A Line Shape Function is a function describing the shape of an Absorption Line (the Line Shape or Spectral Line Shape). Characteristics of the source (such as Temperature) and of the medium through which the Electromagnetic Radiation passes give the Spectral Line width (i.e., prevent it from being infinitely thin, e.g., through Doppler Shifts from the movement of the particles from which the EMR was emitted, Doppler Broadening). The specific shape of the line, i.e., the Intensity at each Wavelength along the width of the line, is determined by characteristics of the source and the medium. By mathematically modeling the factors, the shape of a line given a set of circumstances can be derived, and conversely, an observed line of that shape can reveal the characteristics, or if more than one set of characteristics could create the shape, then constraints on the characteristics.
The Sobolev Approximation is an ingredient to make it tractable to model Line Shape Functions of sufficient accuracy for a particular purpose.
A Voigt Profile is a line shape created by the combination of two broadening mechanisms, one a Gaussian Function (such as for Doppler broadening) and the other Lorentzian. The corresponding mathematical line shape function is a Convolution of mathematical functions for each, or for an approximation, a linear combination of the two.