Effects of linear interpolation of a time series on its frequency spectrum Situation
In order to synchonisize different time series i have to apply linear interpolation on them. After the interpolation and synchronization the
signal is transferred into its frequency domain for further analysis.
The interpolation step should therefore not alter the frequency domain information.
Experiment
Via python I created a signal of white noise.
The second signal is based on the first but shifted by half of the original sampling interval. The third signal was created by oversampling the original signal by the factor 5.0.
Afterward all three signals were transferred to the frequency domain (Bottom of the Figure)
Questions

Why does the curve of the shifted signal decline for higher frequencies?
Why does the curve of the over sampled signal decline for higher frequencies? Though the over sampled signal has the exact same shape as the original signal.
What can i improve in order to interpolate the original signal at specific points but minimize the effects on its frequency spectrum?

My python code for the experiment and the plot: https://pastebin.com/iz08Hiud
Many thanks in advance!
Update
Thank you all for your comprehensive adn helpfull answers, especially robert bristow-johnson and howpow2 for directing me on the right path. I am using now an implementation of the sinc kernel interpolation, which shows an almost perfect frequency response.
See the bottom row of the plot for the results in form of the error between the original frequency spectrum and the frequency spectrums of the interpolated signals
Duane Wise and i wrote a paper back in the 90s that we presented to an AES convention that spelled out how to model time-domain polynomial interpolation (of which linear interpolation is an example) in the frequency domain.

i think you can get a copy here: Performance of Low-Order Polynomial Interpolators in the Presence of Oversampled Input