In Fourier analysis, such as the Fourier transform or the discrete Fourier transform, the sinusoids being fitted to the data are all mutually orthogonal, so there is no distinction between the simple out-of-context dot-product-based projection onto basis functions versus an in-context simultaneous least-squares fit; that is, no matrix inversion is required to least-squares partition the variance between orthogonal sinusoids of different frequencies.
42.
Rather than just taking dot products of the data with sine and cosine waveforms directly, Scargle modified the standard periodogram formula to first find a time delay ? such that this pair of sinusoids would be mutually orthogonal at sample times " t j ", and also adjusted for the potentially unequal powers of these two basis functions, to obtain a better estimate of the power at a frequency, which made his modified periodogram method exactly equivalent to Lomb's least-squares method.
43.
They are written ( hk ! ), and denote the family of planes orthogonal to h \ mathbf { b _ 1 } + k \ mathbf { b _ 2 } + \ ell \ mathbf { b _ 3 }, where \ mathbf { b _ i } are the basis of the reciprocal lattice vectors . ( Note that the plane is not always orthogonal to the linear combination of direct lattice vectors h \ mathbf { a _ 1 } + k \ mathbf { a _ 2 } + \ ell \ mathbf { a _ 3 } because the reciprocal lattice vectors need not be mutually orthogonal . ) By convention, negative integers are written with a bar, as in for " 3.
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