In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of...

existence of a unique reproducing kernel by the Riesz Representation theorem. Mercer's theorem can also be extended to address the vector-valued setting and...

\mathbb {R} ^{m}} , and can be infinite dimensional. This follows from Mercer's theorem, which states that a continuous, symmetric, positive definite kernel...

\operatorname {Tr} (T^{2})=\sum _{i=1}^{\infty }\lambda _{i}(T^{2})<\infty .} Mercer's theorem provides another example of a trace class operator. That is, suppose...

measure ensuring the function k {\displaystyle k} satisfies Mercer's condition. Mercer's theorem is similar to a generalization of the result from linear...

proved Mercer's theorem, which states that positive-definite kernels can be expressed as a dot product in a high-dimensional space. This theorem is the...

The covariance function KX satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set λk, ek(t) of eigenvalues...

Lumer–Phillips theorem (semigroup theory) Marcinkiewicz theorem (functional analysis) Mazur–Ulam theorem (normed spaces) Mercer's theorem (functional analysis)...

linear solve. Mercer's theorem Kernel methods Schölkopf, Bernhard; Herbrich, Ralf; Smola, Alex J. (2001). "A Generalized Representer Theorem". In Helmbold...

kernel matrix are thus non-linear in the location of the data (see Mercer's theorem and the kernel trick for more information). Blind signal separation...

{\displaystyle M} . The infinite-dimensional analog of this statement is Mercer's theorem. If M {\displaystyle M} is the Gram matrix of vectors v 1 , … , v n...

associated three orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient construction of the Cartesian representation of the...

Fundamental theorem of Hilbert spaces Gram–Schmidt process Hellinger–Toeplitz theorem Hilbert space Inner product space Legendre polynomials Matrices Mercer's theorem...

f\in H} , all the φ j {\displaystyle \varphi _{j}} are continuous. Mercer's theorem then states that sup s , t ∈ [ 0 , 1 ] | Σ ( s , t ) − ∑ j = 1 K λ...

) {\displaystyle K(x,y)} is a hermitian kernel which, according to Mercer's theorem, can be represented as K ( x , y ) = ∑ λ n φ n ( x ) φ n ( y ) ¯ ,...

exist since K 11 {\displaystyle K_{11}} is positive semidefinite. By Mercer's theorem, we can decompose the kernel matrix as a Gram matrix: K = X T X {\textstyle...

arguments, continuous and positive semidefinite in a certain sense). Mercer's theorem asserts that K {\displaystyle K} can then be expressed as K ( x , y...

McKay, known for his discovery of monstrous moonshine James Mercer, proved Mercer's theorem. Edward Milne, the study of radiative equilibrium, the structure...

discovered a new class of theorems of Tauberian type, today known as Karamata's tauberian theorems. He also worked on Mercer's theorems, Frullani integral,...

valued reproducing kernel hilbert spaces of integrable functions and mercer theorem," Anal. Appl. (Singap.), 4 Mauricio A. Álvarez, Lorenzo Rosasco, and...

In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: Fejér's Theorem—Let f : R → C {\displaystyle f:\mathbb...

theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two...

Littlewood showed that if the Riemann hypothesis is true, then the prime number theorem follows and obtained the error term. This work won him his Trinity fellowship...

point, if the latter two both exist.: 6  Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker...

mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory...

The original object of Mercer’s paper was to characterize the kernels which are definite in the sense of Hilbert, but Mercer soon found that the class...

the distance exceeds that number; this statement is the Cauchy–Hadamard theorem. Note that r = 1/0 is interpreted as an infinite radius, meaning that f...

The Joe Rogan Experience, where he attempted to debunk the Pythagorean theorem, claimed he can kill gravity, said he does not believe in the number zero...

California, Berkeley alumni Plateau's problem Simons' formula Simons' theorem Coy, Peter (April 11, 2019). "Meet Marilyn Simons, the Bricklayer's Daughter...

function as a counterexample to the converse of the intermediate value theorem: the function takes on every real value in each interval on the real line...