Moderate vs. severe ill-posedness for T = D H and T = H J remains open

Ascertain whether the compact compositions T = D H = H C*: L^2(0,1) → ℓ^2 and T = H J: L^2(0,1) → ℓ^2 exhibit moderate ill-posedness (polynomial singular value decay) or severe ill-posedness (exponential singular value decay), as existing numerical evidence is inconclusive.

Background

The paper provides analytical bounds suggesting that both compositions could be either moderately or severely ill-posed. For T = H J, bounds imply σ_i(T) lies between exp(−c i) and i^−3/2. For T = D H = H C*, bounds imply σ_i(T) lies between (c_0/i) exp(−2 i) and i^−3/2.

Numerical discretizations produce exponential decay in the discretized setting, but due to discretization-dependent behavior and large uncertainty margins, the authors emphasize that numerical calculations cannot resolve the true degree of ill-posedness for the original operators.