Wikipedia has a wild article about the Dirac delta function. Are the things listed correct? Or is there no proof that they are correct? For my master thesis I want to refer to rigorous proofs of these properties if they exist. The problem is that Wikipedia's list of references is meager and in almost every appropriate place, the references are missing. To give you a taste, some properties Wikipedia lists are:
- Fourier transform of delta function,
- delta function composition with another function
- translations of delta function,
- delta function is an even function
- the property $\delta(ax) = \delta(x)/|a|$
- algebraic properties
- integration by parts of integrals containing delta function,
- distributional derivatives
I looked at a few texts, but they were not relevant for two reasons, i.e. Griffel – modern functional analysis, because the space of functions were too small (test functions with compact support). In physics, the convolving function (not the generalized function) is usually any function on $\mathbb{R}^d$, and therefore I am interested in a large a space as possible. And second, they didn't refer to anywhere near all these properties.
Is there a math book written by a mathematician (not a physicist) which treats much of the above rigorously? Alternatively, if you can justify that the above properties are just physics (not math) sufficiently well, then I can let it go and get on with my life. Either is appreciated.
Best Answer
To answer Is there a math book written by a mathematician (not a physicist) which treats much of the above rigorously?
The following references are my favorites:
a) "Mathematics for the Physical Sciences", Laurent Schwartz;
b) "Generalized Functions vol 1", I.M. Gelfand, G. E. Shilov.
These are classics and primary sources in the areas of generalized functions, b), and distributions, a). Generalized functions and distributions are the same thing, see wiki 'generalized functions'. Both are rigorous math books, and are very readable. They both have much information on the Dirac Delta distribution (aka 'Delta Function').
Schwartz is credited with originating the 'theory of distributions' which is also the title of his original book (in French only). "Mathematics for the Physical Sciences" contains much of the material in that book.
Gelfand, a master mathematician, goes into even more detail. A substantial portion of vol 1 is devoted to the Dirac Distribution.
To answer Alternatively, if you can justify that the above properties are just physics (not math) ...
The properties are math not physics.