Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

We present sparse interpolation algorithms for recovering a polynomial with <inline-formula> <tex-math notation="LaTeX">$\le B$ </tex-math></inline-formula> terms from <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> evaluations at distinct values for the variable when <inline-formula> <tex-math notation="LaTeX">$\le E$ </tex-math></inline-formula> of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars <inline-formula> <tex-math notation="LaTeX">$ \mathsf {K}$ </tex-math></inline-formula> and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of <inline-formula> <tex-math notation="LaTeX">$ \mathsf {K}$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$\ne 2$ </tex-math></inline-formula>. Our algorithms return a list of valid sparse interpolants for the <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> support points and run in polynomial-time. For standard power basis our algorithms sample at <inline-formula> <tex-math notation="LaTeX">$N = \lfloor \frac {4}{3} E + 2 \rfloor B$ </tex-math></inline-formula> points, which are fewer points than <inline-formula> <tex-math notation="LaTeX">$N = 2(E+1)B - 1$ </tex-math></inline-formula> given by Kaltofen and Pernet in 2014. For Chebyshev basis our algorithms sample at <inline-formula> <tex-math notation="LaTeX">$N = \lfloor \frac {3}{2} E + 2 \rfloor B$ </tex-math></inline-formula> points, which are also fewer than the number of points required by the algorithm given by Arnold and Kaltofen in 2015, which has <inline-formula> <tex-math notation="LaTeX">$N = 74 \lfloor \frac {E}{13} + 1 \rfloor $ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$B = 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$E \ge 222$ </tex-math></inline-formula>. Our method shows how to correct 2 errors in a block of <inline-formula> <tex-math notation="LaTeX">$4B$ </tex-math></inline-formula> points for standard basis and how to correct 1 error in a block of <inline-formula> <tex-math notation="LaTeX">$3B$ </tex-math></inline-formula> points for Chebyshev Basis.