 1 and x 2 have been found. The theore is now proved. The question now is for what values of N is h(2n 1) negative? To answer this question, we need to look at Daubechies s construction of her copactly supported wavelets, [1]. In Section 4C of her paper, Daubechies defines a specific faily of copactly supported wavelets. The coefficients, h(n) of N φ satisfy the following condition N 1 N 1 2N 1 [1/2(1 + e iξ )] N q(n)e inξ 2 1/2 h(n)e inξ, (23) q(n)e inξ 2 N 1 ( N 1+n n )[ ] n 4 (eiξ + e iξ ). (24) See [1] for details. In Eq. (24), the highest exponential ter on the left hand side is q(n 1)q()e i(n 1)ξ, and on the right hand side, the coefficient on e i(n 1)ξ is negative when N is even. Thus, for N even, q() and q(n 1) have opposite signs. In looking at the lowest and highest exponential ters in (23), we see that h() and h(2n 1) have the sae signs as q() and q(n 1) respectively. Thus, when N is even, h() and h(2n 1) have opposite signs. Since the coefficients can be reversed without effecting the wavelet properties, we can choose h() to be positive, as done in [1]. Thus, h(2n 1) is negative, and we can use Theore (3.5) to get the following result. Corollary 3.6. If N is even, then there exists a Gibbs phenoenon on the right hand side of the origin for the dyadic su wavelet expansion of f(x) generated by N φ. eark. In [1], the coefficients h(n) were listed for the copactly supported wavelets N 2,3,...1 and it can be seen that h() and h(2n 1) do not have opposite signs for the odd N wavelets listed. eark. The proofs for these arguents have only worked for Gibbs effects on the right hand side of the discontinuity. To illustrate why this arguent does not work for the left hand side, we can exaine the wavelets generated by 2 φ and we would need to show that K (a, u) du <, K (a, u) k Z 2φ(a+ k) 2 φ(u+k). Since the support of 2 φ is in [, 3], our work siplifies to showing that 2 φ(a +1) 3 1 2φ(t)dt + 2 φ(a + 2) 3 2 2φ(t) dt <. Using the arguent of the above proof, we would restrict a between 2 and 1 and wish to show 2φ(a +2) 3 2 2φ(t)dt < by showing that the integral is nonzero and that 2 φ(a 1 +2) and 2 φ(a 2 +2) have opposite signs for soe a 1 and a 2 in ( 2, 1). This can not be done with arguents used above. In fact, on the interval (, 1), 2φ does not change sign; this can be seen in Daubechies s paper [1]. In the nuerical estiates of Gibbs effects which follow, Gibbs effects were observed on the left hand side, but the author has been unable to prove the existence. Theore (3.1) has been used to prove the existence of a Gibbs phenoenon. Now, sizes of Gibbs effects for soe of Daubechies copactly supported wavelets will be approxiated by values obtained in FOTAN progras based on Theore (3.1). It is first necessary to deterine a possible Gibbs phenoenon could occur. To do this, Theore (3.1) will be used to deterine a Gibbs effect could not occur for copactly supported wavelets. Let K (a, u) For a>, when is K (a, u) du 1 Nφ(a+n) N φ(u+n). ( ) K (a, u) du true? Since the support of N φ is contained in [, 2N 1], Also, K (a, u) du K (a, u) du n n Nφ(a + n) n Nφ(a + n) (25) Nφ(t) dt. (26) Nφ(t) dt. The two above sus are equal if N φ(a + n) for n 1. This will at least be true if a +n 2N 1, and thus, (25) is satisfied when a. Thus, there is no Gibbs effect, as defined in Theore (3.1), for a for the wavelet expansion generated by N φ. Siilarly, for a<, a Gibbs effect will not occur if K (a, u) du. (27) As seen fro the su of this integral, Eq. (26), Eq. (27) is true if N φ(a + n) for 1 n. This is true if a + n, which iplies that there is no Gibbs effect for a (2N 2). Thus, in searching for a Gibbs effect of dyadic wavelet expansions of f generated by N φ, one only needs to exaine the region {2 a : a ( (2N 2), 2N 2)} as tends to infinity.

6 GIBBS PHENOMENON 77 The next step is to use coputer analysis to approxiate the value of the integral K (a, u) du for values of a in [ (2N 2), 2N 2]. Fro (8), N φ(x) will be approxiated by N η l (x) 2 2N 1 h(n) N η l 1 (2x n) for various values of l, N η (x) χ [ 1/2,1/2](x). Several values of l were used until little change was noted in the output and coputer tie ited going any further. esults fro this coputer analysis are approxiate, but they do give a good idea of the size of the Gibbs effect. For any expansion by copactly supported wavelets, the Gibbs phenoenon on each side of the origin ay differ because of the lack of syetry of these wavelets. This is reflected in the results given in Table 1. It can also be noted that the net Gibbs effect of both sides of the discontinuity sees to be decreasing with higher order wavelets. This agrees with work on periodic spline approxiations done by ichards [7]. He exained higher order splines in approxiating the function 1 1 x< g(x) 1 x<1 (28) in L 2 [ 1, 1]. ichard nuerically calculated the overshoot at g( + ) for splines of degree one through seven, and found that the overshoot was larger than that of the Fourier series. He conjectured that this overshoot approaches the Fourier overshoot as the order of the splines goes to infinity. In a later paper with Foster [2], this conjecture was proved A General Forula for Partial Sus For general wavelet expansions, again, f(x) will be defined as in (12), and S lσ f(x) is defined as its general partial su, S lσ f(x) G lσ (x, y) f(x)+ f(y)g lσ (x, y) dy l ψ,σ(k) (x)ψ,σ(k) (y). (29) k It follows as in the dyadic sus case that { } Slσ f(2 a) 2 K (a, u) du 1 { } + G lσ (a, u) du G lσ (a, u) du. Since ψ, the following result is obtained. Corollary 3.7. For f defined in (12), a, and using the notation of (29), Slσ f(2 a) 2 +2 K (a, u) du 1 G lσ (a, u) du f(2 a)+g lσ (a). The ter G lσ (a) gives the value of the it dependent on which additional ters are added to the dyadic su. The G lσ (a) ter could shift the peak of the Gibbs effect, and could also change the size of it. It is easy to see fro Corollary (3.7) that there is no Gibbs phenoenon for partial su Haar expansions of f. To exaine the addition of ore ters for copactly supported wavelets, let us begin by exaining our new ter. l G lσ (a) 2 Nφ(a + σ(k)) Nψ(t) dt. σ(k) k Since the support of N φ is [ N/2,N], as shown in ([1]), and Nψ(t) dt, σ(k) Nψ(t)dt iplies that N/2 < σ(k) <N, and G lσ (a) iplies that N/2 <a+σ(k)< N. Thus, in looking for values of G lσ (a), the only tie that a nonzero value is obtained is when N N/2 <a< N+N/2, and N/2 <σ(k)<n. For the wavelets generated by 2 φ, σ(k) and 1 are the only values of concern. To illustrate that there is a change in the Gibbs effect, one can exaine the case σ(k) is the only ter added. In this case, in looking at the value of G lσ (a) 2 2 ψ(a) 2ψ(t)dt, when a.99, the right hand Gibbs effect was observed in the nuerical coputations above, it can be seen fro values in [1] that TABLE 1 Approxiate Maxiu Overshoot and Undershoot for Dyadic Su Wavelet Expansions Π f(2 a) Generated by Nφ Left side of origin ight side of origin Nφ Nφ N η 1 a Π f(2 a) a Π f(2 a) 2φ 2η φ 3η φ 4η φ 5η

7 78 SUSAN E. KELLY this ter is nonzero. Thus, either the size of the Gibbs effect is changed, or the value a for which the axiu jup occurs ust be oved. To further look at this case, we shall investigate the nuerical values obtained on the coputer. Fro the above work, when σ(k), the values of 1 <a<2are of interest, and when σ(k) 1, the interval 2 <a<1is what needs to be exained. Thus, the coputer progras for this part will copute values for S lσ f(2 a) over the region 2 <a<2; this includes estiating the values for G lσ (a), σ(k) or 1 or both. esults are given in Table 2. In this case, the data suggests that adding additional ters in the partial su has lessened the Gibbs effect, but that the effect appears to occur in the sae location. 4. GIBBS PHENOMENON AT A GENEAL POINT Because of the translation and dilation procedure used to generate wavelets, wavelets are not translation invariant. With this fact, it is iportant to study Gibbs effects for wavelet expansions of functions with a discontinuity at a general point. We will work with the function (b+1) x, b < x (1 + b) g(x) f(x b) (b 1) x, b 1 x<b (3), else which has a discontinuity at the point b A General Forula for Dyadic Sus A dyadic su expansion for the function g, defined in (3) will now be found. : L p () V (31) g(x) g(y)k (x, y) dy b [(b 1) y]k (x, y) dy b 1 b+1 + [(b +1) y]k (x, y) dy b 2 b + [(b 1) 2 t]k (2 x, t) dt 2 (b 1) 2 (b+1) 2 b [(b +1) 2 t]k (2 x, t) dt. As tends to infinity, points close to b are of interest, so we will let x 2 a+b, a is a fixed real nuber. With soe changes of variables and cobining, one gets 2 g(2 a + b) (1 2 u){k (a +2 b, u +2 b) K (a+2 b, u +2 b)}du. (32) As approaches infinity, the 2 b ter in the arguent of the kernels causes soe difficulty. We want to reove the dependence in the kernels. Note that K (x, y) φ(x+n)φ(y+n) φ(x+n +n)φ(y+n +n), n is any integer. Thus, K (a +2 b, u+2 b)k (a+2 b [[2 b]],u+2 b [[2 b]]), [[x]] is the greatest integer less than or equal to x. Since, as varies, the value of 2 b [[2 b]] ay vary, we will restrict values to a set J such that this expression will be fixed for all in J. Ifbis a rational nuber, there will be a finite nuber of such sets of values; if b is irrational, there will be an infinite nuber of such sets and each will contain one value of. The notation used is the following: b J 2 b [[2 b]], for J. (33) Using the convention of (33), (32) becoes 2 g(2 a + b) (1 2 u){k (a + b J,u+b J ) K (a+b J, u+b J )}du for J. Since the dependence has been taken out of the above kernel s arguent, the it as tends to infinity TABLE 2 Approxiate Maxiu Overshoot and Undershoot for the General Partial Wavelet Sus S lσ f (2 a) Generated by 2 φ for Various Values of l and σ(k) ing lσ (a)( 2 φis Approxiated by 2 η 11 ) Left side of origin ight side of origin l σ(k) a S lσ f(2 a) a S lσ f(2 a) σ(1) σ(1) σ(1) σ(2)

8 GIBBS PHENOMENON 79 TABLE 3 Approxiate Maxiu Overshoot and Undershoot for Dyadic Su Wavelet Expansions Π g(2 a+b) Generated by 2 φ, Where b 1 3 ( 2φ Is Approxiated by 2 η 11 ) Left side of origin ight side of origin J a Π g(2 a + b) a Π g(2 a + b) Even Odd a a Coputer calculated nuber saller, but digits insignificant. can now be taken, as was done in Section 3.1. J g(2 a + b) {K (a+b J,u+b J ) K (a+b J, u+b J )}du b J bj K (a + b J,u)du K (a + b J,u)du. This yields a stateent ore general than that of Theore (3.1). Theore 4.1. If g is defined as in (3), a, and using the notation of (31), J g(2 a + b) 2 K (a+b J,u)du 1, b J 2 b [[2 b]] b J for J. eark. If b 2 k for soe integer k, then Theore (4.1) siplifies to the case b, which is Theore (3.1). Again, using Theore (4.1) it is easy to show that there is no Gibbs phenoenon in this case. The next exaples we look at are the copactly supported wavelets. Again, as done previously, it can be shown that in looking for a Gibbs phenoenon, the region that needs to be checked is (2N 1) <a<2n 1, a is the nuber fro Theore (4.1). Coputer coputations were done for the wavelet expansions of g generated by 2 φ, the point of discontinuity is b 1. In this case, b 3 J 1/3when J { : is even}, and b J 2 when J { : is odd}. The function 3 2φ was approxiated by 2 η 11, and the Gibbs phenoenon was checked for 3 <a<3. esults are given in Table A General Forula for Partial Sus The last type of expansion that will be looked at is general partial sus of a function with a jup discontinuity at any point. This can be done by looking at partial su of g. We will write, S lσ g(x) g(x)+ g(y)g lσ (x, y) dy, (34) G lσ (x, y) l ψ,σ(k) (x)ψ,σ(k) (y). k TABLE 4 Approxiate Maxiu Overshoot and Undershoot for General Partial Wavelet Sus J,, S lσ g(2 a+b) Generated by 2 φ, Where J { : Even} and b 1 3 ( 2φ Is Approxiated by 2 η 11 ) J l σ(n) a S lσ g a a S lσ g a Even Even 1 σ(1) Even 1 σ(1) Even 1 σ(1) Even 2 σ(1) 1 σ(2) Even 2 σ(1) 1 σ(2) Even 2 σ(1) σ(2) Even 3 σ(1) 1 σ(2) σ(3) a Stands for J, S lσ g(2 a + b).

9 8 SUSAN E. KELLY TABLE 5 Approxiate Maxiu Overshoot and Undershoot for General Partial Wavelet Sus J,, S lσ g(2 a+b) Generated by 2 φ, Where J { : Even} and b 1 3 ( 2φ Is Approxiated by 2 η 11 ) J l σ(n) a S lσ g a a S lσ g a Odd b Odd 1 σ(1) b Odd 1 σ(1) b Odd 1 σ(1) b Odd 2 σ(1) 1 σ(2) b Odd 2 σ(1) 1 σ(2) b Odd 2 σ(1) σ(2) b Odd 3 σ(1) 1 σ(2) σ(3) b a Stands for J,, S lσ g(2 a + b). b Last digits ay be insignificant. Siilar to the work done for Π g(2 a + b), and using the notation of Eq. (33), J g(y)g lσ (2 a + b, y) dy 2 b J bj b J G lσ (a + b J,u)du G lσ (a + b J,u)du G lσ (a + b J,u)du. This gives us the following corollary. Corollary 4.2. For g defined in (3), a, b, and using the notation of Eq. (34), J S lσ g(2 a + b) Π g(2 a + b)+gj,b(a), l,σ J GJ,b(a) l,σ 2 G lσ (a + b J,u)du. b J Again it can be verified that there is no Gibbs effect for the general partial Haar su expansion of a function with a jup discontinuity at any point. This shows that in all cases concerning the Haar syste, no Gibbs phenoenon occurs. For the specific case of 2 φ, nonzero GJ,b(a) l,σ ters ay occur for values of σ(k) equal to 1, and 1 and 3 <a< 3. This area for a is the sae area that was checked for a Gibbs phenoenon in the dyadic su case. The coputer progra for this part estiated the G l,σ J,b(a) ter for 3 < a<3,l,1,2or 3, σ(n) taking any one of the values 1,, and 1, and J being the set of odd or even values of. As with the dyadic case, b will be chosen to be 1 3. The results of the coputer estiates are given in Tables 4 and 5. This data sees to suggest that the size and the location of the axiu overshoot and undershoot ay vary with the addition of extra ters in a general partial su. In Section 3.2, the addition of extra ters to the dyadic su seeed to reduce the Gibbs effect. The results in this section see to show that the addition of ters can also increase the Gibbs phenoenon. Another point of interest occurs when J { : odd}. The value for a for the greatest undershoot appears to change. Both of these points appear to be true, but further analysis is needed here. The author hopes to get further details on the behavior of the Gibbs effect with additional ters in future work. To conclude, this paper has given an if and only if condition for a Gibbs phenoenon for wavelets. The existence of Gibbs effects has been deonstrated for soe copactly supported wavelets, and size estiates for Gibbs effects for soe copactly supported wavelets were found. ACKNOWLEDGMENTS The author thanks Dr. ichard ochberg and Dr. Mitchell Taibleson for their help as thesis advisors while working on ost of these results. EFEENCES 1. I. Daubechies, Orthonoral bases of copactly supported wavelets, Co. Pure Appl. Math XLI (1988), J. Foster and F. B. ichards, Gibbs Wilbraha Splines, Constr. Approx., to appear. 3. J. W. Gibbs, Letter to the editor, Nature 59 (1899), 66.

10 GIBBS PHENOMENON S. M. Goes and E. Cortina, Soe results on the convergence of sapling series based on convolution integrals, preprint 1992, SIAM J. Math. Anal., to appear. 5. S. Mallat, Multiresolution approxiations and wavelet orthonoral bases of L 2 (), Trans. Aer. Math. Soc. 315 No. 1, (1989), A. A. Michelson, Letter to the editor, Nature 58 (1898), F. B. ichards, A Gibbs phenoenon for spline functions, J. Approx. Theory, 66 (1991), A. Zygund, Trigonoetric Series, Cabridge Matheatical Library, 2nd ed., Cabridge Univ. Press, Cabridge, UK, 1959.