A NOTE ON COMPREHENSIVE BACKWARD BIORTHOGONALIZATION

We present a backward biorthogonalization technique for giving an orthogonal projection of a biorthogonal expansion onto a smaller subspace, reducing the dimension of the initial space by dropping d basis functions. We also determine which basis functions should be dropped to minimize the L2 distance between a given function and its projection. This generalizes some recent results of Rebollo-Neira.

biorthogonal expansion onto a subspace, reducing the dimension N of the initial space by dropping d = 1 basis function.In this note, we generalize this method to reduce the space by an arbitrary number d of basis functions, d < N. Proposition 3.4 in [3] indicates which single basis function is to be removed in order to minimize the L 2 distance between a function f and its orthogonal projection into the reduced space.We will also generalize this result in Proposition 7. If more than one basis function is to be dropped, Rebollo-Neira recommends iterating the d = 1 process.We show via Example 8 that in some circumstances iterating the d = 1 process k times leads to results inferior to using Proposition 7 and dropping k = d basis functions simultaneously.
We begin with a Hilbert space H and an N-dimensional subspace V .Assume biorthogonal bases of V given by {x i } N i=1 and {x i } N i=1 such that x i ,x j = δ i j .Now drop d basis elements from each set, without loss of generality the first d elements for notational purposes, and form the reduced subspaces We wish to modify the x i so that the projection from V to V is orthogonal.We next recursively construct the sequence We observe that the set {v i } d i=1 forms an orthogonal basis of V by construction.We then construct the sequence { x i } N i=d+1 by and set U = span{ x i } N i=d+1 .We will see that this formula generalizes the dual modification of [3,Theorem 3.1] for d ≥ 1.Note that each x i is created to be orthogonal to V by subtracting from x i its projection onto V .
Proof.Choose i, j such that j ≤ d < i and use the definition of x i and the orthogonality of {v i }, Thus U and V are orthogonal subspaces of V , and their dimensions add to N.
We next verify that U and V are actually the same space.
Lemma 2. The spaces U and V are orthogonal complements in V , and U = V .
Proof.By (1), we can write v j = j n=1 a n x n for some constants a n , so the original biorthogonality condition x i ,x j = δ i j says that, for j < i, v j ,x i = j n=1 a n x n ,x i = 0. Thus V and V are orthogonal subspaces of V , and their dimensions add to N. By the previous proposition, U = V .
Next we give the desired biorthogonal bases of the reduced subspace V .Proposition 3. The reduced spaces U and V are identical and have biorthogonal bases { x i } N i=d+1 and {x j } N j=d+1 .Proof.Using Lemma 2 and (2), we have for i, j > d ≥ , In order to give an explicit method for determining which basis functions to drop to minimize the residual, we give a formula for the projection operator.
Proof.By Proposition 3, P(w) = w for all w ∈ V and Range(P) = V .From Propositions 1 and 3, V is the null space of P, and Range(P) and V = Null(P) are orthogonal, so P is an orthogonal projection.
David K. Ruch 3 The following generalizes [3, Corollary 3.2] to give the coefficients of P( f ) for the case d ≥ 1.
Theorem 5.If f = N i=1 c i x i , where c i = x i , f , then Proof.We calculate, using (2), The following generalizes [3, Corollary 3.3] for the case d ≥ 1.
is the projection of f onto V using the orthogonal basis {v i }.Thus by Parseval and then Lemma 2, we have Next we generalize [3, Proposition 3.4] for the case d ≥ 1.
Proposition 7. By reindexing the original x i and x i to examine all possible N d combinations of d components dropped from the original basis of V and to minimize the L 2 distance between f and P( f ), choose the set of d basis elements x i that minimizes We now give an example demonstrating that iterating the process k times with d = 1 may give a projection considerably farther from the original f than reducing by k = d basis functions simultaneously.Example 8.For simplicity, we consider a function f (t) in the four-dimensional subspace V with basis functions generated from cardinal spline wavelets.Let B 3 (x) be the standard quadratic cardinal spline supported on [−1,2] and let w(t) be the standard associated wavelet for the Riesz basis of L 2 (R) generated by B 3 (x) as mentioned in [1] or [2].
The function f can be expressed as f (t) = 0.7x 1 (t) + 0.5x 2 (t) + 0.4x 3 (t) + x 4 (t).We wish to drop d = 2 basis elements and obtain the best two-dimensional approximation to f .If we iteratively drop one basis element at a time using Proposition 7 with d = 1, then we remove x 3 and then x 2 leaving projection P( f ) = 0.9x 1 + x 4 as shown in Figure 1(a) with residual error f − P( f ) 2 = 0.82.However, if we simultaneously drop two elements with d = 2, then we instead drop x 1 and x 2 leaving projection P( f ) = 1.1x 3 + x 4 as shown in Figure 1(b) with residual error f − P( f ) 2 = 0.03.As can be seen from these errors and the plots in Figure 1, there is a considerable advantage for t ≥ 1.5 in removing two basis elements together, rather than dropping them iteratively.David K. Ruch 5 When the value of N d is large, the computational expense of choosing the optimal set of basis elements to be dropped can be quite large.Investigation of this issue merits further study.