The mathematical framework describing what you want to do is generally called Functional Analysis. 

Just as we can define different distances between two points in Euclidean space, so can we define distances between two functions of a given functional space (for example, continuous functions, twice differentiable functions, etc).

There is SO MUCH to say about this, it is difficult to propose something without knowing in what space the functions you want to compare are "living".  If you don't know anything about what I'm talking about, a good introductory book would be "Introductory Functional Analysis with Applications" by Erwin Kreyszig (see link below). He starts from real analysis going into functional analysis giving plenty of examples. There is also answers to some of the problems at the end of the book.

http://www-personal.acfr.usyd.edu.au/spns/cdm/resources/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications.pdf

By looking at the topics given by you, I assume that you are working with discrete time functions. i.e. the function values are given at discrete points. So, you have two single column arrays of function values. There is one function 'xcorr'. It will be useful. If the two functions are similar, then correlation value will be close to 1. If opposite, then close to -1. If not related, then 0.

Otherwise you can find the total error between two functions. i.e. sum((f(x)-g(x))^2)

Thank you Sanghvi Sir. your guess is right. I am in fact working with digital images. This function I have already tested. I am looking for some approach which are not yet used in signal processing domain. Thank you once again for your reply.

Dear Dr Rout

I send you my work on the similarity in hydraulics (flow on the weir), I hope it is useful for you

Best reagrds