The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

Abstract

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behavior of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composite operator decay faster than those of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

A Note on Open Questions Asked to Analysis and Numerics Concerning the Hausdorff Moment Problem

“…Due to the prominent role of the Legendre polynomials, the Gramian matrices G A n = AL i , e (j) 2 ) n i,j=1 for ûi (t) = L i (t) (0 ≤ t ≤ 1) and vj = e (j) (e (j) k = δ k,j ) are of specific interest. From [7,Lemma 5.4] we have that AL 1 , e (j) 2 = 1 j+1 and for 2 ≤ i ≤ n that AL i , e (j) 2 = 1 j √ 2j+1 L i , h j L 2 (0,1) by using the normalized monomials h j (t) = √ 2j + 1 t j with h j L 2 (0,1) = 1. A plot of the first 20 singular values σ i (G A n ) (i = 1, .…”

“…Proposition 1 (Corollary 3.6 and Theorem 5.1 in [7]). For the composite operator A : L 2 (0, 1) → 2 from (5) there exist positive constants C and C such that, for sufficiently large indices i ∈ N, the inequalities…”

“…We now recall a result from [7], which leaves open whether the operator equation (4) with the composite operator A from ( 5) is moderately or severely (exponentially) ill-posed in the sense of Definition 1.…”

“…Since A is a Hilbert-Schmidt operator, a conceivable alternative approach to the singular value asymptotics of A is via the smoothness of the symmetric and positive kernel k(s, t) for (s, t) ∈ [0, 1] × [0, 1] of the integral operator A * A mapping in L 2 (0, 1). In the appendix of [7] it has been proven that this kernel representation attains the form of an infinite series as…”

“…To the best of our knowledge, this question that we ask to analysis and numerics in this note has not yet been answered satisfactorily, despite considerable efforts in the articles [3,4,7]. In Section 2 we formulate our current knowledge about analytical results and corresponding suggestions with respect to Open Question 2.…”