The orthogonality condition implies the relation known as Coulson's theorem:
2.
One can find the coefficients either through the application of an inner product or by the discrete orthogonality condition.
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In particular, the orthogonality conditions that you mention is really an extension of the orthonormality of spherical harmonics, i . e .:
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Thus, in advanced mathematics, the word " perpendicular " is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal.
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Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives an often useful result for " approximate " coefficients,
6.
Among the 2 " A " & minus; 1 possible solutions of the algebraic equations for the moment and orthogonality conditions, the one is chosen whose scaling filter has extremal phase.
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However, in practice, \ vec { r } _ 1, \ vec { r } _ 2 are noisy and the orthogonality condition of the attitude matrix ( or the direction cosine matrix ) is not preserved by the above procedure.
8.
The weights w _ i and v _ j are determined such that the function interpolates N given points ( \ mathbf { c } _ i, f _ i ) ( for i = 1, 2, \ dots, N ) and fulfills the d + 1 orthogonality conditions
9.
Where ~ \ delta _ { kk'} is the Kronecker delta . ( In the last step, the summation is trivial if k = k', where it is 1 + 1 + ?" ?" ?" = " N ", and otherwise is a geometric series that can be explicitly summed to obtain zero . ) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.
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