OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES AND MULTIWAVELETS IN HILBERT SPACES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 2, Pages S (99) Article electronically published on September 27, 1999 OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES AND MULTIWAVELETS IN HILBERT SPACES WAI-SHING TANG (Communicated by David R. Larson) Abstract. In this paper, we obtain equivalent conditions relating oblique projections to biorthogonal Riesz bases angles between closed linear subspaces of a Hilbert space. We also prove an extension theorem in the biorthogonal setting, which leads to biorthogonal multiwavelets. 1. Introduction In recent developments in wavelet analysis, many significant results have been obtained using the powerful tool of Fourier transform on the Hilbert space L 2 (R d )of square integrable complex-valued functions on R d. However, it is not clear whether analogous results are still valid in domains, other than L 2 (R d ), where Fourier transform is no longer available. In an effort to get a better understing of the problem, in a series of papers ([5], [9] [6]) we try to extract the essence of the underlying problem by considering wavelets in a general Hilbert space. This paper is another step in this direction. A recent paper [3] by Dai Larson is in the same spirit, though with different emphasis. One of the earliest papers concerning wavelets in Hilbert spaces is that by J. B. Robertson [10], before the birth of modern wavelet theory. While our papers [9] [6] can be viewed as an extension of the results in [10], the present paper is in turn an extension of [9] [6] in the following respect. Theorem 2 in [10] leads to orthonormal wavelets associated with orthonormal multiresolutions, whereas an analogous result, [6, Theorem 2.5] (see also [9, Theorem 3.2]), leads to prewavelets. One of our main results in this paper, namely Theorem 3.6, leads to biorthogonal wavelets associated with biorthogonal multiresolutions. We recall some definitions set up some notation. Throughout this paper, H denotes a complex Hilbert space. A sequence {v n } in H is a Riesz basis for its closed linear span V := span{v n } if there exist positive constants A B such that (1.1) A a n 2 a n v n 2 B a n 2, {a n } l 2 (Z). Two sequences {v n } {ṽ n } in H are biorthogonal if (1.2) v n, ṽ m = δ n,m n, m. Received by the editors March 23, Mathematics Subject Classification. Primary 46C99, 47B99, 46B15. Key words phrases. Riesz basis, biorthogonal system, oblique projection, multiwavelets. 463 c 1999 American Mathematical Society

2 464 WAI-SHING TANG Chapters 1 4 of [12] provide much information on these concepts. If V W are closed linear subspaces of H such that V W = {0} V + W = H, then we write H = V W call this a direct sum. Inthiscase,we can define a map P : H H by (1.3) P (v + w) =v, v V,w W, call P the projection (sometimes oblique projection) of H on V along W. For the special case when W = V, the orthogonal complement of V in H, we shall call P the orthogonal projection of H on V, which is related to the orthogonal direct sum V V. Theorem 2.3, a main result in this paper, gives equivalent conditions relating oblique projections to biorthogonal Riesz bases angles between closed linear subspaces of H. We now give a summary of the contents of this paper. In Section 2, we discuss various relations between oblique projections biorthogonal Riesz bases. In Section 3, in the biorthogonal setting we obtain an extension theorem (Theorem 3.6), which leads to the existence of biorthogonal multiwavelets. The final section describes briefly the construction of biorthogonal wavelets associated with biorthogonal scaling vectors. 2. Oblique projections biorthogonality Let us first state a probably folk result on the vector sum of two closed linear subspaces of a Hilbert space. We leave its proof to the reader. Theorem 2.1. Let V W be closed linear subspaces of a complex Hilbert space H. The following conditions are equivalent: (i) sup{ v, w : v V,w W, v = w =1}<1. (ii) There exists a positive constant C such that v + w 2 C( v 2 + w 2 ), v V,w W. (iii) V + W is closed in H, V W={0}. Moreover, if V W = {0}, X Y are Riesz bases for V W respectively, then (i), (ii) (iii) are each equivalent to: (iv) X Y is a Riesz basis for V + W. The proof of (iii)= (i) in the above theorem is implicit in [4, pp ]. Note that there exist closed linear subspaces V W of a Hilbert space H for which V W = {0}, but V + W is not closed in H (see [7, pp ]). Corollary 2.2. Let U 1 U 2 be closed linear subspaces of H U 1 U 2 = {0}. If U 1 + U 2 is closed, then V 1 + V 2 is closed for any closed linear subspace V i of U i,i=1,2. The expression in condition (i) of Theorem 2.1 is closely related to the concept of the angle θ(v,w) (0 θ(v,w) π 2 ) between two closed linear subspaces V W of a Hilbert space H, which is defined by (2.1) cos(θ(v,w)) := inf v V v =1 P W v, where P W is the orthogonal projection of H on W. (See [1], [4, pp ] [11] for related properties engineering interpretations of θ(v,w).) Itiseasyto

3 OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES 465 see that (2.2) cos 2 (θ(v,w)) = 1 sup v V,w W v = w =1 v, w 2. Note that in general θ(v,w) θ(w, V ) are not necessarily equal, but by (2.2) we always have θ(v,w) =θ(w,v ). If V W = H, then θ(v,w) =θ(w, V ). Theorem 2.3. Let V Ṽ be closed linear subspaces of H. The following conditions are equivalent: (i) V Ṽ = H. (ii) Ṽ V = H. (iii) There exist Riesz bases {v n } {ṽ n } for V Ṽ respectively such that {v n } is biorthogonal to {ṽ n }. (iv) cos(θ(v,ṽ )) > 0 cos(θ(ṽ,v)) > 0. Proof. (iii)= (ii): Suppose that {v n : n Z} {ṽ n : n Z} are Riesz bases for V Ṽ respectively such that (2.3) Then v n, ṽ k = δ n,k, n,k Z. (2.4) V = {f H : f = a n v n, a n 2 < } (2.5) Let g Ṽ V. By (2.5) (2.3), Ṽ = {g H : g = b n ṽ n, b n 2 < }. g = g, v n ṽ n =0. Hence Ṽ V = {0}. Let f H. By the Riesz basis property of {v n } {ṽ n }, (2.6) Pf := f,v n ṽ n is a well-defined vector in Ṽ. By (2.3) (2.6), f Pf,v k =0, k Z. Hence f Pf V,so f=pf +(f Pf) Ṽ +V. Therefore Ṽ + V = H. (ii)= (iii): Suppose that Ṽ V = H. Let{v n } {u n } be dual Riesz bases for V, i.e., {v n } {u n } are both Riesz bases for V v n,u k =δ n,k, n, k. Recall that ([12], p. 185 p. 188) if F : V V is the frame operator defined by (2.7) then (2.8) F (v) = v, v n v n, v V, u n = F 1 (v n ), n.

4 466 WAI-SHING TANG Consider the map P V : H H defined by (2.9) P V f := f,v n u n, f H. Then P V is the orthogonal projection of H on V.LetG:= P V Ṽ be the restriction of P V to Ṽ.Iff Ṽ G(f) =0,thenf Ṽ V ={0}.So G is injective, G(Ṽ )=PV(Ṽ)=P V(Ṽ +V )=P V (H)=V. Hence G is an invertible bounded operator from Ṽ onto V. Define (2.10) ṽ n = G 1 (u n ), n. Then {ṽ n } is a Riesz basis of Ṽ,foreveryn, u n =G(ṽ n )= k ṽ n,v k u k. Hence ṽ n,v k =δ n,k, n, k. Therefore (ii) (iii). Interchanging the roles of V Ṽ,wealsohave(i) (iii). (i)= (iv): Suppose that (i) holds. Since V + Ṽ is closed V Ṽ = {0}, by Theorem 2.1, sup{ v, w : v V,w Ṽ, v = w =1}<1. Hence by (2.2), cos(θ(v,ṽ )) > 0. Since (ii) also holds, interchanging the roles of V Ṽ in the above argument, cos(θ(ṽ,v)) > 0, too. (iv)= (i): Suppose that (iv) holds. Since cos(θ(v,ṽ )) > 0, sup{ v, w : v V,w Ṽ, v = w =1}<1. By Theorem 2.1, V + Ṽ is closed V Ṽ = {0}. Since cos(θ(ṽ,v)) > 0also, by the above argument, Ṽ + V is closed Ṽ V = {0}. Hence V + Ṽ =(V +Ṽ ) =(V Ṽ) =H. Let U =(U 1,..., U d ) be an ordered d-tuple of distinct unitary operators on a Hilbert space H such that U k U j = U j U k,k,j=1,...,d. We shall use the multiindex notation U m = U m1 1 U m d d for m =(m 1,..., m d ) Z d, with the convention that Uj 0 is the identity operator on H, j =1,..., d. We also assume that U m is the identity operator only if m =0. Corollary 2.4. Suppose that one of the conditions in Theorem 2.3 holds for the closed linear subspaces V Ṽ of H. (a) Given a Riesz basis {v n } for V, there exists a Riesz basis {ṽ n } for Ṽ such that {v n } is biorthogonal to {ṽ n }. (b) If {U n w j : n Z d,j =1,...,r} is a Riesz basis for V for some positive integer r, U k (Ṽ) Ṽ, k=1,...,d, then there exist w 1,..., w r in Ṽ such that {U n w j : n Z d,j =1,...,r} is a Riesz basis for Ṽ, it is biorthogonal to {U n w j : n Z d,j =1,...,r}.

5 OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES 467 Proof. We follow the proof of the implication (ii)= (iii) in Theorem 2.3, which gives (a) directly. For (b), suppose now that V has a Riesz basis of the form {U n w j : n Z d,j = 1,...,r}. By (2.7), F commutes with each U k,k=1,...,d.using this observation (2.8), the dual Riesz basis in V is given by {U n z j : n Z d,j =1,...,r}, where z j = F 1 (w j ),j = 1,...,r. Likewise, by (2.9), G commutes with each U k,k=1,...,d. Using (2.10), {U n w j : n Z d,j =1,...,r} is then the desired Riesz basis for Ṽ,where w j =G 1 (z j ),j =1,...,r. Corollary 2.5 ([1, Theorem 3.2 (iii)]). Let T be a unitary operator on H. LetV W be closed linear subspaces of H, let {T n v j : n Z,j =1,...,r} {T n w j : n Z,j =1,...,r} be Riesz bases for V W respectively, for some positive integer r. Then the oblique projection of H on V along W is well defined (i.e., V W = H) if only if cos(θ(v,w)) > 0. Proof. This follows from Theorem 2.3 the result [1, Theorem 3.2(i)] that for such V W,cos(θ(V,W)) = cos(θ(w, V )). Corollary 2.6. Let V,Ṽ,W W be closed linear subspaces of H, let V W W Ṽ. Let {v n}, {ṽ n }, {w n }, { w n } be Riesz bases for V,Ṽ,W W respectively, let {v n } be biorthogonal to {ṽ n }, let {w n } be biorthogonal to { w n }. Then V W = {0},V +W is closed, {v n } {w n } is a Riesz basis for V W. Proof. By Theorem 2.3, V Ṽ = {0}, V +Ṽ =H. In particular, V + Ṽ is closed. Since W Ṽ, we have V W = {0}, by Corollary 2.2, V + W is closed. Hence the desired result follows from Theorem An extension theorem biorthogonal multiwavelets Let V 0, Ṽ0,V 1 Ṽ1 be closed linear subspaces of a Hilbert space H, letv 0 V 1,Ṽ0 Ṽ1, for i =0,1, let V i Ṽi have biorthogonal Riesz bases. By Theorem 2.3, (3.1) V 0 Ṽ 0 V 0 +Ṽ 0 (3.2) Define V 1 Ṽ1 V 1 +Ṽ1 (3.3) W 0 = V 1 Ṽ 0 (3.4) W 0 = Ṽ1 V0 We collect below some properties of W 0 W 0. Proposition 3.1. W 0 W 0 have the following properties: (i) W 0 is the unique closed linear subspace of H satisfying the conditions V 1 = V 0 W 0 W 0 Ṽ0. (ii) W 0 = V 0 + Ṽ 1. (iii) W 0 W 0 = H. (iv) If P, Q R are the projection on V 1 along Ṽ 1, the projection on V 0 along Ṽ0, the projection on W 0 along W 0 respectively, then P = Q + R.

6 468 WAI-SHING TANG Proof. We omit the proof of (i), which follows from some simple arguments using (3.1) (3.3). Since V 0 V 1, by (3.2) Corollary 2.2, V 0 + Ṽ 1 is closed. Hence W 0 =(Ṽ1 V 0 ) =Ṽ 1 +V 0 =V 0 +Ṽ 1. Now W 0 W 0 =(V1 Ṽ 0 ) (V0+Ṽ 1 ).Let x V 1 Ṽ 0 x = v + w for some v in V 0 w in Ṽ 1. Then w is in Ṽ0,v=x w Ṽ 0 V 0 ={0}.Hence x = w V 1 Ṽ 1 = {0}. ThusW 0 W 0 ={0}, W 0 + W 0 = W 0 +(V 0 +Ṽ 1 )=V 1 +Ṽ 1 =H. Finally, (iv) is an easy consequence of the above properties. We recall some notation terminology from [6]. Let U := (U 1,..., U d )bean ordered d-tuple of commuting distinct unitary operators on a Hilbert space H. For a subset S of H, let S denote the closed linear span of S, U Zd (S):={U n s:n Z d,s S}. If V = {v 1,..., v r } W = {w 1,..., w p } are finite subsets of H such that (3.5) { v k,u n w j } n Z d l 2 (Z d ), k =1,...,r, j =1,...,p, then the function Φ V,W defined almost everywhere on R d by ( ) (3.6) Φ V,W (θ):= n Z d v k,u n w j e in θ 1 k r, 1 j p is an r p matrix function with entries in L 2 ([0, 2π) d ). It is easy to verify the following properties: (3.7) (Φ V,W (θ)) =Φ W,V (θ); (3.8) if r = p, then V U Zd (W ) Φ V,W (θ)=0a.e.; (3.9) U n v k,u m w j =δ n,m δ k,j,k,j=1,...,r, n,m Z d Φ V,W (θ)=i r a.e., where I r is the r r identity matrix. A deeper result is the following [6, Theorem 2.1]: Theorem 3.2. If a finite subset V = {v 1,..., v r } of H satisfies (3.10) v k,u n v j 2 <, k,j =1,...,r, n Z d then U Zd (V ) is a Riesz basis for U Zd (V ) if only if there exist positive constants C 1 C 2 such that (3.11) C 1 Φ V,V (θ) C 2 for almost every θ R d.

7 OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES 469 Assume now that Y = {y 1,..., y s } Ỹ = {ỹ 1,..., ỹ s } are finite subsets of H such that U Zd (Y )U Zd (Ỹ) are Riesz bases for U Zd (Y ) U Zd (Ỹ ) respectively, U Zd Zd (Y )U (Ỹ) are biorthogonal, i.e., (3.12) U n y k,u m ỹ j =δ n,m δ k,j, k,j =1,...,s, n,m Z d. In particular, since U Zd Zd (Y )U (Ỹ) are Bessel sequences in H (see [12, pp ]), the entries of the matrix functions Φ V,Y Φ V, Ỹ are L2 -functions for any finite subset V of H. By (3.9) (3.12), (3.13) Φ Y, Ỹ =Φ Ỹ,Y =I s a.e., where I s is the s s identity matrix, we have the biorthogonal expansions (3.14) f = s j=1 n Z d f,u n ỹ j U n y j s (3.15) g = g, U n y j U n ỹ j, j=1 n Z d We also have the following results. f U Z d (Y), g U Z d (Ỹ). Proposition 3.3. Let V = {v 1,..., v r } W = {w 1,..., w p } be finite subsets of H satisfying condition (3.5). (i) (3.16) If V U Zd (Y),then Φ V,W =Φ V, Ỹ Φ W,Y, Φ W,V =Φ W,Y Φ V,Ỹ (3.17) Φ V,Y =Φ V, Ỹ Φ Y,Y. (ii) (3.18) (iii) (3.19) Zd If V U (Ỹ),then Φ V,W =Φ V,Y Φ, Φ W,Ỹ W,V =Φ W, Ỹ Φ V,Y. If V U Zd (Y) condition (3.10) holds, then Φ V,V =Φ V,Y Φ =Φ V,Ỹ V,Ỹ Φ V,Y =Φ V,Ỹ Φ Y,Y Φ. V,Ỹ Proof. By (3.14), for ν Z d,k=1,...,r, l=1,...,p, s v k,u ν w l = v k,u n ỹ j y j,u ν n w l. j=1 n Z d Hence Φ V,W =Φ V, Ỹ Φ Y,W =Φ V, Ỹ Φ W,Y. The second equality in (3.16) follows by taking adjoints. For (3.17), take W = Y. (ii) is proved similarly using (3.15), (iii) follows directly from (i). We should emphasis that our present setting is more general than that of [6]. Thereweassumedthat U Zd Zd (Y) = U (Ỹ),but here we don t. Hence some of the results in [6] may not be valid in this new setting. Using Theorem 3.2 Proposition 3.3 above, the proof of [9, Proposition 3.3] can be carried over verbatim for our present setting to give

8 470 WAI-SHING TANG Proposition 3.4. Let V = {v 1,..., v r } U Zd (Y),wherer s. The following conditions are equivalent: (i) U Zd (V ) is a Riesz basis for U Zd (V ). (ii) There exist positive constants A B such that A Φ V,Y Φ V,Y B a.e. (iii) There exist positive constants Ã B such that Ã Φ V, Ỹ Φ V,Ỹ B We record one more useful result from [6, Theorem 2.4] (see also [9, Theorem 3.1]) that can still be applied here. Theorem 3.5. Let V = {v 1,..., v r } Y = {y 1,..., y s } be finite subsets of H suppose that U Zd (V ) U Zd (Y ) are Riesz bases for U Zd (V ) U Zd (Y ) respectively. If U Zd (V ) U Zd (Y) r = s, then U Zd (V) = U Zd (Y). We now state prove the main result of this section, which is an extension of [6, Theorem 2.5] to the biorthogonal setting. Theorem 3.6. Let X = {x 1,..., x r }, X = { x1,..., x r },Y ={y 1,..., y s } Ỹ = {ỹ 1,..., ỹ s } be finite subsets of H. Let U Zd (X),U Zd ( X),U Zd (Y) U Zd (Ỹ ) be Riesz bases for their closed linear spans V 0, Ṽ0,V 1 Ṽ1 respectively, U Zd (X) biorthogonal to U Zd ( X), U Zd (Y ) biorthogonal to U Zd (Ỹ ), V 0 V 1, Ṽ 0 Ṽ1. Let W 0 = V 1 Ṽ 0 W 0 = Ṽ1 V0. If r<s,then (i) there exists a subset Γ:={z 1,..., z s r } of W 0 such that U Zd (Γ) is a Riesz basis for W 0 U Zd (X Γ) is a Riesz basis for V 1, (ii) there exists a subset Γ :={ z 1,..., z s r } of W 0 such that U Zd ( Γ) is a Riesz basis for W 0 U Zd ( X Γ) is a Riesz basis for Ṽ1, U Zd ( Γ) is biorthogonal to U Zd (Γ). Proof. Since the r s matrix (θ) has at most rank r, choosean(s r) s Φ X,Y matrix Y (θ) such that (3.20) Y (θ) (θ)φ X,Y =0 (3.21) Y (θ)y (θ) = I s r. (The measurability of one such function Y is assured by some stard arguments in measure theory; see, e.g., [8, Lemma 2.4].) Let the (k, j)-entry of Y (θ) be Y k,j (θ), k=1,..., s r, j =1,..., s. By (3.21), s Y k,j (θ) 2 =1, k =1,..., s r, j=1 so all Y k,j are bounded functions in L 2 ([0, 2π) d ). For k =1,..., s r, j =1,..., s, let (3.22) Y k,j (θ) = a k,j (n)e in θ, n Z d a.e.

9 OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES 471 where {a k,j (n)} n Z d l 2 (Z d ), let s (3.23) z k = a k,j (n)u n y j. j=1 n Z d Then z k is in V 1, by (3.22), (3.23) (3.6), (3.24) Y (θ) =Φ Γ, Ỹ (θ) a.e., where Γ := {z 1,..., z s r }. By (3.16) (3.20), Φ Γ, X =Φ Γ, Ỹ Φ X,Y =0 Hence by (3.8), Γ U Zd ( X). By (3.21) (3.24), Φ Γ, Ỹ Φ = I Γ,Ỹ s r a.e. a.e. Hence by Proposition 3.4, U Zd (Γ) is a Riesz basis for U Zd (Γ). By Proposition 3.1, V 0 W 0 = V 1, which is closed in H. Since U Zd (Γ) V 1 Ṽ 0 =W 0, using Corollary 2.2, V 0 + U Zd (Γ) is also closed in H V 0 U Zd (Γ) = {0}. Hence by Theorem 2.1, U Zd (X Γ) is a Riesz basis for V 0 + U Zd (Γ). Since U Zd (X Γ) = V 0 + U Zd (Γ) V 1 = U Zd (Y) #(X Γ) = #(Y )=s, by Theorem 3.5 we have V 0 U Zd (Γ) = V 1 = V 0 W 0. Since U Zd (Γ) W 0,we then actually have U Zd (Γ) = W 0, as proven above, U Zd (X Γ) is a Riesz basis for V 1. This completes the proof of (i). For each k =1,..., d, since U k is unitary, U k (V 1 )=V 1, U k (Ṽ 0 )=Ṽ 0, we have U k ( W 0 )= W 0. By Proposition 3.1, W 0 W 0 = H. Hence the results of (ii) follow from (i), Corollary 2.4 Theorem 2.1. Remark 3.7. Letting V 0 = Ṽ0 V 1 = Ṽ1 in Theorem 3.6, we recover the results of [6, Theorem 2.5]. In this case, W 0 = W 0,V 1 is the orthogonal direct sum of V 0 W Biorthogonal wavelets We follow the notation in Section 3. Let U := (U 1,..., U d ) be an ordered d-tuple of commuting distinct unitary operators on a Hilbert space H. Let X = {x 1,..., x r } X = { x 1,..., x r } be finite subsets of H such that U Zd (X) U Zd ( X) are biorthogonal Riesz bases for their closed linear spans V 0 Ṽ0 respectively. Now suppose that there is a unitary operator D on H such that (4.1) V 0 V 1 := D(V 0 ), Ṽ 0 Ṽ1 := D(Ṽ0) (4.2) U n D = DU Mn, n Z d, where M is a d d matrix with integer entries det(m) > 1. Under this setting, Theorem 3.6 yields analogous results to parts (1) (2) of [6, Theorem

10 472 WAI-SHING TANG 3.1], leading to the existence of biorthogonal multiwavelets in the Hilbert space setting. A special case of this is when H = L 2 (R d ), (U k f)(x) =f(x e k ), where e k =(δ k,j ) j=1,...,d,k=1,...,d, (Df)(x) = det(m) 1 2 f(mx), for x in R d f in L 2 (R d ). We shall not go into this any further. Instead, let us consider the special case when d =1,r =1s=2. Let T D be unitary operators on a Hilbert space H such that (4.3) TD =DT 2. Let T Z ({φ}) T Z ({ φ}) be biorthogonal Riesz bases for their closed linear spans V 0 Ṽ0 respectively such that (4.1) holds. For j =0,1, let φ j := DT j φ φ j := DT j φ. Then T Z ({φ 0,φ 1 })={DT n φ : n Z} T Z ({ φ 0, φ 1 })={DT n φ : n Z} are biorthogonal Riesz bases for V 1 Ṽ1 respectively. Since φ is in V 1 φ is in Ṽ1, (4.4) φ = n Z c n DT n φ (4.5) φ = n Z d n DT n φ for some sequences { c n } {d n } in l 2 (Z). Theorem 4.1. Let (4.6) η := n Z( 1) n+1 d n+1 DT n φ (4.7) η := n Z( 1) n+1 c n+1 DT n φ. Then T Z ({η}) T Z ({ η}) are biorthogonal Riesz bases for W 0 := V 1 Ṽ 0 W 0 := Ṽ1 V0 respectively. We omit the details here. The proof is a minor modification of the arguments in Section 4 of [9], by first establishing analogous results for our present biorthogonal setting. Remark 4.2. If H = L 2 (R), Tf(x)=f(x 1), Df(x) = 2f(2x), for x in R f in L 2 (R), then Theorem 4.1 recovers the explicit formulae of the biorthogonal wavelets, constructed in [2], that are associated with biorthogonal scaling functions.

11 OBLIQUE PROJECTIONS, BIORTHOGONAL RIESZ BASES 473 Acknowledgement This research was supported by the Wavelets Strategic Research Programme, National University of Singapore, under a grant from the National Science Technology Board the Ministry of Education, Republic of Singapore. We would like to thank Professor S. L. Lee for his many helpful suggestions on this paper. After the completion of the first draft of this manuscript, Professor T. N. T. Goodman communicated orally to us that he his co-authors had also obtained parts of Theorem 2.1 of this paper. References [1] A. Aldroubi, Oblique projections in atomic spaces, Proc. Amer. Math. Soc. 124 (1996), MR 96i:42020 [2] A. Cohen, I. Daubechies J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. XLV (1992), MR 93e:42044 [3] X.DaiD.R.Larson,Wering vectors for unitary systems orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no MR 98m:47067 [4] I.C.GohbergM.G.Krein,Introduction to The Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, MR 39:7447 [5] T. N. T. Goodman, S. L. Lee W. S. Tang, Wavelets in wering subspaces, Trans. Amer. Math. Soc. 338 (1993), MR 93j:42017 [6] T. N. T. Goodman, S. L. Lee W. S. Tang, Wavelet bases for a set of commuting unitary operators, Adv. Comput. Math. 1 (1993), MR 94h:42057 [7] P. R. Halmos, Introduction to Hilbert Spaces Spectral Multiplicity, Chelsea, New York, MR 13:563a [8] R. Q. JiaZ. W. Shen, Multiresolution wavelets, Proc. Edinburgh Math. Soc. 37 (1994), MR 95h:42035 [9] S. L. Lee, H. H. Tan W. S. Tang, Wavelet bases for a unitary operator, Proc. Edinburgh Math. Soc. 38 (1995), MR 96g:42019 [10] J. B. Robertson, On wering subspaces for unitary operators, Proc. Amer. Math. Soc. 16 (1965), MR 30:5165 [11] M. Unser A. Aldroubi, A general sampling theory for non-ideal acquisition devices, IEEE Trans. on Signal Processing 42 (1994), [12] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, MR 81m:42027 Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, , Republic of Singapore address: mattws@math.nus.edu.sg

On Riesz-Fischer sequences and lower frame bounds

On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition