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Does ArcGIS API provide interpolation functions, like if I want to interpolate from my own set of values?

Edit:
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?

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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.

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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.

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