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${\lambda }_j=2\pi j/T$ is a frequency of the length-$T$ discrete Fourier transform. Distinct interior frequencies give orthogonal sine and cosine vectors; for a Gaussian white noise record this makes the corresponding Fourier components independent.

The inverse discrete Fourier transform reconstructs samples from their discrete Fourier transform. Under one common normalization,moving the factor $1/n$ between the forward and inverse transforms gives other standard conventions.

A Laurent operator on the whole integer lattice is a translation-invariant bi-infinite matrix. Under the discrete Fourier transform it becomes multiplication by its symbol, so its $2$-operator norm is the essential supremum of the symbol's modulus.

For $\omega =e^{2\pi i/N}$,This finite geometric sum is the orthogonality relation underlying the discrete Fourier transform.