Quantitative conditioning of quasi-Chebyshev feature matrices

Determine quantitative conditioning estimates for the feature matrices associated with the fixed-interval quasi-Chebyshev parameter sets used in the linearized shallow-network constructions.

Background

The paper constructs linearized shallow-network approximation spaces whose inner parameters are fixed in advance and whose one-dimensional parameters are distributed according to quasi-Chebyshev sets on fixed intervals. These constructions achieve sharp Sobolev approximation rates for specified analytic activations, including tanh, and are lifted to multidimensional domains through products of quasi-uniform directions.

Although the fixed-interval geometry avoids the factorial-scale clustering arising in an earlier finite-difference construction, the numerical conditioning of the resulting feature matrices is not analyzed. Establishing quantitative conditioning estimates would clarify the stability and practical computability of the proposed approximation spaces.