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The results presented in this paper are refinements of some results presentedin a previous paper. Three such refined results are presented. The first onerelaxes one of the basic hypotheses assumed in the previous paper, and thusextends the results obtained there to a wider class of real functions. Theother two relate to a closer examination of the issue of the representabilityof real functions by their Fourier coefficients. As was shown in the previouspaper, in many cases one can recover the real function from its Fouriercoefficients even if the corresponding Fourier series diverges almosteverywhere. In such cases we say that the real function is still representableby its Fourier coefficients. Here we establish a very weak condition on theFourier coefficients that ensures the representability of the function by thosecoefficients. In addition to this, we show that any real function that isabsolutely integrable can be recovered almost everywhere from, and hence isrepresentable by, its Fourier coefficients, regardless of whether or not itsFourier series converges. Interestingly, this also provides proof for aconjecture proposed in the previous paper.

Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients