Dear allFor algorithmic trading purposes I need to convert the "actual time" to "volume weighted time" for stock-instruments, i.e. to calculate the intra-day historical "volume time-function". In order to do so I chose the "actual time" to start at 0 and finish at 1, and of course the same for "volume-time" ("real time" opening hours are 9:00-17:30). Volume here refers to the volumes executed during continuous trading of a certain stock.Within this intervall I have a total of 34 points (based in 50days rolling data), which are evualuated from intra-day market-data and they express volumes traded in timebuckets of size 15 minutes. I center these points in the timebuckets respectively and interpolate.In order to retrieve an analytical expression I use FFT, which is pretty convinient and easy to integrate and differentiate. However, this method experiences two problems:1) How do I solve the problem of the boundaries? (when the exchange open and close, time=0 and 1)2) How do I avoid the function to have "over-shoots", which are typical for Fourier Series? Using for example Hermite interpolation for sure looks better, but is not analytical. And using least square interpolation seems dangerous, and it is not easily integrable. Any ideas?Bonus question:1) What is a good way to measure the volatility in the volume profile? I measure the volume in the timebuckets each day, perhaps rolling 50days.Many thanks for replies!Best regardsMikael