Estimating Low Frequencies (including amplitude and phase) For the past month I've been chasing the DFT/FFT rabbit.
Along the way I've been cut-n-pasting the name of other rabbits to chase later:
Autocorrelation, Phase Vocoder, Parabolic Approximation, Wavelet Transform

Here's my problem.
When using an FFT, the lower frequencies are always going to be missing.
I actually found the same problem described here:
Problems with the FFT/IFFT

Frankly it is worse than that. The energy of these frequencies are not missing at all.
They are dissolved into the first several bins, making these bins semi-useless for interpolation.

For now, I simply want a general approximation of the Low Frequencies, including amplitude and phase. In FFT terms, I want bin #0.05, 0.10, 0.15, 0.20, 0.25 ...up to say around bin 3.50. Realistically, I understand that anything to small, say 0.05, would almost amount to Linear Regression. But in my mind, 0.5 or 0.75 should be doable.

I even toyed with changing k to a double (which was 'cough' "interesting"):
best window for close frequency components

When Moses came down from the mountain, he did not carry tablets of stone on which it was commanded that in the equation: F = k*(df) = k*sample_rate/N, that k has to be an integer.

Anyway, I've decided it is time to try something else.
So, what rabbit should I chase next?
(even if it is close approximation)

PS: recommending a bigger sample is not the answer - I'm already sampling my available limit.

Edited to add pics (hoping for a possible new answer):

Test Case (I want a Blue approximation, even if lousy):


Rectangle (of course, lousy Green and so-so Red):


Gaussian 3.0 (only Red can be Interpolated, but decent):


Smaller Gaussian gives a lousy Green Interpolation and decent Red.


Edited to answer general questions (I'm sorry I did not do this to begin with):

What's the mathematical description of the data in the plot?
I'm guessing Sum of Sinusoids (with noise).

What's the sample rate?
All of my samples are at different rates, as in, each entire file size is used.
Rather than Hertz/periods/cycles, the measurements are in deserialized data units.
Most often I work with 8-byte doubles, but sometimes 4-byte integers.
Therefore, with doubles, you can say that my Fs = fileBytes/8

How many samples can you capture?
One each per file.

What's the application?
I have no clue - it's for a client.
In the frequency representation given by the FFT there are no frequencies missing and their is no binning of frequencies.

FFT decomposes a discrete signal in to a sum of sinusoids of certain frequency and phase. This representation is exact and has the same energy as the original signal. No information is lost, as you can do an inverse FFT and recover the original signal exactly. The binning is coming from the discretisation of the signal, not the FFT.

Also for a finite length signal, there are no frequencies lower than the one with the period equal to the length of the signal. This is from the mathematical definition of the spectrum. Put more simply, the DFT/FFT 'assumes' the signal is repeating. Zero padding of the signal therefore increases the period and therefore the lowers the first frequency.

If there are lower frequencies due to some prior knowledge, e.g. if we sampled a sine wave over half its period, then this information must be included.

One solution may be to roll your own spectrum and scale the frequency components with period longer than that of the signal. The assumption being that the low frequency component continues.

For example, let $f_S$ be the frequency you are interested in, with period = $1/f_s &gt; 1$ longer than the period of the signal which is normalised to 1.

Now you convolve the signal with the sine and cosine waves at that frequency. No zero padding.

$$f_e = \sum_t^N f[t] \times cos[2\pi f_s t]$$
$$f_o = \sum_t^N f[t] \times sin[2\pi f_s t]$$

Then multiply by $1/ f_s$ to account for the assumption the signal outside of the sample continues like this, and multiply by 1/N to account for number of samples.

$$f_e' = f_e \times 1/ f_s \times 1/N$$
$$f_o' = f_o \times 1/ f_s \times 1/N$$

Amplitude and phase are then given in the usual way.

This can be done at whatever frequency graduations you desire.

Edit:
Changes with regard to first comment.

Edit 2:
Answer riddled with errors. Updated.

Edit 3: Add factoring for N

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Edits welcome if I have this wrong.