Representation of sparse Legendre expansions

Rapidly computing sparse Legendre expansions via sparse Fourier transforms

“…Although highly accurate for small values of N (see [26]), the method outlined in §2.2 has a relatively low accuracy for the large degree polynomials considered herein, achieving only two or three digits of accuracy per Legendre coefficient on average. Figure 2a compares its average ℓ 2 -error on the true Legendre coefficients, a m with m ∈ S, over all 100 trials at each data point against Algorithm 1's average ℓ 2 -error over the at least 70 trials at each data point for which it correctly identified a superset of S from (36). As one can see, Algorithm 1 is generally more accurate when it manages to identify S. The error graphs for the other values of N considered herein were similar.…”

The Generalized Operator Based Prony Method

“…The needed samples F (A ℓ f ) are in this case of the form F (A ℓ f ) = F (S ℓ 1 f ) = F (f (· + ℓ)) = f (ℓ). There have been other attempts to generalize the idea of Prony's method to different expansions, including sparse polynomials [2], piecewise sinusoidal signals [3], sparse expansions into Legendre polynomials [14] or Chebyshev polynomials [21] and into Lorentzians [1]. All these expansions can be also recovered directly using the approach in [13].…”

Sparse sums with bases of Chebyshev polynomials of the third and fourth kind

“…There are also some efficient reconstruction algorithms for this work. One of them is a random recovery method such as Legendre expansion with M nonzero coefficients; see [8,11]. Moreover, there are some deterministic methods for the reconstruction…”