Inner Products, Norms and Orthogonality

We’ve derived several “Fourier series” (including the trigonometric Fourier series, the Fourier cosine series and the Fourier sine series), and we will be discussing yet another Fourier series (the exponential Fourier series) in the next chapter. That’s a lot of series, with a lot of different formulas to learn. Fortunately, there is a fairly general framework for describing all of these (and other) Fourier series. This framework is based on an operation with pairs of functions analogous to the dot product operation in vector analysis. Using this, and other ideas from vector analysis, we can then easily describe all Fourier series and derive a single simple formula for computing “the components of ...

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