I think it is kind of window such as morlet in wavelet application. Go through detailed description, you can do it.

Best wishes,

Saravanan

Interesting question. So you have a bump shape, and you ask what you can do with it? I think you have to revisit the rationale for wavelets. It's signal analysis, and decomposition on a base of functions.

To make it closer to grasp, think of Euclidian spaces of dimension N: what is nice to have there is a base e(1)... e(N). If this base is not orthogonal, then you orthogonalise it by looking at the Gram-Schmidt matrix m(i,j)= and analysing eigenvalues, eigenvectors. Gram-Schmidt orthogonalisation.

Now make the dimension infinite. You have Hilbert spaces.

Assume you have a signal S, with successive values (time series), S(1)...S(N)...

You form vectors of length/dimension L, then you compare such vectors (sliding: S(1+j)...S(L+j)) that's signal analysis.

You can do many things, AR, ARMA, and other models to exploit potential statistics inside the signal.

Now take the bump, call it B(x), with B(x) equal zero outside of compact set C (call C support set of B).

It could be a step up -stationary -step down function. In other terms B(x) =1 on [0,1], and 0 outside, for example.

Define as the integral of function f(x)B(x)dx on a set where these are integrable, say X.

Now take B define as 1 on C, and zero outside. Make B depend on parameter n, with C(n)= [0, 1/n], B(n)(x) =1 if 0