# Discrete Spectrum Reconstruction using Integral Approximation Algorithm

## TL;DR

This paper introduces an integral approximation algorithm for reconstructing discrete spectra from observed spectral data, effectively enhancing spectrometer resolution by solving a complex inverse problem.

## Contribution

The paper presents a novel integral approximation algorithm that simplifies solving a nonlinear inverse spectral problem by avoiding nonlinear equations.

## Key findings

- Effective enhancement of spectrometer resolution demonstrated
- Algorithm successfully applied to synthetic and experimental spectra
- Improved accuracy in reconstructing discrete spectral lines

## Abstract

An inverse problem in spectroscopy is considered. The objective is to restore the discrete spectrum from observed spectrum data, taking into account the spectrometer's line spread function. The problem is reduced to solution of a system of linear-nonlinear equations (SLNE) with respect to intensities and frequencies of the discrete spectral lines. The SLNE is linear with respect to lines' intensities and nonlinear with respect to the lines' frequencies. The integral approximation algorithm is proposed for the solution of this SLNE. The algorithm combines solution of linear integral equations with solution of a system of linear algebraic equations and avoids nonlinear equations. Numerical examples of the application of the technique, both to synthetic and experimental spectra, demonstrate the efficacy of the proposed approach in enabling an effective enhancement of the spectrometer's resolution.