I want to find some type of analytical chemistry method, such as spectroscopy, that is capable of analysis the chemical composition of air. If it can be used for high resolution analysis would be preferable.
Headspace GC-MS is the way to go. It's used to detect vapors coming off a solution. For example, it is used in the fragrance industry to test the time-dependent development of a perfume, since the lighter molecules will evaporate first and form a higher concentration in the headspace initially.
Unfortunately, for detection of noble gases in an extremely low concentration this is difficult. The technology has just been created and involves concentrating the rare gases via adsorption over high surface area carbon before analysis. Each gas adsorbs in a specific temperature range that is different.
From an article on the International Noble Gas Experiment, a modern method to detect trace amounts of Xenon from nuclear explosions:
“Xenon is isolated from the air by a charcoal-containing purification device resulting in higher concentrations of the gas in its stable and unstable, i.e. radioactive, forms. The isolated xenon is then measured for its level of radioactivity which helps to quantify the radioactive xenon contained in the sample.” Source
I know that a company called Air Products liquifies air before distilling Nitrogen and Oxygen, and then purifies Xenon for medical use. An in-depth process is described here.
You can use gas chromatography attached to a mass spectrometer. I've used headspace GC-MS to analyze the vapor above a sample. If you're looking to analyze oxygen/nitrogen/etc, this may not work (it might, but I'm not positive), but to test the vapors in the air, it should work.