The analysis of a complex waveform expressed as a series of sinusoidal functions, the frequencies of which form a harmonic series.
- His theorem, now called the Riesz - Fischer theorem, which he proved in 1907, is fundamental in the Fourier analysis of Hilbert space.
- The nonclassified areas of focus were pure mathematics, applied mathematics, statistics, logistics, applied areas of analysis such as Fourier analysis, and computation.
- He did considerable work developing new mathematical tools particularly in the area of Fourier analysis and generalised functions.
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