Design a pass-band function (filter) with no transition bands (sharp edges impulse response)

This is a simple project and this problem might be a typical textbook problem however since it is not in my field I just do not know its solution. The project might seem complicated since I do not know all the terminology and the terminology are different in my field. We need to communicate with graphs, Mathematics and using my pdf notations and terminologies. Here are few notes on terminology: There are two domains in the problem the theta domain (usually correspond to frequency) and n-domain (usually correspond to time). The ideal rectangular function is in the theta domain and the sinc is in the n-domain. Your function in n-domain will have a rectangular like function in theta domain and it will allow ripples but have a sharp edges (the jump discontinuity). The ideal function will be rectangular in theta and its transform sinc however since our n=-N:+N and N<40 there will be errors. We could tolerate ripple errors. There are many methods for designing the function (filter) to obtain a particular response since this is not my field of study I will offer some suggestions based on my readings of signal processing and how to design a filter (function in this case). They are just some suggestions. I suggest using the window design method or equi-ripple methods also the elliptic or Chebyshev function (filter). The design constraints are detailed in the pdf but here is a brief list: - I have mixed indices changing from -N:+N and N<40 - A sharp jump and almost a null transition band -A pass-band filter with its start beta_1 and end beta_2 in theta-domain. The values of beta_1 are usually (but not always) beta_1=90:180 deg and beta_2=180:270 degree