Does ArcGIS API provide interpolation functions, like if I want to interpolate from my own set of values?
I know that ArcGIS provides functions for interpolation like IDW, Spline and Kriging etc, but can I use those functions to interpolate against my own dataset?
Yes, you can use them against your own code. Perhaps the easiest way is via arcpy, embedded within the documentation for IDW you can see a stand-alone implementation of that particular function. Based on your previous question though, it sounds like you have a specific problem: you want to interpolate the values for a few unknown locations within an existing raster layer. If that is the case, then IDW and the like are not the right approach: they're intended for use against layers where you have a limited number of data points and want to interpolate the surface entirely.
For just filling a few holes, a simpler method would be to just perform a focal operation on the raster and replace any NoData cells with the average value of its surroundings. There's another question which goes through the logic of using this method.
IDW, Kriging etc is available with Spatial Analyst extension and not with the core product. You need to have a license of SA in order to use these functionalities out-of-the-box.
