Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.

However, isn't there a flaw? When I integrate over the continuous signal, there is something in between the samples (i.e., the sinc interpolation). So the document above shows only that they are equivalent after the continuous signal has been sampled (which is intuitively trivial since the sampled signal contains only Diracs)!

Is it possible to derive a relationship without the signal being sampled? Suppose for simplicity only the DC component. $\tau$ is the period, $T_s$ the (hypothetical) sampling rate, $N=\tau/T_s$ the number of samples and $x[n]=x(n T_s)$:

$$ \int_0^{\tau} x(t) dt = \int_0^{\tau} \sum_{n=0}^{N-1} x(nT_s) \mathrm{sinc}(t-nT_s) dt = \dots \sum_{n=0}^{N-1} x[n] $$

Obviously this is not true, due to sinc interpolation, so tomething is missing on the right side.

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.

However, isn't there a flaw? When I integrate over the continuous signal, there is something in between the samples (i.e., the sinc interpolation). So the document above shows only that they are equivalent after the continuous signal has been sampled (which is intuitively trivial since the sampled signal contains only Diracs)!

Is it possible to derive a relationship without the signal being sampled? Suppose for simplicity only the DC component. $\tau$ is the period, $T_s$ the (hypothetical) sampling rate, $N=\tau/T_s$ the number of samples and $x[n]=x(n T_s)$:

$$ \int_0^{\tau} x(t) dt = \int_0^{\tau} \sum_{n=0}^{N-1} x(nT_s) \mathrm{sinc}(t-nT_s) dt = \dots \sum_{n=0}^{N-1} x[n] $$

Obviously this is not true, due to sinc interpolation, so something is missing on the right side.

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.

However, isn't there a flaw? When I integrate over the continuous signal, there is something in between the samples (i.e., the sinc interpolation). So the document above shows only that they are equivalent after the continuous signal has been sampled (which is intuitively trivial since the sampled signal contains only Diracs)!

Is it possible to derive a relationship without the signal being sampled? Suppose for simplicity only the DC component. $\tau$ is the period, $T_s$ the (hypothetical) sampling rate, $N=\tau/T_s$ the number of samples and $x[n]=x(n T_s)$:

$$ \int_0^{\tau} x(t) dt = \int_0^{\tau} \sum_{n=0}^{N-1} x(nT) \mathrm{sinc}(t-nT) dt = \dots \sum_{n=0}^{N-1} x[n] $$

Obviously this is not true, due to sinc interpolation, so tomething is missing on the right side.

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.

However, isn't there a flaw? When I integrate over the continuous signal, there is something in between the samples (i.e., the sinc interpolation). So the document above shows only that they are equivalent after the continuous signal has been sampled (which is intuitively trivial since the sampled signal contains only Diracs)!

Is it possible to derive a relationship without the signal being sampled? Suppose for simplicity only the DC component. $\tau$ is the period, $T_s$ the (hypothetical) sampling rate, $N=\tau/T_s$ the number of samples and $x[n]=x(n T_s)$:

$$ \int_0^{\tau} x(t) dt = \int_0^{\tau} \sum_{n=0}^{N-1} x(nT_s) \mathrm{sinc}(t-nT_s) dt = \dots \sum_{n=0}^{N-1} x[n] $$

Obviously this is not true, due to sinc interpolation, so tomething is missing on the right side.