Splitting Methods for SU(N) Loop Approximation

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1 Splitting Methods for SUN Loop Approximation arxiv:math/061645v1 [math.na] 1 Dec 006 Peter Oswald International University Bremen p.oswald@iu-bremen.de Abstract Tatiana Shingel DAMTP, Cambridge University t.shingel@damtp.cam.ac.uk The problem of finding the correct asymptotic rate of approximation by polynomial loops in dependence of the smoothness of the elements of a loop group seems not well-understood in general. For matrix Lie groups such as SUN, it can be viewed as a problem of nonlinearly constrained trigonometric approximation. Motivated by applications to optical FIR filter design and control, we present some initial results for the case of SUN-loops, N. In particular, using representations via the exponential map and ideas from splitting methods, we prove that the best approximation of an SUN-loop belonging to a Hölder-Zygmund class Lip α, α > 1/, by a polynomial SUN- loop of degree n is of the order On α/1+α as n. Although this approximation rate is not considered final and can be improved in special cases, to our knowledge it is the first general, nontrivial result of this type. 1 Introduction The study of classes of periodic functions with values in a Lie group G is of theoretical importance as a simple example of infinite-dimensional Lie groups [8], but also occurs in a more practical context. For example, polarization mode dispersion in fiber-optical communication systems is studied using SU-valued functions [7], lossless multi-port communication involves para-unitary transfer matrices in other words, SUN-valued functions [14, 16], the theory of orthogonal wavelet constructions involves polyphase symbol representations that are SUN-valued Laurent polynomials [1, 11]. In all these areas, approximation problems by trigonometric polynomials with values in these matrix Lie groups arise in a natural way - they are, e.g., the key to constructing and operating optical FIR filter architectures for polarization mode dispersion compensation [7]. The classical Jackson-Bernstein theory of approximation by trigonometric polynomials gives a quantitative answer to the connection between approximation rates and smoothness properties of periodic functions: A function f belongs to the Hölder- Zygmund class Lip α T IC, α > 0, if and only if its best approximations E n f Work done while at IU Bremen 1

2 by complex trigonometric polynomials of degree n satisfy E n f = On α as n. After searching in the literature, we did not find a similar result on quantitative approximation for general loop groups. The only reference is to [8] where the density of the subgroup of polynomial loops in the loop group C T G is proved for any compact semi-simple Lie group G. In particular, the polynomial loops are dense in C T SUN for all N, see also [4, 5]. In the present note we provide a Jackson-type estimate for SUN loops. Theorem 1 Let Ut Lip α T SUN, α > 1/. Then there exists a sequence of polynomial loops U n t Π n T SUN of degree n such that U U n C C α,n,u n + 1 α/1+α, n 0. The approach for establishing this result is straightforward. By suitable factorization, we reduce the problem to studying the special case N = and Ut = e At, where At Lip α T su. Next we approximate At componentwise by linear methods, and then use the splitting method for the exponential map to obtain a polynomial SU-valued loop. The proof is carried out in detail in Section 3, and uses preliminary facts collected in a separate Section. We do not consider the approximation rate obtained in Theorem 1 final in any respect. Rather we think that by publishing a result of possibly only temporary value we will attract further attention to a widely open and challenging area of nonlinear approximation: constructive approximation of manifold-valued functions. This field is currently fueled by several communities symplectic integration of dynamical systems [3, 6], manifold subdivision [9, 15], etc. where mostly local approximation schemes are developed. We add another angle by focusing on traditional polynomial approximations here for periodic functions, i.e., loops. Note that the approximation problem for SU-valued functions is a partial case d = 4 of the problem of approximating loops on the unit sphere S d 1 IR d by trigonometric loops. This is another simple test case of manifold-valued periodic functions for which we do not know any quantitative approximation results for general d 3 with the exception of some partial results for d = 4 following from the present paper. Definitions and auxiliary results Throughout this paper, we consider various loop spaces X T M, where M IR d or M IC d is some manifold, and the topology is induced from the coordinate spaces X T IR d via a topology on M. In particular, if M = G is one of the complex matrix Lie groups of dimension N, we have G IC N, and will use the topology induced by the Frobenius norm F i.e., by the Euclidean norm in IC N or the spectral norm i.e., the operator norm for linear maps induced by the Euclidean norm in IC N. We will drop the subscript in the norm whenever it does not matter which one we use. The Lie algebra associated with a Lie group G will be denoted by g. See [10] for the basic facts on Lie groups and algebras used below.

3 We define the class of continuous loops Ut C T G by requiring that Ut : T G is continuous, and set dist C U 1, U := U 1 U C := max t T U 1t U t. Similarly, the function classes C k T G are introduced. The Hölder-Zygmund classes Lip α T G C T G, α > 0, are defined by the finiteness of the semi-norm { U Lipα := sup h>0 h α U + h U C, 0 < α < 1, sup h>0 h 1 U + h U + U h C, α = 1, and by recursion for α > 1 by requiring Ut C k T G and setting U Lipα := U k Lipα k, where k is the largest integer k < α. Obviously, C k T G Lip α T G. For further use we introduce the notation U Lipα := U C + U Lipα. We note that for the matrix groups considered in this paper, the above introduced Hölder-Zygmund classes of loops form algebras, i.e., the Lip α property is preserved under multiplication. This fact will be used below without further mentioning. We are mostly interested in G-valued trigonometric polynomials also called polynomial loops in G, and their g-valued cousins. I.e., we specifically consider loops P n t : T G of degree n whose entries belong to Π n T IC := {p n t = c k z k }, z e it. The nonlinear set of all these P n t is denoted by Π n T G. An analogous definition holds for Π n T g, which is obviously a linear set. Most of the paper deals with SU and su loops. The following simple result about Π n T su will be used below. Lemma 1 The space Π n T su of polynomial loops of degree n in su possesses a basis B n over IR given by the 3n + 1 elements 0 z k 0 iz k B 1,k t := z k, B 0 3,k t := iz k, k = 0,..., n, 1 0 k n 0 iz k, B 4,k t := iz k 0 0 z k B,k t := z k 0 i coskt sin kt B 5,k t := sin kt i cos kt, k = 1,..., n,, k = 0,...,n, 3 3

4 and B 6,k t := i sin kt coskt coskt i sin kt, k = 1,..., n, 4 with the following properties: B l,k t F = B l,k t =, and e cb l,kt = cosc I + sin c B l,k t Π k T SU 5 for all c IR, l = 1,...,6, and all k. Moreover, J e c jb l,kj t Π n T SU, l = 1,..., 6, 6 for any product of this form with 0 k 1 < k <... < k J n. Proof. Evidently, since A su is equivalent to A = A and tra = 0, we can always write i β γ + i δ A =, β, γ, δ IR. γ + i δ i β Moreover, A = φ I and one easily verifies by definition of the matrix exponential that e A = cosφ I + sin φ φ A, φ := β + γ + δ = A F. 7 For all three loop types, we have φt 1 and 7 implies for real c e cb l,kt = cosc I + sin c B l,k t. The properties B l,k t = 1 resp. B l,k t F = are obvious from 1-4. Due to 5, the property 6 follows if we can show that J B l,kj t, 0 < k 1 <... < k J n, J > 0, is a matrix polynomial of degree n for each l = 1,...,6. Since 0 ρ 1 z k 0 ρ z k ρ1 ρ ρ 1z k 0 ρ z k = k 0 zk 0 0 ρ 1ρ z k k, setting ρ 1 = ρ = 1 we obtain for even J = J J J B 1,kj t = B 1,kj 1 tb 1,kj t = 1 J z m 0 0 z m, where 0 < J m = J k j k j 1 k J k 1 < n. For odd J = J + 1, we obtain a similar result is since J B 1,kj t = 1 J z m 0 0 z m B 1,kJ +1 t = 1 J 0 z m+k J z m k J 0 4

5 leads to 0 m k J n. I.e., in either case the result is a matrix polynomial of degree n. This settles the case l = 1. The case l = is quite similar just replace m by m. Repeating the argument with ρ 1 = ρ = i gives the cases l = 3, 4. To treat l = 5, 6, observe that B 5,k t = UB 1,k tu 1, k 0, B 6,k t = UB 3,k tu 1, k > 0, where This, e.g., implies U = SU. J J B 5,kj t = U B 1,kj tu 1 k j > 0, and the reduction to the previous cases is achieved, similarly for products of B 6,kj, k j > 0. Finally, the basis property follows from observing that an arbitrary loop A n t Π n T su can be written in the form iβ A n t = n t γ n t + iδ n t = γt + iδ n t iβn t βn t β n t iβ n t iβ n t + 0 p n t p n t 0 where β n t = n n a k coskt + b k sin kt, βn t = k=0 n b k coskt a k sin kt are a real-valued trigonometric polynomial β n t associated with the diagonal entries of A n t, similarly γ n t, δn are the real-valued trigonometric polynomials corresponding to the off-diagonal entries and its conjugate β n t, and p n t = c k z k a k + i b kz k, c k IC, a k, b k IR. k n k n is some complex-valued trigonometric polynomial of degree n. Thus, by setting c 1,k = a k, c 3,k = b k, c 5,k = a k for k = 0,...,n, and c,k = a k, c 4,k = b k, c 6,k = b k for k = 1,...,n, we obtain 3 n n 6 A n t = c l 1,kB l 1,kt + c l,kb l,kt A n,l t 8 l =1 k=0 with c l,k IR. Since A n t was arbitrarily chosen from Π n T su, and the number of elements in B n coincides with the dimension of Π n T su, this proves the basis property. 5

6 We note that, by construction, the entries of the terms A n,l t in the decomposition 8 can be obtained from the entries of A n t by applying elementary operations and conjugation map f f. E.g. for l = 5, 6 this statement follows from separating even and odd parts of β n t, β n t + β n t = n a k coskt, k=0 β n t β n t = n b k sin kt. Similarly, for l = 1,..., 4 we observe that as a function of t, p n t = γ n t β n t+ iδ n t, and use conjugation to split into k 0 and k < 0 parts, p n t i p n t = c n 0 + c k z k, k=0 p n t + i p n t = c 0 + n in combination with separating c k into real and imaginary parts: p n t + p n t The lemma is established. = a k zk, k n p n t p n t i k= n = b k zk. k n c k z k, The next two lemmas follow by applying classical results from trigonometric approximation theory for Lip α classes, see [] or [17]. Lemma Let At Lip α T su, α > 0. Then there exists A n t Π n T su such that A A n C C α n + 1 α A Lipα, n 0. 9 and A n Lipα C α A Lipα, n Lemma 3 Let ft k Z c kz k Lip α T IC, α > 0. Then and f S n f C C α lnn + n + 1 α f Lip α, S n ft = k n c k z k, n 0, 11 c n C α 1 n + 1 α f Lip α, n 0. 1 Our strategy is to use factorization techniques and the exponential map to obtain similar approximation estimates for arbitrary Ut Lip α T SUN. Lemma 4 For any α > 1/ and any Ut Lip α T SUN, there exist constant matrices U 0,l SUN and loops A l t Lip α T su such that Ut = L U 0,l eâlt, t T, L := NN 1/. 13 6

7 Here, Â l t = T ij A l t denotes the canonical extension of A l t to a sun loop by the map I i a11 a A = 1 0 a 11 0 a 1 0 T a 1 a ij A = 0 0 I j i a 1 0 a I N j for some index pair i, j with 1 i < j N I k denotes the k k identity matrix. Moreover, A l Lipα Cα, N, U U Lipα, l = 1,..., L. 14 Proof. In a first step, the QR factorization with complex Givens rotations is adapted to the loop case. The complex Givens rotation U SU of a matrix X which annihilates the sub-diagonal entry x 1 is defined as follows: U X x 11 /d x 1 /d x 1 /d x 11 /d x11 x 1 = d where d := x 11 + x 1. This definition of U is subject to the assumption that d = x 11 + x 1 > 0, i.e., that the first column of X is non-degenerate. Let us assume for a moment that the loop Ut Lip α T SUN satisfies u 11 t 0, t T. 16 Then u 11 t + u 1 t u 11 t c 0 > 0 on T. Let U [1,] t be the complex Givens rotation for the sub-matrix of Ut corresponding to row/column indices 1 and defined according to 15. Then Û[1,]t := T 1 U [1,] t and U [1] t := Û[1,] tut belong to Lip α T SUN, with norms controlled by a constant depending on α, Ut, and N. Note that due to 15 the leading diagonal element of U [1] t is real and positive, and thus automatically satisfies 16. With this in mind, we can recursively construct Û[1,j] = T 1j U [1,j] t and U [j 1] t, j =,..., N such that N Ut = Û [1,j] tu [N 1] t, U [N 1] t = j= Ũ [N 1] t where all loops have inherited the Lip α property. In particular, we have Ũ[N 1] t Lip α T SUN 1. Assuming that its leading diagonal entry satisfies again 16, it can be further factorized using the same method. Continuing this way, we would eventually arrive at a factorization of the form, Ut = L U l,0 Û l t, 17 7

8 where each Ûlt is the lift of a SU loop to a SUN loop by some T ij, i.e. some Û [i,j] t for 1 i < j N, and under the above assumptions U 0,l = I N. It remains to remove the assumption 16. We establish the following auxiliary fact: For any loop Ut Lip α T SUN with α > 1/ and N, there is a constant matrix U 0 SUN such that U0 Ut satisfies 16. Evidently, by applying this auxiliary result to the original Ut, then to Ũ[N 1] t, an so on, we establish 17 in full generality after this step, some of the U 0,l are not identity matrices anymore. To see the auxiliary result, consider the loop vt Lip α T S N 1 obtained from rewriting real and imaginary parts of the first column ut of Ut into a IR N vector. Similarly, let v 0 S N 1 be the real unit vector corresponding to the first column u 0 of U 0. Since the statement is equivalent to proving u 0 ut 0, t T, for at least one v 0 S N 1, we are done if we establish that the closed subset Σ := {v 0 S N 1 : v 0 vt for at least one t T} has zero surface measure on S N 1, i.e., possesses a non-empty open complement. Here, v 0 vt is a short-cut notation for orthogonality of the corresponding unit vectors in IC N : u 0 ut = 0. Let Cr, v = {w S N 1 : w v r} be a spherical cap of small radius r with center at v. It is easy to check that measσ Cr,v r, r 0, where Σ Cr,v := {w 0 S N 1 : w 0 w for some w Cr, v}. Here is a sketch of the argument. Any w 0 in this set is at distance r from some element v 0 from the set {v 0 S N 1 : v 0 v}, i.e., from the intersection of S N 1 with a N -dimensional subspace of IR N. Thus, Σ Cr,v can be generated by moving a two-dimensional ball of radius r along a copy of S N 3 embedded into S N 1 which gives the formula. But any loop ut Lip α T S N 1 can be covered by n spherical caps Cr, v m of radius r Cn α and with center at v m = vπm/n, m = 1,..., n. The constant C depends on the Lipschitz constant of the loop. In other words, Σ is covered by the union of n sets Σ Cr,vm of measure Cn α, i.e., has overall area Cn 1 α. Since α > 1/, letting n we get the desired result. This establishes the factorization 17 in full generality. The second step is to represent each of the Ûlt by the exponential map. Because Û l t = T i,j U l t for some index pair i, j and some U l t Lip α T SU, α > 1/, it is enough to consider the case N =. It is well-known that there are continuous SU loops that cannot be represented as the exponential of a continuous su loop. We will therefore prove the desired representation only up to a constant SU factor, i.e., we will show that for any Ut Lip α T SU, α > 1/3, there is a At Lip α T su, and a U 0 SU such that Ut = U 0 e At, A Lipα Cα, U U Lipα. 18 8

9 We again use a covering argument for Lipschitz loops, by identifying SU with the unit ball S 3 in IR 4. Since Ut is Lip α, we see that the associated loop on S 3 is contained in the union of n closed spherical caps C m of radius Cn α on S 3. Let us consider the system of the n caps {C m, C m, m = 1,...,n}. Evidently, each of these caps has S 3 surface area Cn 3α. Thus, the union of all these caps covers an area Cn 1 3α and, since α > 1/3, we can find a finite n 0 such that the open complement S 3 \ n 0 m=1 C m C m is not empty. By construction, this means that there is a U 0 SU such that Ut does not intersect with some ball around U 0 and some ball around U 0. Equivalently, the loop V t := U0 Ut does not intersect with some balls around ±I. So, if we write at + i bt ct + i dt V t =, a t + b t + c t + d t = 1, ct + i dt at i bt then these four real-valued component functions are Lip α and satisfy at r 0 < 1, b t + c t + d t 1 r 0 > 0, t T, with some r 0 < 1. Thus, t := arccosat [arccos r 0, π arccos r 0 ] belongs to Lip α T IR, and so do the three functions βt := tbt sin t, tct γt := sin t, tdt δt := sin t. It remains to verify that the element At Lip α T su given by i βt γt + i δt At :=, γt + i δt i βt satisfies V t = e At as desired. Since by construction β t + γ t + δ t = t sin t b t + c t + d t = = t sin t 1 cos t = t, t sin t 1 a t this follows from the elementary formula 7 which proves the formula in 18. The proof of the Lip α estimate is obvious because the transformations from Ut to At are all diffeomorphisms preserving the Lip α property since the transformation is nonlinear, the constant depends in general on U. Together with 17, this proves the statement of Lemma 4. It should be noted that it is impossible to use the argument in the second step of the proof with α < 1/3, due to the fact that there exist space-filling curves on S 3 that are Lip α -continuous for any such α. This is analogous to the corresponding statements for Peano and Hilbert curves for the unit cube in IR d, where the critical Hölder exponent is 1/d. Similarly, we cannot drop the assumption α > 1/ in the first step of the proof. However, we believe that statements of the type any Lip α 9

10 loop in SU is, up to a simple correction, representable as the exponential of a Lip α loop in su or by a product of a few such exponentials can be deduced from the available factorization theorems for loop groups, see [8, 13], and hold for more general Lie groups. Unfortunately, we could not yet locate these statements in the literature. To work with factorizations such as in Lemma 4, we will often use the following simple estimate. Lemma 5 For any U k t, Ũkt C T SUN, k = 1,..., K, we have K K Ũ k U k C K Ũk U k C. 19 Proof. By definition of C, we have U C = 1 for arbitrary Ut C T SUN, and k X k C k X k C for arbitrary X k t C T GLN. Thus, K Ũ k K K K U k C Ũ1 U 1 Ũ k C + U 1 Ũ k k= Ũ1 U 1 C +... K Ũk U k C. K Ũ k k= k= K U k C k= K U k C k= Finally, we need some technical estimates for exponentials of the form e P A j which are well-known in the theory of splitting methods. Although they hold in much more generality, we formulate them only for su. Lemma 6 a Let A, Ã su. Then e A eã C A, Ã A Ã. 0 b If A j su, j = 1,..., m, and M 1. Then e λ P j A j m e λa j λ m 1 [A j, where [A, Ã] := AÃ ÃA and λ IR. In particular, m e P j A j e 1 M A j M 1 m 1 M [A j, m l=j+1 A l ], 1 m l=j+1 A l ], Inequality 0 follows from applying formula 7 while 1 and can be found in [1]. 10

11 3 Proof of Theorem 1 We will not follow the dependencies of the constants occuring in the estimates below, in general, they will depend on α, N, and U. They are independent of other parameters, and in particular, of the final degree n of the polynomial loop U n t to be constructed. By Lemmas 4 and 5, it is enough to concentrate on the case Ut = e At, where At Lip α T su. For such At, we use linear approximation methods see Lemma and construct polynomial loops i β A m t := m t γ m t + i δ m t Π γ m t + i δ m t i β m t m T su with the optimal approximation rate By Lemma 6 a this gives At A m t Cm α, m. U Ũm C = e At e Amt C Cm α, m, 3 where Ũmt := e Amt. The remaining effort goes into approximating the loop Ũmt := e Amt which is typically not an element of the polynomial loop group by a suitable element U n t Π n T SU. We proceed in several steps. Using Lemma 1, we write A m t = 6 A m,l, and apply Lemma 6 b: e Amt 6 e A m,lt/m M 1 M 6 [A m,l t, m j=l+1 A m,j t]. Now, by 10 and the remark at the end of the proof of Lemma 1, we have A m,l Lipα C A m Lipα C A Lipα C U Lipα. 4 because the algebraic operations and trigonometric conjugation leading from A m t to representations for A m,l t all preserve the Lip α property. Estimating the commutators in the above estimate leads to Ũmt 6 e A m,lt/m M C M max l The integer M will be chosen later. The next step is to deal with approximating each of the factors Ũ m,l := e A m,lt/m = e P m k=k 0 c k,l B k,l t/m, appearing in the left-hand side of 5 by the polynomial loop U m,l t := m k=k 0 e ck,l/mbk,lt 11 A m,l C C M. 5

12 of degree m here k 0 = 0 for odd l and k 0 = 1 for even l. That U m,l Π m T SU follows from Lemma 1. For estimating the error we rely on 1 with λ = 1/M for notational convenience we do the case of odd l: Ũm,lt U m,l t C 1 m 1 c M k,l [B k,l t, k=0 m j=k+1 c j,l B j,l t] C. Now Lemma 3 will be used in conjunction with 4: On the one hand, c k,l Ck + 1 α, on the other m m lnk + [B k,l t, c j,l B j,l t] C c j,l B j,l t] C C k + 1, α j=k+1 j=k+1 because m j=k+1 c j,lb j,l t is the difference between A m,l and its k-th partial sum. Since α > 1/, we obtain after substitution where Ũm,lt U m,l t C C m 1 lnk + M k + 1 C α M, l = 1,...,6. 6 k=0 It remains to use 6 in conjunction with Lemma 5: 6 Ũ m,l M U n C M U n := 6 Ũm,l U m,l C C M, 6 U m,l M Π n T SU, is a polynomial loop of degree n 6mM. Together with 3 and 5, this shows that we have constructed a polynomial loop U n of degree at most 6mM such that U U n C C 1 m α + 1 M. Choosing m as the integer part of n 1/α+1 and M as the integer part of 1 6 nα/α+1, we arrive at an error estimate of the form Cn α/1+α for large enough n. This proves Theorem 1. Further Remarks. For 0 < α 1/, some weaker results are possible. Then in the proof of 6 the estimate lnk + k + 1 C < α k m is not valid anymore, and needs to be replaced by an upper bound that depends on m. This leads to other choices for m and M, and an overall weaker approximation rate, compared to the result stated in Theorem 1. Such results are subject to the availability of factorizations such as proved in Lemma 4 for α > 1/. 1

13 If Ut Lip α T SU is diagonal and 0 < α 1 then we can establish the improved bound U U n C C nα, n. 7 Here is a sketch of the elementary argument. The diagonal element ut of modulus 1 of the matrix Ut can be approximated by an appropriate polynomial p n t of degree n such that and u p n C C n α, 0 1 p n t Cn α. The crucial fact is the upper bound for the deviation of p n t from ut = 1, which is stronger than the trivial On α estimate following from 1 p n t 1 p n t u p n C. With this at hand, and q n t determined from q n t = 1 p n t via spectral factorization, we see that pn t q U n = n t qnt p Π n T SU, nt satisfies 7. At the moment, we do not know how to close the gap between the bound of Theorem 1 and the obvious lower bound of Cn α for general non-constant Ut Lip α T SUN. References [1] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Series in Applied Math., vol. 61, SIAM, 199. [] R.A. DeVore, G. G. Lorentz, Constructive Approximation, Springer-Verlag, [3] A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, and A. Zanna, Lie-group methods, Acta Numerica 9, , 000. [4] W. M. Lawton, Conjugate quadrature filters, In: Advances in Wavelets, Ka- Sing Lau ed., Springer, pp , [5] W. M. Lawton, Hermite interpolation in loop groups and conjugate quadrature filter approximation, Acta Applicandae Mathematicae 84, , 004. [6] R.I. McLachlan, G.R.W. Quispel, Splitting methods, Acta Numerica 11, ,

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