Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions

By Christian Berg, J. Christensen, P. Ressel. 1984

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions (2016, background): “Importantly, the weak and weak-∗ topologies of (cid:77) f coincide on (cid:80) (but not on (cid:77) f ) (Berg et al., 1984, Chap.”
Spatio-temporal variograms and covariance models (2005, background): “For references, see Gangolli (1967a), (1967b), Berg et al. (1984), and Ma (2003b).”
Energy statistics: A class of statistics based on distances (2013, methodology): “According to a characterization theorem of Schoenberg (Berg et al., 1984, Theorem 3.2.2), a function is conditionally negative definite, continuous, symmetric, and takes the value 0 at 0 if and only if it is the negative logarithm of an infinitely divisible characteristic function.”
Some classes of completely monotonic functions (2002, background): “A detailed collection of the most important properties of completely monotonic functions can be found in [35, Chapter IV], and in an abstract setting in [7].”
Radial kernels and their reproducing kernel Hilbert spaces (2010, background): “For a thorough discussion of this topic see [7].”
Kernel-Based Data Fusion and Its Application to Protein Function Prediction in Yeast (2003, background): “Basic algebraic operations such as addition, multiplication and exponentiation preserve the key property of positive semidefiniteness, and thus allow a simple but powerful algebra of kernels.(19) For example, given two kernels K1 and K2, inducing the embeddings Φ1(x) and Φ2(x), respectively, it is p”
Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity (2007, methodology): “Conditionally negative definite kernels and functions play an important role in probability theory, Dirichlet spaces, potential theory, and in relation with isometric embeddings of metric spaces into Hilbert spaces, see Schoenberg [42] and the treatment in [13].”
Clustering with Bregman Divergences (2005, methodology): “The crux of the argument relies on results in harmonic analysis connecting positive definiteness to integral transforms (Berg et al., 1984). In particular, we use a result due to Devinatz (1955) that relates exponentially convex functions to Laplace transforms of bounded non-negative measures.”
A statistical framework for genomic data fusion (2004, background): “Basic algebraic operations such as addition, multiplication and exponentiation preserve the key property of positive semidefiniteness, and thus allow a simple but powerful algebra of kernels (Berg et al., 1984).”
Random Laplace Feature Maps for Semigroup Kernels on Histograms (2014, methodology): “Exponential-Semigroup [1, 9] e ∑d j=1 √ xj+zj ⊕s j=1 √ 1 s e −xwj β 2 √ π w− 3 2 e −β2 4w (Lévy) This Paper”
Some covariance models based on normal scale mixtures (2011, unknown): “However, C1(x, y) = exp ∗(γ(x,0)+ γ(y,0)− γ(x, y)) and C2(x, y) = exp ∗(γ(x,0) + γ(y,0)−Dxy − γ(x, y)), (3) (Dxy)jk = γjj(x,0) + γkk(y,0), are always positive definite functions in Rd, cf. Theorem 2.2 in Berg et al. (1984) for the univariate case.”