Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse

By S. Bochner. 1933

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 Mean Embedding of Distributions: A Review and Beyonds (2016, background): “When the kernel k is translation invariant on Rd, i.e., k(x,x′) = ψ(x − x′), x,x′ ∈ Rd, the kernel can be characterized by Bochner’s theorem (Bochner 1933), which states that any bounded continuous kernel k is the inverse Fourier transform of some finite non-negative Borel measure.”
Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels (2014, background): “The starting point of Rahimi & Recht (2007) is a celebrated result that characterizes the class of positive definite functions: Theorem 1 (Bochner (1933)).”
Kernel methods in machine learning (2007, neutral): “A variation of Bochner’s theorem, stated by Wendland [152], then tells us that the measure corresponding to h has a nonvanishing density υ with respect to the Lebesgue measure, that is, that k can be written as k(x,x′) = ∫ e−i〈x−x ′,ω〉υ(ω)dω = ∫ e−i〈x,ω〉e−i〈x′,ω〉υ(ω)dω.”
Random Laplace Feature Maps for Semigroup Kernels on Histograms (2014, methodology): “Their construction is based on a classical characterization of the family of positive-definite functions on Rd given by Bochner’s Theorem [2].”
On potentially negative space time covariances obtained as sum of products of marginal ones (2008, methodology): “Instead of using Bernstein’s formula (5), one can use the Lévy-Khinchine formula for conditionally positive definite functions (see e.”
Local Group Invariant Representations via Orbit Embeddings (2016, methodology): “Bochner’s theorem establishes Fourier transform as a bijective map from finite non-negative Borel measures on Rd to positive definite functions on Rd. Applying it to shift-invariant positive definite kernels one gets k(x, x′) = k̃(x− x′) = ∫ Rd e−i(x−x ′)>ωp(ω)dω, ∀x, x′, (5) where p(·) is the uniqu”
Convolution roots of radial positive definite functions with compact support (2004, background): “for all nite systems of complex numbers c1;::: ;cn and points x1;::: ;xn 2 R d . Conversely, Bochner’s theorem [ 10 ] states that every such function ’ is the Fourier transform of a probability measure F on R d . If ’ is integrable then F has a continuous density given by f = F 1 d ’. Since a positi”
Randomized numerical linear algebra: Foundations and algorithms (2020, methodology): “A classic result from analysis, Bochner’s theorem (Bôchner 1933), gives a characterization of these kernels.”
Characterization and construction of radial basis functions (2001, background): “On the other hand, Bochner’s characterization provides direct proofs for translation–invariant and possibly nonradial functions, but is not applicable to conditionally positive definite functions.”
Stationary covariances associated with exponentially convex functions (2003, background): “By a classical theorem of Bochner (1933), a continuous function is positive definite if and only if it is of the form j(t) ¼ ð eir t dF(r), t 2 Rd , (3) where F is a non-negative measure on Rd .”
Solving PDEs with radial basis functions * (2015, unknown): “Following Bochner (1933), we will first show that the RBF matrix A = φ(‖x1 − x1‖) φ(‖x1 − x2‖) · · · φ(‖x1 − xN‖) φ(‖x2 − x1‖) φ(‖x2 − x2‖) · · · φ(‖x2 − xN‖) ... ... ... φ(‖xN − x1‖) φ(‖xN − x2‖) · · · φ(‖xN − xN‖) (2.5) is guaranteed to be non-singular for GA RBFs, no matter how the nodes…”