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Which existing methods are capable of performing a smooth curve interpolation with constraints on derivative? What I need it to smooth my evenly spaced (with few missing points) data to get a smooth curve which will have positive first and second derivatives.

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Your question is a bit unfocused and not very clear, it would help with more context. But assuming this is about nonparametric curve fitting, you could try with splines. Since you need positive first and second derivative, you could look into monotone splines, see Monotone, Bounded Local/Spline Regression in R and Are there any inverse distribution graph that looks like this?, Looking for function to fit sigmoid-like curve. Within this posts you will find several useful R packages.

You could also look into generalized additive models (GAM's), based on splines. For instance, R has the cgam package (on CRAN) which fits constrained gam's.

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