Sparse polynomial interpolation in Chebyshev bases

We study the problem of reconstructing a sparse polynomial in a basis of Chebyshev polynomials (Chebyshev basis in short) from given samples on a Chebyshev grid of [−1, 1]. A polynomial is called M-sparse in a Chebyshev basis, if it can be represented by a linear combination of M Chebyshev polynomials. For a polynomial with known and unknown Chebyshev sparsity, respectively, we present efficient reconstruction methods, where Prony-like methods are used. The reconstruction results are mainly presented for bases of Chebyshev polynomials of first and second kind, respectively. But similar issues can be obtained for bases of Chebyshev polynomials of third and fourth kind, respectively.