Better interpolation of nonuniform data to uniform grid than with spline

I am resampling non-uniformly sampled data to a uniform grid with a spline interpolation. However, if the signal is of high frequency (no aliasing), the spline interpolation does not perform well. Any ideas how to achieve a better resampling? Thanks!

[Walter Roberson]

Some of the ideas in https://www.mathworks.com/matlabcentral/fileexchange/25135-nufft--nfft--usfft might help.

[Vilnis Liepins]

One more option is to use Extended DFT code available on fileexchange http://se.mathworks.com/matlabcentral/fileexchange/11020-extended-dft

You should run the program NEDFT.m for non-uniformly sampled data and get the DFT on uniform frequency set which corresponds to Matlab FFT set [-ceil((NFFT-1)/2):floor((NFFT-1)/2)]/NFFT where NFFT is the length of the DFT. As there is no aliasing you can select NFFT equal to the length of data segment, means, just one NEDFT iteration will be performed to calculate the DFT from non-uniform data.

To get resampled data apply the Matlab IFFT to the output of NEDFT.