On exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions

TL;DR

The paper studies exact L2-norm discretization for the span of the first N Rademacher functions, focusing on the minimal number of nodes and weight signs. Using a matrix formulation, it connects the minimum-node case to Hadamard matrices and characterizes when positive weights are impossible.

• Problem

  The paper investigates the minimal number of nodes and admissible weights needed for exact L2-norm discretization in the Rademacher-function space.

• Method

  The authors reduce the discretization condition to a matrix equation involving only columns associated with nonzero weights, then analyze its relationship with Hadamard matrices.

• Results

  The minimum number of nodes is N when a Hadamard matrix of order N exists and N + 1 otherwise; for N ≡1 (mod 4), N ≥5, or N ≡2 (mod 4), N ≥6, positive weights require more than the minimum.

• Takeaways & Limitations

  At N nodes, the weights are positive and equal, whereas an N + 1-node construction can use one negative weight and N equal weights.

• Takeaways & Limitations

  The positive-weight impossibility argument is stated under the assumption that all N + 1 weights are strictly positive.