Positive definite functions and generalizations, an historical survey

By James Stewart. 1976

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Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “8 in Wendland (2005) states that: if (supp(Λ))◦ 6= ∅, then k(x, y) = ψ(x− y) is strictly pd.”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “The above definition is a generalization of integrally strictly positive definite functions (Stewart, 1976, Section 6): ∫∫ M k(x, y)f(x)f(y) dx dy > 0 for all f ∈ L2(Rd), which is the strictly positive definiteness of the integral operator given by the kernel.”
Kernel methods in machine learning (2007, neutral): “For further historical details, see the review of Stewart [130] or Berg at al.”
Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere (2020, methodology): “Definition (Strict positive definiteness (Bochner, 1992; Stewart, 1976)).”
A Kernel Loss for Solving the Bellman Equation (2019, background): “More discussion on ISPD kernels can be found in Stewart (1976) and Sriperumbudur et al. (2010).”
Regularization Networks and Support Vector Machines (2000, background): “In [91] it is reported that the positivity of the operator associated to K is equivalent to the statement that the kernel K is positive definite, that is the matrix Kij = K(xi,xj) is positive definite 15 We remind the reader that positive definite operators in L2 are self-adjoint operators such that”
A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations (2006, unknown): “[42].”
Kernel techniques: From machine learning to meshless methods (2006, background): “More precisely, any symmetric positive definite kernelK generates a distance metric d : Ω×Ω → [0,∞) via d2(x, y) := K(x, x) − 2K(x, y) +K(y, y) for all x, y ∈ Ω (2.10) on a general set (Schoenberg 1937, Stewart 1976).”
Reconstruction of Multivariate Functions from Scattered Data (2003, background): “6 Conditionally Positive Definite Functions Positive definite functions (formerly defined in a slightly different way) have a long history that is nicely surveyed by Stewart [29](Stewart:76-1).”
Measuring Sample Quality with Kernels (2017, background): “4.1) or integrally strictly positive deﬁnite (ISPD, Stewart, 1976, Sec.”
Kernel Choice and Classifiability for RKHS Embeddings of Probability Distributions (2009, background): “The above definition is a generalization of integrally strictly positive definite functions [ 17, Section 6 ]: R R k(x;y)f(x)f(y)dxdy > 0 for all f 2 L2(Rd),”
Universality , Characteristic Kernels and RKHS Embedding of Measures (2011, background): “This definition is a generalization of integrally strictly positive definite functions on R d (Stewart, 1976, Section 6): R d k(x, y) f (x) f (y) dx dy > 0 for all f ∈ L 2 (R d ), which is the strictly positive definiteness of the integral operator given by the kernel. c-, cc-, c 0 -and L p…”