Trigonometric identities and sums of separable functions

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A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Only in exceptional cases, it will be possible to represent f exactly by such a sum, see [26,195,208].”
Structure of the Hessian matrix and an economical implementation of Newton’s method in the problem of canonical approximation of tensors (2010, background): “instance, the tensor with the components s i = sin( ) (see [ 12 ]).”
Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation (2012, background): “Let us give an example, taken from [16], for which Assumption 1 does not hold.”
On approximation of functions by exponential sums (2005, background): “This approximation provides an example of an analytic construction of a separated representation as introduced in [9] and discussed in [10,23].”
Algorithms for Numerical Analysis in High Dimensions (2005, unknown): “A proof of this identity and some of its generalizations can be found in [32], along with a discussion of its relationship to the work of Calogero [10, 11] and Milne’s identity [31, 7].”
Multivariate Regression and Machine Learning with Sums of Separable Functions (2009, background): “for all choices of {αj} such that sin(αk − αj) = 0 for all j = k (see [23]), and, thus, only r = d is needed.”