# Explicit eigenvalue bounds of differential operators defined by symmetric positive semi-definite bilinear forms

## TL;DR

This paper extends eigenvalue bounding theorems from positive definite to positive semi-definite bilinear forms, providing new explicit bounds and applying them to polynomial projection error estimation.

## Contribution

It introduces extended eigenvalue bounds for semi-definite bilinear forms and demonstrates their application in polynomial projection error analysis.

## Key findings

- Extended eigenvalue bounds for semi-definite forms
- Application to polynomial projection error estimation
- Improved explicit eigenvalue bounds

## Abstract

Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi-definite bilinear forms. As an application, the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections.