## Approximating eigenvalues of boundary value problems by using the Hermite-Gauss sampling method

### Abstract

The Hermite-Gauss sampling operator was introduced by Asharabi and Prestin (2015) to approximate a function from a wide class of entire functions, using few samples from the function and its first derivative. This operator converges at the rate $\mathrm{e}^{-(2\pi-\sigma h)N}/\sqrt{N}$, and has been applied to construct a new sampling method for approximating the eigenvalues of boundary value problems whose eigenvalues are real and simple. In this paper, we use the first derivative of this operator to approximate non-real and non-simple eigenvalues of boundary value problems. For this task, we estimate two types of errors associated with the first derivative of the Hermite-Gauss operator. These error estimates give us the possibility to establish the error analysis when the eigenvalues are not real or not algebraically simple. Illustrative examples are discussed and show the effectiveness of the proposed method. Our numerical results are compared with the results of sinc-Gaussian sampling method.

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### Key words

sinc methods, approximating eigenvalues, boundary value problems, error bounds, rate of convergence

### AMS subject classifications

34L16, 94A20, 65L15, 65N15