In this video, is it noted that Georg Cantor concluded it impossible to decompose all functions as an infinite linear combination of sines and cosines, "because there are only countably many...degrees of freedom in a Fourier Series, and uncountably many degrees of freedom in a function." I would like to ask some questions to understand this better.
Obviously, there are uncountably many possible Fourier Series representations of a function, since there are uncountably many possible expressions for each coefficient in a Fourier Series. It is fairly clear that what is meant by there being only countably many degrees of freedom in a Fourier Series is that a Fourier Series has only countably many terms. However, I don't understand what is meant by there being uncountably many degrees of freedom in a function, in this context. In general, a function's number of degrees of freedom is at most the number of independent variable it contains, which is finite. In fact, the Fourier Series I'm familiar with attempt to represent functions of a single variable (perhaps there is a multivariable extension). What is meant here by the number of degrees of freedom in the function a Fourier Series attempts to represent?
According to Wikipedia, "Through Fourier's research the fact was established that an arbitrary (at first, continuous and later generalized to any piecewise-smooth) function can be represented by a trigonometric series." How are piecewise-smooth functions countable, whereas functions in general are uncountable, so as to solve Cantor's problem?
Why does it even matter that a Fourier Series has only countably many terms for the purpose of a bijection between functions and Fourier Series representations being possible, since as stated above, there are still uncountably many possible Fourier Series representations? It would seem that a countable family of uncountable sets would be more than large enough to stand in a bijective relation to an uncountable set.
A Fourier series is determined by a bisequence (two-sided sequence) of complex numbers. You have one degree of freedom for each of those coefficients: once you’ve chosen each of the countably many coefficients, you’ve completely determined the series, but the series isn’t determined until you’ve made all of those choices. An arbitrary real- or complex-valued function on $\Bbb R$ or $\Bbb C$, however, has $\mathfrak{c}$ degrees of freedom: it isn’t pinned down completely until you have specified its value at every real or complex number.
There are $\mathfrak{c}=2^{\aleph_0}$ complex numbers, so there are $$\left(2^{\aleph_0}\right)^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}=\mathfrak{c}$$ such bisequences. There are also $\mathfrak{c}$ real numbers, so pinning down an arbitrary function from $\Bbb R$ or $\Bbb C$ to $\Bbb R$ or $\Bbb C$ requires specifying $\mathfrak{c}$ real or complex numbers, something that can be done in
$$\mathfrak{c}^{\mathfrak{c}}=\left(2^{\aleph_0}\right)^{\mathfrak{c}}=2^{\aleph_0\cdot\mathfrak{c}}=2^{\mathfrak{c}}$$
ways. And $2^{\mathfrak{c}}>\mathfrak{c}$ by Cantor’s theorem, so there are more functions than there are Fourier series: there are altogether $2^{\mathfrak{c}}$ possible real- or compled-valued functions on $\Bbb R$ or $\Bbb C$, but only $\mathfrak{c}$ of them are nice enough to be represented by Fourier series.
Note that the claim is not that there are only countably many piecewise smooth functions; in fact there are $\mathfrak{c}$ of them. The claim is that each is completely determined by countably many complex parameters.