- #1
hamsterman
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A book I'm reading says that the set of continuous functions is an Euclidean space with scalar product defined as [itex]<f,g> = \int\limits_a^bfg[/itex] and then defines Fourier series as [itex]\sum\limits_{i\in N}c_ie_i[/itex] where [itex]c_i = <f, e_i>[/itex] and [itex]e_i[/itex] is some base of the vector space of continuous functions.
What I want to know is why that scalar product function was chosen. Would any function that has the properties of scalar multiplication do? Are there any other possible definitions and do they change some properties of the series?
Another thing I find odd is, why do I see differences on the Internet? Only trigonometric Fourier series are talked about. Does general decomposition of a function into a series go by some other name?
Thanks for your time.
What I want to know is why that scalar product function was chosen. Would any function that has the properties of scalar multiplication do? Are there any other possible definitions and do they change some properties of the series?
Another thing I find odd is, why do I see differences on the Internet? Only trigonometric Fourier series are talked about. Does general decomposition of a function into a series go by some other name?
Thanks for your time.