Polynomial approximation on Lissajous curves in the d-cube

For a∈Z>0d we let ℓa(t):=(cos⁡(a1t),cos⁡(a2t),⋯,cos⁡(adt)) denote an associated Lissajous curve. We study such Lissajous curves which have the quadrature property for the cube [−1,1]d that ∫[−1,1]dp(x)dμd(x)=∫0πp(ℓa(t))dt for all polynomials p(x)∈V where V is either the space of d-variate polynomials of degree at most m or else the d-fold tensor product of univariate polynomials of degree at most m. Here dμd is the product Chebyshev measure (also the pluripotential equilibrium measure for the cube). Among such Lissajous curves with this property we study the ones for which maxp∈V⁡deg(p(ℓa(t))) is as small as possible. In the tensor product case we show that this is uniquely minimized by g:=(1,(m+1),(m+1)2,⋯,(m+1)d−1). In the case of m=2n we construct discrete hyperinterpolation formulas which are easily evaluated with, for example, the Chebfun system ([6]).