## Probability against condition number and sampling of multivariate trigonometric random polynomials

### Summary

Summary: The difficult factor in the condition number of a large linear system is the $\textcent$###$\sterling$###$\textcent$###$\textcent$###$\sterling \ddot$§$\copyright \textcent \sterling$ spectral norm of . To eliminate this factor, we here replace worst case analysis by a probabilistic argument. To $\sterling$###§$\copyright$be more precise, we randomly take from a ball with the uniform distribution and show that then, with a certain probability close to one, the relative errors and satisfy with a constant that involves $\textcent$###$\textcent \textcent$###$ !\textcent \textcent$###$\textcent$#"% textcent$###$ !textcentonly the Frobenius and spectral norms of . The success of this argument is demonstrated for Toeplitz systems and £for the problem of sampling multivariate trigonometric polynomials on nonuniform knots. The limitations of the argument are also shown.

### Mathematics Subject Classification

65F35, 15A12, 47B35, 60H25, 94A20

### Keywords/Phrases

condition number, probability argument, linear system, Toeplitz matrix, nonuniform sampling, multivariate trigonometric polynomial