Minimum norm interpolation in the ℓ1(ℕ) space.

Cheng, Raymond; Xu, Yuesheng

We consider the minimum norm interpolation problem in the ℓ 1 (ℕ) space, aiming at constructing a sparse interpolation solution. The original problem is reformulated in the pre-dual space, thereby inducing a norm in a related finite-dimensional Euclidean space. The dual problem is then transformed into a linear programming problem, which can be solved by existing methods. With that done, the original interpolation problem is reduced by solving an elementary finite-dimensional linear algebra equation. A specific example is presented to illustrate the proposed method, in which a sparse solution in the ℓ 1 (ℕ) space is compared to the dense solution in the ℓ 2 (ℕ) space. This example shows that a solution of the minimum norm interpolation problem in the ℓ 1 (ℕ) space is indeed sparse, while that of the minimum norm interpolation problem in the ℓ 2 (ℕ) space is not.

INTERPOLATION; LINEAR algebra; LINEAR equations; SPACE; LINEAR programming

Analysis & Applications, 2021, Vol 19, Issue 1, p21

0219-5305

Academic Journal

10.1142/S0219530520400059