Reconstruction of sparse Legendre and Gegenbauer expansions

Transcription

1 BIT Numer Math 06 56: DOI 0.007/s Reconstruction of sparse Legendre and Gegenbauer expansions Daniel Potts Manfred Tasche Received: 6 March 04 / Accepted: 8 December 05 / Published online: 9 December 05 Springer Science+Business Media Dordrecht 05 Abstract We present a new deterministic approximate algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Pronylike methods. The method proposed is robust with respect to noisy sampled data. Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order. Keywords Legendre polynomials Sparse Legendre expansions Gegenbauer polynomials Ultraspherical polynomials Sparse Gegenbauer expansions Sparse recovering Sparse Legendre interpolation Sparse Gegenbauer interpolation Asymptotic formula Prony-like method Mathematics Subject Classification 65D05 33C45 4A45 65F5 Introduction In this paper, we present a new deterministic approximate approach to the reconstruction of sparse Legendre and Gegenbauer expansions, if relatively few samples on a special grid are given. Communicated by Michael S. Floater. B Daniel Potts potts@mathematik.tu-chemnitz.de Manfred Tasche manfred.tasche@uni-rostock.de Department of Mathematics, Technische Universität Chemnitz, 0907 Chemnitz, Germany Institute of Mathematics, University of Rostock, 805 Rostock, Germany

2 00 D. Potts, M. Tasche Recently the reconstruction of sparse trigonometric polynomials has attained much attention. There exist recovery methods based on random sampling related to compressed sensing see e.g. [5,6,,8] and the references therein and methods based on deterministic sampling related to Prony-like methods see e.g. [6] and the references therein. Both methods are already generalized to other polynomial systems. Rauhut and Ward [9] presented a recovery method of a polynomial of degree at most N given in Legendre expansion with M nonzero terms, where OM log N 4 random samples are taken independently according to the Chebyshev probability measure of [, ]. Some recovery algorithms in compressive sensing are based on weighted l -minimization see [9,0] and the references therein. Exact recovery of sparse functions can be ensured only with a certain probability. Peter et al. [5] have presented a Prony-like method for the reconstruction of sparse Legendre expansions, where only M + function resp. derivative values at one point are given. We mention that sparse Legendre expansions are important for the analysis of spherical functions which are represented by expansions of spherical harmonics. In [], it was observed that the equilibrium state admits a sparse representation in spherical harmonics see [, Section 4.3], i.e., the equilibrium state can be represented by a very short Legendre expansion see [, Figure 4.5]. Recently, the authors have described a unified approach to Prony-like methods in [6] and applied it to the recovery of sparse expansions of Chebyshev polynomials of first and second kind in [7]. Similar sparse interpolation problems for special polynomial systems have been formerly explored in [3,4,8,] and also solved by Prony-like methods. A very general approach for the reconstruction of sparse expansions of eigenfunctions of suitable linear operators was suggested by Peter and Plonka [4]. New reconstruction formulas for M-sparse expansions of orthogonal polynomials using the Sturm Liouville operator, were presented. However one has to use sampling points and derivative values. In this paper we present a new method for the reconstruction of sparse Legendre expansions which is based on a local approximation of Legendre polynomials by cosine functions. Therefore this algorithm is closely related to [7]. Note that fast algorithms for the computation of Fourier Legendre coefficients in a Legendre expansion see [,7] are based on similar asymptotic formulas of the Legendre polynomials. However the key idea is that the conveniently scaled Legendre polynomials behave similar to the cosine functions near zero. Therefore we use a sampling grid located near zero. Finally we generalize this method to sparse Gegenbauer expansions. The outline of this paper is as follows. In Sect., we collect some useful properties of Legendre polynomials. In Sect. 3, we present the new reconstruction method for sparse Legendre expansions. We extend our recovery method in Sect. 4 to the case of sparse Gegenbauer expansions of low positive order. Finally we show some results of numerical experiments in Sect. 5. In Example 5.5, it is shown that the method proposed is robust with respect to noisy sampled data.

3 Reconstruction of sparse Legendre and Gegenbauer expansions 0 Properties of Legendre polynomials As known, the Legendre polynomials P n are special Gegenbauer polynomials C n α of order α = so that the properties of P n follow from corresponding properties of the Gegenbauer polynomials see e.g. [, pp ]. For each n N 0, the Legendre polynomials P n can be recursively defined by P n+ x := n + 3 n + xp n+x n + n + P nx x R with P 0 x := and P x := x see [, p. 8]. The Legendre polynomial P n can be represented in the explicit form P n x = n/ n j=0 see [, p. 84]. Hence it follows that for m N 0 j n j! j! n j! n j! xn j P m 0 = m m! m m!, P m+ 0 = 0,. P m+ 0 = m m +! m m!, P m 0 = 0.. Further, Legendre polynomials of even degree are even and Legendre polynomials of odd degree are odd, i.e. P n x = n P n x..3 Moreover, the Legendre polynomial P n satisfies the following homogeneous linear differential equation of second order x P n x x P n x + n n + P nx = 0.4 see [, p. 80]. In the Hilbert space L / [, ] with the constant weight,the normalized Legendre polynomials form an orthonormal basis, since L n x := n + P n x n N 0.5 L n x L m x dx = δ n m m, n N 0, where δ n denotes the Kronecker symbol see [, p. 8]. Note that the uniform norm max L nx = L n = L n =n + / x [,] is increasing with respect to n see [, p.64].

4 0 D. Potts, M. Tasche Let M be a positive integer. A polynomial Hx := d b k L k x.6 k=0 of degree d with d M is called M-sparse in the Legendre basis or simply a sparse Legendre expansion, ifm coefficients b k are nonzero and if the other d M + coefficients vanish. Then such an M-sparse polynomial H can be represented in the form M 0 M Hx = c 0, j L n0, j x + c,k L n,k x.7 j= with c 0, j := b n0, j = 0 for all even n 0, j with 0 n 0, < n 0, < < n 0,M0 and with c,k := b n,k = 0 for all odd n,k with n, < n, < < n,m.the positive integer M = M 0 + M is called the Legendre sparsity of the polynomial H. The numbers M 0, M N 0 are the even and odd Legendre sparsities, respectively. Remark. The sparsity of a polynomial depends essentially on the chosen polynomial basis. If k= T n x := cosn arccos x x [, ] denotes the nth Chebyshev polynomial of first kind, then the nth Legendre polynomial P n can be represented in the Chebyshev basis by P n x = n n/ j! n j! δ n j j! n j! T n jx j=0 see [, p. 90]. Thus a sparse polynomial in the Legendre basis is in general not a sparse polynomial in the Chebyshev basis. In other words, one has to solve the reconstruction problem of a sparse Legendre expansion without change of the Legendre basis. As in [9], we transform the Legendre polynomial system {L n ; n N 0 } into a uniformly bounded orthonormal system. We introduce the functions Q n :[, ] R for each n N 0 by Q n x := π 4 n + π x L n x = 4 x P n x..8 Note that these functions Q n have the same symmetry properties.3 as the Legendre polynomials, namely Q n x = n Q n x x [, ]..9

5 Reconstruction of sparse Legendre and Gegenbauer expansions 03 Further the functions Q n are orthonormal in the Hilbert space L w [, ] with the Chebyshev weight wx := π x /, since for all m, n N 0 Q n x Q m xwx dx = L n x L m x dx = δ n m. Note that wx dx =. In the following, we use the standard substitution x = cos θθ [0, π] and obtain Q n cos θ = π n + π sin θ Ln cos θ = sin θ Pn cos θ θ [0, π]. Lemma. For all n N 0, the functions Q n cos θ are uniformly bounded on the interval [0, π], i.e. Q n cos θ < θ [0, π]..0 Since Lemma. is a special case of Lemma 4. for α =, we abstain here from a separate proof. Now we describe the asymptotic properties of the Legendre polynomials. Theorem. For each n N 0, the function Q n cos θ can be represented by the asymptotic formula [ Q n cos θ = λ n cos n + θ π ] + R n θ θ [0, π]. 4 with the scaling factor and the error term R n θ := 4n + λ n := θ π/ 4m+π π 4m+3 m! m m! n = m, m+! m m! n = m + sin [ n + ] θ τ sin τ Q n cos τdτ θ 0, π.. = 0 and has the symmetry The error term R n θ fulfills the conditions R n π = R n π property R n π θ = n R n θ..3 Further, the error term can be estimated by R n θ cot θ..4 n +

6 04 D. Potts, M. Tasche Since Theorem. is a special case of Theorem 4. for α =, we leave out a separate proof. Remark. From Theorem. it follows the asymptotic formula of Laplace see [, p. 94] that for θ 0, π Q n cos θ = [ cos n + θ π ] + On n..5 4 The error bound holds uniformly in [ε, π ε] with ε 0, π. Remark.3 For arbitrary m N 0, the scaling factors λ n in. can be expressed in the following form with 4m+π α m n = m, λ n = π 4m+3 m + α m+ n = m + α 0 :=, α n := 3 n + 4 n n N. In Fig., we plot the expression tanθ R n θ for some polynomial degrees n. We have proved the estimate.4 in Theorem.. In the Table, one can see that the maximum value of tanθ R n θ on the interval [0, π] is much smaller than n+. We observe that the upper bound.4 of R n θ is very accurate in a small neighborhood of θ = π. Substituting t = θ π [ π, π ], we obtain by.9 and. that x 0 3 x pi/4 pi/ 3 pi/4 pi 0 0 pi/4 pi/ 3 pi/4 pi 0 0 pi/4 pi/ 3 pi/4 pi 0 0 pi/4 pi/ 3 pi/4 pi Fig. Expression tanθ R n θ for n {3,, 5, 0} Table Maximum values of tanθ R n θ for some polynomial degrees n n max tanθr nθ θ [0,π]

7 Reconstruction of sparse Legendre and Gegenbauer expansions 05 [ Q n sin t = n λ n cos n + nπ = n λ n cos sin [ n + t t + nπ ] + n R n t + π [ cos n + ] nπ t n λ n sin ] + n R n t + π..6 3 Prony-like method In a first step we determine the even and odd indexes n 0, j, n,k in.7, similar to [7]. We use.8 and consider the function π 4 x Hx = M 0 c 0, j Q n0, j x + c,k Q n,k x 3. j= on [, ]. In the following we use the asymptotic formulas of Q n0, j and Q n,k from Theorem.. We substitute x = cost + π = sin t for t [ π, π ].Nowwehave to determine indexes n 0, j and n,k from sampling values of the function π [ cos th sin t t π, π ]. 3. We introduce the function M 0 [ Ft:= d 0, j cos n 0, j + ] t + j= with the coefficients M k= M k= [ d,k sin n,k + [ t] t π, π ] d 0, j := n 0, j / c 0, j λ n0, j, d,k := n,k+/ c,k λ n,k. 3.3 By.9 and.6, the new function 3.3 approximates the sampled function 3. in a small neighborhood of t = 0. Then we form Ft + F t M 0 = j= [ d 0, j cos n 0, j + ] t 3.4 and Ft F t M = k= [ d,k sin n,k + ] t. 3.5 Now we proceed similar to [7], but we use only sampling points near 0, due to the small values of the error term R n t + π in a small neighborhood of t = 0 [see.4].

8 06 D. Potts, M. Tasche Let N N be sufficiently large such that N > M and N is an upper bound of π the degree of the polynomial.6. For u N := sin N we form the nonequidistant kπ sine grid {u N,k := sin N ; k = M,...,M } in the interval [, ]. We consider the following problem of sparse Legendre interpolation: For given sampled data h k := π cos kπ N H sin kπ N k = M,...,M determine all parameters n 0, j j =,...,M 0 of the sparse cosine sum 3.4, determine all parameters n,k k =,...,M of the sparse sine sum 3.5 and finally determine all coefficients c 0, j j =,...,M 0 and c,k k =,...,M of the sparse Legendre expansion Sparse even Legendre interpolation For a moment, we assume that the even Legendre sparsity M 0 of the polynomial.7 is known. Then we see that the above interpolation problem is closely related to the interpolation problem of the sparse, even trigonometric polynomial h k + h k M 0 f k := d 0, j cos n 0, j + /kπ N j= k = M 0,...,M 0, 3.6 where the sampled values f k k = M 0,...,M 0 are approximately given by h k+h k. Note that f k = f k k = 0,...,M 0. We introduce the Prony polynomial Π 0 of degree M 0 with the leading coefficient M0, whose roots are cos n 0, j +/π N j =,...,M 0, i.e. M 0 Π 0 x = M 0 j= x cos n 0, j + /π. N Then the Prony polynomial Π 0 can be represented in the Chebyshev basis by Π 0 x = M 0 l=0 p 0,l T l x p 0,M0 :=. 3.7 The coefficients p 0,l of the Prony polynomial 3.7 can be characterized as follows: Lemma 3. See [7, Lemma 3.] For all k = 0,..., M 0, the sampled data f k and the coefficients p 0,l of the Prony polynomial 3.7 satisfy the equations M 0 f k+l + f l k p 0,l = f k+m0 + f M0 k. l=0

9 Reconstruction of sparse Legendre and Gegenbauer expansions 07 Using Lemma 3., one obtains immediately a Prony method for sparse even Legendre interpolation in the case of known even Legendre sparsity. This algorithm is similar to [7, Algorithm.7] and omitted here. In practice, the even/odd Legendre sparsities M 0, M of the polynomial.7 of degree at most N are unknown. Then we can apply the same technique as in [7, Section 3]. We assume that an upper bound L N of max {M 0, M } is known, where N N is sufficiently large with max {M 0, M } L N. In order to improve the numerical stability, we allow to choose more sampling points. Therefore we introduce an additional parameter K with L K N such that we use K + L sampling points of.7, more precisely we assume that sampled data f k k = 0,...,L + K from 3.6 are given. With the L + K sampled data f k R k = 0,...,L + K, we form the rectangular Toeplitz-plus-Hankel matrix H 0 K,L+ := f k+l + f l k K,L k,l= Note that H 0 K,L+ is rank deficient with rank M 0 see [7, Lemma 3.]. 3. Sparse odd Legendre interpolation First we assume that the odd Legendre sparsity M of the polynomial.7 is known. Then we see that the above interpolation problem is closely related to the interpolation problem of the sparse, odd trigonometric polynomial h k h k M g k := d, j sin n, j + /kπ N j= k = M,...,M, 3.9 where the sampled values g k k = 0,...,M are approximately given by h k h k. Note that g k = g k k = 0,...,M. We introduce the Prony polynomial Π of degree M with the leading coefficient M, whose roots are cos n, j +/π N j =,...,M, i.e. Π x = M M j= x cos n, j + /π. N Then the Prony polynomial Π can be represented in the Chebyshev basis by Π x = M l=0 p,l T l x p,m :=. 3.0 The coefficients p,l of the Prony polynomial 3.0 can be characterized as follows:

10 08 D. Potts, M. Tasche Lemma 3. For all k = 0,...,M, the sampled data g k and the coefficients p,l of the Prony polynomial 3.0 satisfy the equations M g k+l + g k l p,l = g k+m + g k M. 3. l=0 Proof Using sinα + β + sinα β = sinα cos β, we obtain by 3.9 that M g k+l + g k l = d, j sin n, j + /k + lπ + sin n, j + /k lπ N N j= M = d, j sin n, j + /kπ N j= Thus we conclude that cos n, j + /lπ. N M g k+l +g k l p,l = d, j sin n, j + /kπ N l=0 M j= M = d, j sin n, j +/kπ N j= M l=0 p,l cos n, j + /lπ N Π cos n, j +/π N =0. By p,m =, this implies the assertion 3.. Using Lemma 3., one can formulate a Prony method for sparse odd Legendre interpolation in the case of known odd Legendre sparsity. This algorithm is similar to [7, Algorithm.7] and omitted here. In general, the even/odd Legendre sparsities M 0 and M of the polynomial.7 of degree at most N are unknown. Similarly to Sect. 3., letl N be a convenient upper bound of max {M 0, M }, where N N is sufficiently large with max {M 0, M } L N. In order to improve the numerical stability, we allow to choose more sampling points. Therefore we introduce an additional parameter K with L K N such that we use K + L sampling points of.7, more precisely we assume that sampled data g k k = 0,...,L + K from 3.9 are given. With the L + K sampled data g k R k = 0,...,L + K we form the rectangular Toeplitz-plus-Hankel matrix H K,L+ := g k+l + g k l K,L k,l=0. 3. Note that H K,L+ is rank deficient with rank M. This is an analogous result to [7, Lemma 3.].

11 Reconstruction of sparse Legendre and Gegenbauer expansions Sparse Legendre interpolation In this subsection, we sketch a Prony-like method for the computation of the polynomial degrees n 0, j and n,k of the sparse Legendre expansion.7. Mainly we use singular value decompositions SVD of the Toeplitz-plus-Hankel matrices 3.8 and 3.. For details of this method see [7, Section 3]. We start with the singular value factorizations H 0 K,L+ = U 0 K D0 K,L+ W 0 L+, H K,L+ = U K D K,L+ W L+, where U 0 K, U K, W 0 L+ and W L+ are orthogonal matrices and where D0 D K,L+ are rectangular diagonal matrices. The diagonal entries of D0 K,L+ singular values of 3.8 arranged in nonincreasing order σ 0 σ 0 σ 0 M 0 σ 0 0 M 0 + σ L+ 0. K,L+ and are the We determine the largest M 0 such that σ 0 M 0 /σ 0 >ε, which is approximately the rank of the matrix 3.8 and which coincides with the even Legendre sparsity M 0 of the polynomial.7. Similarly, the diagonal entries of D K,L+ are the singular values of 3. arranged in nonincreasing order σ σ σ M σ M + σ L+ 0. We determine the largest M such that σ M /σ >ε, which is approximately the rank of the matrix 3. and which coincides with the odd Legendre sparsity M of the polynomial.7. Note that there is often a gap in the singular values, such that we can choose ε = 0 8 in general. Introducing the matrices D 0 K,M 0 := D 0 K,L+ : K, : M 0 = W 0 0 M 0,L+ := W L+ : M 0, : L +, D K,M := D K,L+ : K, : M = W M,L+ := W L+ : M, : L +, diag σ 0 j M 0 j= O K M0,M 0 diag σ j M j= O K M,M we can simplify the SVD of the Toeplitz-plus-Hankel matrices 3.8 and 3. as follows,, H 0 K,L+ = U 0 K D0 K,M 0 W 0 M 0,L+, H K,L+ = U K D K,M W M,L+.

12 030 D. Potts, M. Tasche Using the known submatrix notation and setting W 0 0 M 0,Ls := W M 0,L+ : M 0, + s : L + s s = 0,, 3.3 W M,Ls := W M,L+ : M, + s : L + s s = 0,, 3.4 we form the matrices F 0 M 0 := F M := W 0 M 0,L 0 0 W M 0,L, 3.5 W M,L 0 W M,L, 3.6 where W 0 M 0,L 0 denotes the Moore Penrose pseudoinverse of W 0 M 0,L 0. Finally we determine the nodes x 0, j [, ] j =,...,M 0 and x, j [, ] j =,...,M as eigenvalues of the matrix F 0 M 0 and F M, respectively. Thus the algorithm reads as follows: Algorithm 3. Sparse Legendre interpolation based on SVD Input: L, K, N N N, 3 L K N, L is upper bound of max{m 0, M }, sampled values H sin N kπ k = L K,...,L + K of the polynomial.7 ofdegreeatmostn.. Compute h k := and form π cos f k := h k + h k kπ N H sin, g k := h k h k kπ k = L K,...,L + K N k = L K,...,L + K.. Compute the SVD of the rectangular Toeplitz plus Hankel matrices 3.8 and 3.. Determine the approximate rank M 0 of 3.8 such that σ 0 M 0 /σ 0 > 0 8 and form the matrix 3.3. Determine the approximate rank M of 3. such that σ M /σ > 0 8 and form the matrix Compute all eigenvalues x 0, j [, ] j =,...,M 0 of the square matrix 3.5. Assume that the eigenvalues are ordered in the following form x 0, > x 0, > > x 0,M0. Calculate n 0, j := [ N π arccos x 0, j ] j =,...,M 0, where [x] := x means rounding of x R to the nearest integer. 4. Compute all eigenvalues x, j [, ] j =,...,M of the square matrix 3.6. Assume that the eigenvalues are ordered in the following form x, > x, > > x,m. Calculate n, j := [ N π arccos x, j ] j =,...,M. 5. Compute the coefficients c 0, j R j =,...,M 0 and c, j R j =,...,M as least squares solutions of the overdetermined linear Vandermonde like systems

13 Reconstruction of sparse Legendre and Gegenbauer expansions 03 M 0 kπ c 0, j Q n0, j sin = f k k = 0,...,L + K, N j= M kπ c, j Q n, j sin = g k k = 0,...,L + K. N j= Output: M 0 N 0, n 0, j N 0 0 n 0, < n 0, < < n 0,M0 < N, c 0, j R j =,...,M 0. M N 0, n, j N n, < n, < < n,m < N, c, j R j =,...,M. Remark 3. The Algorithm 3. is very similar to [7, Algorithm 3.5]. Note that one can also use the QR decomposition of the rectangular Toeplitz-plus-Hankel matrices 3.8 and 3. instead of the SVD. In that case one obtains an algorithm similar to [7, Algorithm 3.4]. 4 Extension to Gegenbauer polynomials In this section we show that our reconstruction method can be generalized to sparse Gegenbauer expansions of low positive order. The Gegenbauer polynomials C n α of degree n N 0 and fixed order α>0can be defined by the recursion relation see [, p.8]: C α α + n + n+ x := xc α α + n n+ x n + n + Cα n x n N 0 with C α 0 x := and C α x := α x. Sometimes, C n α are called ultraspherical polynomials too. In the case α =, one obtains again the Legendre polynomials P n = C n /. By[, p. 84], an explicit representation of the Gegenbauer polynomial reads as follows C α n n/ C n α j Γn j + α x = ΓαΓj + Γn j + xn j. j=0 Thus the Gegenbauer polynomials satisfy the symmetry relations Further one obtains that for m N 0 C α d dx Cα m+ C n α x = n C n α x. 4. m 0 = m Γ m + ΓαΓm +, Cα m+ 0 = m Γα+ m +, ΓαΓm + 0 = 0, 4. d dx Cα m 0 =

14 03 D. Potts, M. Tasche Moreover, the Gegenbauer polynomial C n α satisfies the following homogeneous linear differential equation of second order see [, p.80] x d dx Cα n x α + x d dx Cα n x + n n + αc α n x = Further, the Gegenbauer polynomials are orthogonal over the interval [, ] with respect to the weight function w α x = Γα+ πγα+ x α / x,, i.e. more precisely by [, p. 8] C m α x Cα n xw α x dx = Note that the weight function w α is normalized by αγα + n n + α Γ n + Γα δ m n m, n N 0. w α x dx =. Then the normalized Gegenbauer polynomials L α n x := n + α Γ n + Γα C n α x n N αγα + n form an orthonormal basis in the weighted Hilbert space L w α[, ]. Let M be a positive integer. A polynomial Hx := d k=0 b k L α k x of degree d with d M is called M-sparse in the Gegenbauer basis or simply a sparse Gegenbauer expansion, ifm coefficients b k are nonzero and if the other d M + coefficients vanish. Then such an M-sparse polynomial H can be represented in the form M 0 Hx = c 0, j L α n 0, j x + j= M k= c,k L α n,k x 4.6 with c 0, j := b n0, j = 0 for all even n 0, j with 0 n 0, < n 0, < < n 0,M0 and with c,k := b n,k = 0 for all odd n,k with n, < n, < < n,m. The positive integer M = M 0 + M is called the Gegenbauer sparsity of the polynomial H. The integers M 0, M are the even and odd Gegenbauer sparsities, respectively.

15 Reconstruction of sparse Legendre and Gegenbauer expansions 033 Now for each n N 0, we introduce the functions Q α n Q α n x := Γα+ π Γ α + x α/ L α n x x [, ] 4.7 These functions Q α n polynomials, namely possess the same symmetry properties 4. as the Gegenbauer Q α n x = n Q α x x [, ]. 4.8 n Further the functions Q α n are orthonormal in the weighted Hilbert space L w [, ] with the Chebyshev weight wx = π x /, since for all m, n N 0 Q α m x Qα n xwx dx = by L α m x Lα n xwα x dx = δ m n. In the following, we use the standard substitution x = cos θθ [0, π] and obtain Q α n cos θ := Γα+ π Γα+ sin θ α L α n cos θ θ [0, π]. Lemma 4. For all n N 0 and α 0,, the functions Q α n cos θ are uniformly bounded on the interval [0, π], i.e. Q α n cos θ < θ [0, π]. 4.9 Proof For n N 0 and α 0,, we know by [3]thatforallθ [0, π] sin θ α C n α cos θ < α Γα n + αα. Then for the normalized Gegenbauer polynomials L α n, we obtain the estimate sin θ α L α α n cos θ < Γα Using the duplication formula of the Gamma function ΓαΓ n + α Γn + Γα n + α α. αγα + n α + = α πγα, 4.0 we can estimate

16 034 D. Potts, M. Tasche For α = see [9] Q α n cos θ < Γn + Γα + n n + αα /., we obtain Q/ n cos θ <. In the following, we use the inequalities n + σ σ Γn + < n Γn + σ < σ + σ for all n N and σ 0,. In the case 0 <α<, the estimate 4. with σ = α implies that Q α n cos θ < n + α + 4 n + α α+/. Since n α < n + α for all n N, we conclude that Q α n cos θ < α. In the case <α<, we set β := α 0,. By4. with σ = β, we can estimate Γn + Γn + α = Γn + n n + βγ n + β < β n + β + β + 4. Hence we obtain by n + β + 4 < n + β that Q α n cos θ < n β/ n + β + β + n + α α / 4 < n β/ + β + n + β β/ <. 4 Finally, by Q α αγα π 0 cos θ = Γ α + sin θ α and Q α 0 cos θ 4 π, we see that the estimate 4.9isalsotrueforn = 0.

17 Reconstruction of sparse Legendre and Gegenbauer expansions 035 By 4.4, the function Q α n cos θsatisfies the following linear differential equation of second order see [, p.8] d dθ Qα n cos θ+ n + α + α α sin θ Q α n cos θ = 0 θ 0, π. 4. By the method of Liouville Stekloff, see [, pp. 0 ], we show that for arbitrary n N 0, the function Q α n cos θ is approximately equal to some multiple of cos[n + α θ απ ] in a small neighborhood of θ = π. Theorem 4. For each n N 0 and α 0,, the function Q α n cos θ can be represented by the asymptotic formula [ Q α n cos θ = λ n cos n + α θ απ with the scaling factor λ n := and the error term R n α α θ := α n + α m+α Γ m+ Γα+m Γm+ m++α Γ α+m+ θ π/ ] α / Γ m+ + R n α θ θ [0, π] 4.3 Γm+ n = m, α+/ Γα+m+ Γm+ n = m + sin[n + αθ τ] sin τ The error term R n α θ satisfies the conditions π d R n α = dθ Rα n and has the symmetry property R α n Further, the error term can be estimated by Q α n cos τdτ θ 0, π. 4.4 π = 0 π θ = n R α θ. 4.5 R n α α θ α cot θ. 4.6 n + α Proof. Using the method of Liouville Stekloff see [, pp. 0 ], we derive the asymptotic formula 4.3 from the differential equation 4., which can be written in the form d dθ Qα n cos θ+ n + α Q α α n cos θ = α sin θ n Qα n cos θ θ 0, π. 4.7

18 036 D. Potts, M. Tasche Since the homogeneous linear differential equation has the fundamental system [ cos n + α θ απ d dθ X θ + n + α X θ = 0 ] [, sin n + α θ απ ], the differential equation 4.7 can be transformed into the Volterra integral equation [ Q α n cos θ = λ n cos n + α θ απ α α n + α θ π/ ] [ + μ n sin n + α θ απ sin[n + αθ τ] sin τ Q α n cos τdτ θ 0, π with certain real constants λ n and μ n. Introducing R n α θ by 4.4, we see immediately that R n α = 0. From d π dθ Rα n θ = α α θ π/ cos[n + αθ τ] sin τ Q α n ] cos τdτ it follows that dθ d Rα n π = 0.. Now we determine the constants λ n and μ n. For arbitrary even n = m m N 0, the function Q α m cos θ can be represented in the form [ Q α m cos θ=λ m cos m + α θ απ ] [ +μ m sin m + α θ απ ] + R α m θ Hence the condition R α m π = 0 means that Qα m 0 = m λ m.using4.7, 4.5, 4., and the duplication formula 4.0, we obtain that λ m = m + α Γ m + Γα + m α / Γ m +. Γm + From d for θ = π 0 = 0by4.3 it follows that the derivative of Qα m cos θ vanishes d. Thus the second condition dθ Rα m π = 0 implies that dx Cα m i.e. μ m = 0. 0 = μ m m + α m,

19 Reconstruction of sparse Legendre and Gegenbauer expansions If n = m + m N 0 is odd, then [ Q α m+ cos θ = λ m+ cos m + + αθ απ ] [ + μ m+ sin m + + α θ απ ] + R α m+ θ. Hence the condition R α m+ π = 0 implies by Cα m+ 0 = 0[see4.] that 0 = μ m+ m, i.e. μ m+ = 0. The second condition dθ d Rα m+ π = 0 reads as follows n + α Γ α Γ α Γ m + π Γα+ Γα + m + Γα = λ m+ m + + α m. Thus we obtain by 4.3 and the duplication formula 4.0 that λ m+ = Γm + m + + α Γ α + m + d dx Cα m+ 0 α+/ Γα+ m +. Γm + 4. As shown, the error term R α n θ has the explicit representation 4.4. Using 4.9, we estimate this integral and obtain R n α α θ α n + α θ π/ sin τ dτ The symmetry property 4.5 of the error term R n α θ = Qα cos θ λ n cos n = α α n + α [ n + αθ απ cot θ. ] follows from the fact that Q α n cos θ and cos[n + αθ απ symmetry properties as 4.8. This completes the proof. ] possess the same Remark 4. The following result is stated in [0]: If α, then πγ α + sin θ α L α n cos θ Γα+ α /6 + α / n for all n 6 and θ [0, π]. Using4., we can see that sin θ α L α n cos θ is uniformly bounded for all α, n 6 and θ [0, π]. Using above estimate, one can extend Theorem 4. to the case of moderately sized order α.

20 038 D. Potts, M. Tasche We observe that the upper bound 4.6 of R n α θ is very accurate in a small neighborhood of θ = π. By the substitution t = θ π [ π, π ] and 4.8, we obtain Q α n [ sin t = n λ n cos n + α t + nπ ] + n R n t + π nπ nπ = n λ n cos cos[n+α t] n λ n sin + n R n t + π. sin[n+α t] Now the Algorithm 3. can be straightforward generalized to the case of a sparse Gegenbauer expansion 4.6: Algorithm 4. Sparse Gegenbauer interpolation based on SVD Input: L, K, N N N, 3 L K N, L is upper bound of max{m 0, M }, sampled values H sin N kπ k = L K,...,L + K of polynomial 4.6 ofdegreeatmostn and of low order α>0.. Compute for k = L K,...,L + K h k := Γα+ π Γα+ cos kπ α H sin N kπ N and form f k := h k + h k, g k := h k h k.. Compute the SVD of the rectangular Toeplitz plus Hankel matrices 3.8 and 3.. Determine the approximate rank M 0 of 3.8 such that σ 0 M 0 /σ 0 > 0 8 and form the matrix 3.3. Determine the approximate rank M of 3. such that σ M /σ > 0 8 and form the matrix Compute all eigenvalues x 0, j [, ] j =,...,M 0 of the square matrix 3.5. Assume that the eigenvalues are ordered in the following form x 0, > x 0, > > x 0,M0. Calculate n 0, j := [ N π arccos x 0, j α] j =,...,M Compute all eigenvalues x, j [, ] j =,...,M of the square matrix 3.6. Assume that the eigenvalues are ordered in the following form x, > x, > > x,m. Calculate n, j := [ N π arccos x, j α] j =,...,M. 5. Compute the coefficients c 0, j R j =,...,M 0 and c, j R j =,...,M as least squares solutions of the overdetermined linear Vandermonde like systems

21 Reconstruction of sparse Legendre and Gegenbauer expansions 039 M 0 j= M j= c 0, j Q α kπ n 0, j sin = f k k = 0,...,L + K, N c, j Q α kπ n, j sin = g k k = 0,...,L + K. N Output: M 0 N 0, n 0, j N 0 0 n 0, < n 0, < < n 0,M0 < N, c 0, j R j =,...,M 0. M N 0, n, j N n, < n, < < n,m < N, c, j R j =,...,M. 5 Numerical examples Now we illustrate the behavior and the limits of the suggested algorithms. Using IEEE standard floating point arithmetic with double precision, we have implemented our algorithms in MATLAB. In Example 5., an M-sparse Legendre expansion is given in the form.7 with normed Legendre polynomials of even degree n 0, j j =,...,M 0 and odd degree n,k k =,...,M, respectively, and corresponding real non-vanishing coefficients c 0, j and c,k, respectively. In Examples 5.3 and 5.4,an M-sparse Gegenbauer expansion is given in the form 4.6 with normed Gegenbauer polynomials of even/odd degree n 0, j resp. n,k and order α>0 and corresponding real non-vanishing coefficients c 0, j resp. c,k. We compute the absolute error of the coefficients by ec := max j=,...,m 0 k=,...,m { c 0, j c 0, j, c,k c,k } c := c 0,,...,c 0,M0, c,,...,c,m T, where c 0, j and c,k are the coefficients computed by our algorithms. The symbol + in the Tables, 3, 4 and 5 means that all degrees n j are correctly reconstructed, the Table Results of Example 5. for exact sampled data N K L Algorithm 3. ec e e e e e 3

22 040 D. Potts, M. Tasche symbol indicates that the reconstruction of the degrees fails. We present the error ec in the last column of the tables. Example 5. We start with the reconstruction of a 5-sparse Legendre expansion.7 which is a polynomial of degree 00. We choose the even degrees n 0, = 6, n 0, =, n 0,3 = 00 and the odd degrees n, = 75, n, = 77 in.7. The corresponding coefficients c 0, j and c,k are equal to. Note that for the parameters N = 400 and K = L = 5, due to roundoff errors, some eigenvalues x 0, j resp. x,k are not contained in [, ]. But we can improve the stability by choosing more sampling values. In the case N = 500, K = 9 and L = 5, we need only K + L = 7 sampled values of 3. for the exact reconstruction of the 5-sparse Legendre expansion.7. Example 5. Now we show that the Algorithm 3. is robust with respect to noisy sampled data. We reconstruct a 5-sparse Legendre expansion.7 with the even degrees n 0, =, n 0, = 50 and the odd degrees n, = 75, n, = 77 and n,3 = 33, where the corresponding coefficients c 0, j and c,k are equal to. Then we add noise of size 0 δ η, where η is uniformly distributed in [, ], to each sampling value. The corresponding results of Algorithm 3. are shown in Table 3. Note that for certain parameters, some eigenvalues x 0, j resp. x,k are not contained in [, ].But we can improve the stability of Algorithm 3. by choosing more sampling values see Table 3. Note that we can also reconstruct the polynomials for higher noisy level, if we replace the identification of the rank M 0 of 3.8 and M of 3. instepof Algorithm 3. by a gap condition. For the results see the last four lines of Table 3. Example 5.3 We consider now the reconstruction of a 5-sparse Gegenbauer expansion 4.6oforderα>0which is a polynomial of degree 00. Similar as in Example 5., we choose the even degrees n 0, = 6, n 0, =, n 0,3 = 00 and the odd degrees n, = 75, n, = 77 in 4.6. The corresponding coefficients c 0, j and c,k are equal to. Here we use only L + K = 9 sampled values for the exact recovery of the 5-sparse Gegenbauer expansion 4.6 of degree 00. Despite the fact, that we show in Theorem 4. only results for α 0,, we show also some examples for α>. But the suggested method fails for α = 3.5. In this case our algorithm cannot exactly detect the smallest degrees n 0, = 6 and n 0, =, but all the higher degrees are exactly detected. Example 5.4 We consider the reconstruction of a 5-sparse Gegenbauer expansion 4.6 of order α>0which does not consist of Gegenbauer polynomials of low degrees. Thus we choose the even degrees n 0, = 60, n 0, = 0, n 0,3 = 00 and the odd degrees n, = 75, n, = 77 in 4.6. The corresponding coefficients c 0, j and c,k are equal to. In Table 5, we show also some examples for α.5. But the suggested method fails for α = 8. In this case, Algorithm 4. cannot exactly detect the smallest degree n 0, = 60, but all the higher degrees are exactly recovered. This observation is in perfect accordance with the very good local approximation near θ = π/, see Theorem 4. and Remark 4..

23 Reconstruction of sparse Legendre and Gegenbauer expansions 04 Table 3 Results of Example 5. for noisy sampled data N K L δ Algorithm 3. ec e e e e e e e 04 Table 4 Results of Example 5.3 α N K L Algorithm 4. ec e e e e e e e Table 5 Results of Example 5.4 α N K L Algorithm 4. ec e e e e e e e e e Example 5.5 We stress again that the Prony-like methods are very powerful tools for the recovery of a sparse exponential sum Sx := M c j e f j x j= x 0

24 04 D. Potts, M. Tasche with distinct numbers f j [ δ, 0] +i [ π, π 0 δ and complex non vanishing coefficients c j, if only finitely many sampled data of S are given. In [7, Example 5.5], we have presented a method to reconstruct functions of the form Fθ = M c j cosν j θ+ d j sinμ j θ θ [0, π]. j= with real coefficients c j, d j and distinct frequencies ν j, μ j > 0 by sampling the function F. Now we reconstruct a sum of sparse Legendre and Chebyshev expansions Hx := M M c j L n j x + d k T mk x x [, ]. 5. j= k= Here we choose c j = d k =, M = M = 5 and n j 5 j= = 6, 3, 65, 68, 90T and m k 5 k= = 60, 0, 75, 78, 00T. Substituting x = sin t for t [ π, π ], we obtain π M M π cos th sin t= c j Q n j sin t+ d k j= By Theorem., the function k= cos t cos m k t + m kπ. 5. M c j λ n j j= [ cos n j + t + n ] jπ M π + d k cos m k t + m kπ k= 5.3 approximates 5. in the near of 0. We apply Algorithm 3. with N = 00 and K = L = 0. In step 3 of Algorithm 3., we calculate all frequencies n j + and m k of 5.3, if n j and m k are even: N 7 arccos x 0, j =99.99, 90.50, 78.00, 68.49, 0.00, , 6.5 T. π j= In step 4 of Algorithm 3., we determine all frequencies n j + and m k of 5.3, if n j and m k are odd: N 3 arccos x, j = , , T. π j= Thus the Legendre polynomials L n j x in 5. have the even degrees 6, 68, and 90 and the odd degrees 3 and 65. Thus the Chebyshev polynomials T mk x in 5.

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