Lectures on Fourier Integrals

By R. V. Churchill, S. Bochner, Morris Tenenbaum, H. Pollard. 1959

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Taking the Human Out of the Loop: A Review of Bayesian Optimization (2016, positive): “However, using recent spectral sampling techniques [20], [94], [125], we can draw an”
Predictive Entropy Search for Efficient Global Optimization of Black-box Functions (2014, methodology): “pose to sample and optimize an analytic approximation to f. We will brieﬂy derive this approximation below, but more detail is given in Appendix A. Given a shift-invariant kernel k, Bochner’s theorem [4] asserts the existence of its Fourier dual s(w), which is equal to the spectral density of k. Let”
Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs (2015, methodology): “We use Bochner’s theorem (Bochner, 1959) to reformulate the covariance function in terms of its frequencies.”
Vector valued reproducing kernel Hilbert spaces and universality (2008, methodology): “The above equation is precisely the content of Bochner’s theorem for operator valued functions of positive type [2, 15].”
An Entropy Search Portfolio for Bayesian Optimization (2014, background): “Given a shift-invariant kernel k , Bochner’s theorem [3] asserts the existence of its Fourier dual s ( w ) , which is equal to the spectral density of k .”