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Theorem of Fourier transforms of Borel measures
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier-Stieltjes transform of a positive finite Borel measure on the real
Bochner's_theorem
Sufficiency theorem for reconstructing signals from samples
1949, p. 448. Nemirovsky, Jonathan; Shimron, Efrat (2015). "Utilizing Bochners Theorem for Constrained Evaluation of Missing Fourier Data". arXiv:1506.03300
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Austrian mathematician (1899–1982)
arguments. In 1933, he defined the Bochner integral, as it is now called, for vector-valued functions. Bochner's theorem on Fourier transforms appeared in
Salomon_Bochner
Isometry group of a compact Riemannian manifold with negative Ricci curvature is finite
Consequently the isometry group of the manifold must be finite. The theorem is a corollary of Bochner's more fundamental result which says that on any connected
Bochner's theorem (Riemannian geometry)
Bochner's_theorem_(Riemannian_geometry)
Characterization theorem found in 1929
In the theory of orthogonal polynomials, Bochner's theorem is a characterization theorem of certain families of orthogonal polynomials as polynomial solutions
Bochner's theorem (orthogonal polynomials)
Bochner's_theorem_(orthogonal_polynomials)
Bimodal function
function g on the real line with g(y) ≥ 0. The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real
Positive-definite_function
Mathematical theorem
x , y ) = K ( x − y ) {\displaystyle K(x,y)=K(x-y)} ) is given by Bochner's theorem. It states that a continuous function K ( x − y ) {\displaystyle K(x-y)}
Mercer's_theorem
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
functional on a nuclear space A , {\displaystyle A,} the Bochner–Minlos theorem (after Salomon Bochner and Robert Adol'fovich Minlos) guarantees the existence
Nuclear_space
Concept in mathematics
{\displaystyle E\in \Sigma } . A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if f n : X → B {\displaystyle f_{n}\colon
Bochner_integral
Mathematical transform that expresses a function of time as a function of frequency
at infinity, i.e., the Riemann–Lebesgue lemma fails for measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform
Fourier_transform
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Formula in differential geometry
^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u)} . Bochner used this formula to prove the Bochner vanishing theorem. As a corollary, if ( M , g ) {\displaystyle
Bochner's_formula
Theorem about holomorphic functions of several complex variables
In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in C n {\displaystyle \mathbb
Bochner's_tube_theorem
}a_{m-n}\lambda _{m}{\overline {\lambda _{n}}}=2(1-|z|^{2})\,\Re \,f(z).} Bochner's theorem Carathéodory, C. (1907), "Über den Variabilitätsbereich der Koeffizienten
Positive_harmonic_function
Function in probability theory
single-argument version of the covariance function can be checked by Bochner's theorem. For a given variance σ 2 {\displaystyle \sigma ^{2}} , a simple stationary
Covariance_function
Singularities of holomorphic functions extend infinitely outward
theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several
Hartogs's_extension_theorem
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Fourier transform of the probability density function
The central result here is Bochner’s theorem, although its usefulness is limited because the main condition of the theorem, non-negative definiteness
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. Function f is almost surely separably valued (or essentially
Bochner_measurable_function
Technique and process of creating visual representations of the interior of a body
Imaging. March 19, 2010. Nemirovsky J, Shimron E (2015). "Utilizing Bochners Theorem for Constrained Evaluation of Missing Fourier Data". arXiv:1506.03300
Medical_imaging
Algebras arising in harmonic analysis
algebra A ( G ) {\displaystyle A({\mathit {G}})} . The generalized Bochner theorem states that a measurable function on G {\displaystyle {\mathit {G}}}
Fourier_algebra
Bounded operators with sub-unit norm
v〉 where Ug is a (strongly continuous) unitary representation (see Bochner's theorem). Replacing v, a rank-1 projection, by a general projection gives
Contraction_(operator_theory)
The isometry group of a Riemannian manifold is a Lie group
Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod. The first
Myers–Steenrod_theorem
Relative importance of certain frequencies in a composite signal
instead of functions. If R x x {\displaystyle R_{xx}} is continuous, Bochner's theorem can be used to prove that its Fourier transform exists as a positive
Spectral_density
Group theory theorem
In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is
Closed-subgroup_theorem
Operation in calculus
this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides
Integral
Bochner's theorem Hamburger moment problem Moment problem Orthogonal polynomials on the unit circle Spectral measure Schur class Szegő limit theorems
Trigonometric_moment_problem
Machine learning technique
is a stronger convergence guarantee by Hoeffding's inequality. By Bochner's theorem, the above construction can be generalized to arbitrary positive definite
Random_feature
Generalization of finite measure to Banach spaces
Lyapunov's theorem has been proved by using the Shapley–Folkman lemma, which has been viewed as a discrete analogue of Lyapunov's theorem. Bochner measurable
Vector_measure
Polynomial sequence
+\beta +1)} . The other solution involves the logarithm function. Bochner's theorem states that the Jacobi polynomials are uniquely characterized as polynomial
Jacobi_polynomials
Duality for locally compact abelian groups
of envelope of topological algebra. Peter–Weyl theorem Cartier duality Stereotype space Bochner's theorem Joint continuousness means here that the map G
Pontryagin_duality
Generalization of the Cauchy integral formula
by Enzo Martinelli (1938) and Salomon Bochner (1943). Formula (53) of the present paper and a proof of theorem 5 based on it have just been published
Bochner–Martinelli_formula
Type of operator in Fourier analysis
problem for Bochner–Riesz multipliers is only partially solved; see also Bochner–Riesz conjecture. Calderón–Zygmund lemma Marcinkiewicz theorem Singular
Multiplier_(Fourier_analysis)
Provides integral formulas for all derivatives of a holomorphic function
Parseval–Gutzmer formula Bochner–Martinelli formula Helffer–Sjöstrand formula Titchmarsh 1939, p. 84 "Gauss's Mean-Value Theorem". Wolfram Alpha Site. Pompeiu
Cauchy's_integral_formula
Theorem about holomorphic functions of several complex variables
This theorem can be seen as a generalization of Bochner's tube theorem to functions defined on tube-like domains whose base is not an open set. Theorem Let
Malgrange–Zerner_theorem
measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. A function f {\displaystyle f} is said to be almost surely
Weakly_measurable_function
Class of stochastic process
positive definiteness of the autocovariance function, it follows from Bochner's theorem that there exists a positive measure μ {\displaystyle \mu } on the
Stationary_process
German mathematician (1930–2011)
In the compact case, the theorem is due to Morrey. The case when there is an analytic Riemannian metric is due to Bochner. Grauert was awarded a fellowship
Hans_Grauert
Mathematical theorem
In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization
Kōmura's_theorem
design Blocking (statistics) Blumenthal's zero–one law BMDP – software Bochner's theorem Bonferroni correction Bonferroni inequalities – redirects to Boole's
List_of_statistics_articles
In functional analysis, a Hilbert space
field of statistical learning theory because of the celebrated representer theorem which states that every function in an RKHS that minimises an empirical
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
American mathematician (1943–2024)
implicit function theorem, and many authors have attempted to put the logic of the proof into the setting of a general theorem. Such theorems are now known
Richard_S._Hamilton
Hungarian and American mathematician and physicist (1903–1957)
the application of this work was instrumental in his mean ergodic theorem. The theorem is about arbitrary one-parameter unitary groups t → V t {\displaystyle
John_von_Neumann
Class of nonparametric methods
x')=h(x-x')} with x ∈ R b {\displaystyle x\in \mathbb {R} ^{b}} . Then Bochner's theorem guarantees the existence of a unique finite Borel measure μ {\displaystyle
Kernel embedding of distributions
Kernel_embedding_of_distributions
Theorem in differential geometry
splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian
Splitting_theorem
Expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold
In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic
Bochner–Kodaira–Nakano identity
Bochner–Kodaira–Nakano_identity
This is a list of misnamed theorems in mathematics. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object)
List_of_misnamed_theorems
Branch of differential geometry
to Z×Z. Myers theorem. If a complete Riemannian manifold has positive Ricci curvature then its fundamental group is finite. Bochner's formula. If a compact
Riemannian_geometry
Japanese mathematician
mathematician working on differential geometry who introduced the Bochner–Yano theorem. He also published a classical book about geometric objects (i.e
Kentaro_Yano_(mathematician)
Concept in mathematics
the Eells−Sampson theorem together with an extension of the Siu–Corlette Bochner formula, they were able to prove new rigidity theorems for lattices. Existence
Harmonic_map
Type of mathematical functions
inverse function theorem, and implicit function theorems also hold. The Weierstrass preparation theorem serves as an implicit function theorem for complex
Function of several complex variables
Function_of_several_complex_variables
(f(A))}}} This is a consequence of the Hahn-Banach theorem and generalizes the mean value theorem for integrals of real-valued functions: If V = R {\displaystyle
Pettis_integral
Mathematical concept
-z|\geq \delta }{f(\zeta )-f(z) \over \zeta -z}\,d\zeta .} By Cauchy's theorem the right-hand side tends to 0 uniformly as ε, and hence δ, tends to 0
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
American-Israeli mathematician
famous theorem that there are infinitely many primes. Furstenberg pursued his doctorate at Princeton University under the supervision of Salomon Bochner. In
Hillel_Furstenberg
Icelandic mathematician (1927–2023)
proved the principal theorems for this transform, the inversion formula, the Plancherel theorem and the analog of the Paley–Wiener theorem. Sigurdur Helgason
Sigurður Helgason (mathematician)
Sigurður_Helgason_(mathematician)
Theorem in harmonic analysis
→ ∞ {\displaystyle |\xi |\to \infty } due to the dominated convergence theorem. Now, if f {\displaystyle f} is an arbitrary integrable function, it may
Riemann–Lebesgue_lemma
Chinese-American mathematician (born 1949)
partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors
Shing-Tung_Yau
American mathematician (1915–1994)
MR 0008629. Loomis, Lynn H. (1943). "The converse of the Fatou theorem for positive harmonic functions". Trans. Amer. Math. Soc. 53 (2): 239–250
Lynn_Harold_Loomis
Function that "converges" to periodicity
with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any
Almost_periodic_function
quotes as of August 2026[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
Normed vector space that is complete
(1998) Theorem 1.12.11, p. 112 in Megginson (1998) Theorem 2.5.16, p. 216 in Megginson (1998). see II.A.8, p. 29 in Wojtaszczyk (1991) see Theorem 2.6.23
Banach_space
American mathematician (born 1950)
domain leading to the same conclusion. These rigidity theorems are complemented by their existence theorem for harmonic maps on noncompact domains, as a simple
Richard_Schoen
American mathematician
1948 from Princeton University under Salomon Bochner with thesis Generalizations of Some Classic Theorems to the Case of Functions of Several Variables
Harry_Rauch
American mathematician (1911–2004)
known for the Cameron–Martin theorem and for his 1948 book Several complex variables, co-authored with Salomon Bochner. He was born on June 4, 1911,
W._T._Martin
Mode of convergence of an infinite series
} , or the divergent harmonic series. According to the Riemann series theorem, any conditionally convergent series can be permuted so that its sum is
Absolute_convergence
Rådström's embedding theorem to identify convex and compact valued correspondences with subsets of a real Banach space, over which Bochner integration is straightforward
Integral_of_a_correspondence
Awarded every year by the American Mathematical Society
Academic Press. ISBN 9780080873732. Edwards, Harold M. (1977). Fermat's Last Theorem. Graduate Texts in Mathematics. Vol. 50. Springer New York. ISBN 978-0-387-90230-2
Leroy_P._Steele_Prize
Mathematical problem in classical harmonic analysis
Hardy spaces, and another proof, due to Salomon Bochner relies upon the Riesz–Thorin interpolation theorem. For p = 1 and infinity, the result is not true
Convergence_of_Fourier_series
Japanese mathematician (1915–1997)
vanishing theorem Kodaira–Spencer mapping Kodaira dimension Kodaira surface Kodaira embedding theorem Kodaira's classification of singular fibers Bochner–Kodaira–Nakano
Kunihiko_Kodaira
French mathematician (1921–2016)
mathématique. 4 vols., Springer-Verlag 1998–2001. Commutation theorem for traces Plancherel theorem for spherical functions Standard L-function "Décès de Roger
Roger_Godement
Mathematical Reviews (Moedomo and J. J. Uhl, Jr. "Radon-Nikodym theorems for the Bochner and Pettis integrals" published in the Pacific Journal of Mathematics
Moedomo_Soedigdomarto
American mathematician (1912–1992)
lions in the Sahara desert. One "proof" parodies the Bolzano–Weierstrass theorem: The Bolzano–Weierstrass Method. Bisect the desert by a line running N–S
Ralph_P._Boas_Jr.
American mathematician
(graph theory) Cheeger–Müller theorem Collapsing manifold L² cohomology Riemannian geometry Soul theorem Splitting theorem Faculty Profile 1984 U.S. and
Jeff_Cheeger
Kind of mathematical function
However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of a map Y → π X , {\displaystyle
Measurable_function
Generalization of a positive-definite matrix
between positive definite kernels and RKHS is given by the following theorem Theorem— Every reproducing kernel is positive-definite, and every positive
Positive-definite_kernel
American tennis player and mathematician (1916-2008)
students include Robert McCallum Blumenthal and Richard M. Dudley. Hunt's theorem states that for a large class of positive kernels V {\displaystyle V} satisfying
Gilbert_Hunt
On weak solutions of differential equations
solution is smooth enough to be qualified as a classical solution. Several theorems have been proposed for different types of PDEs. Let U {\displaystyle U}
Regularity_theory
2001 film by Ron Howard
geometry and partial differential equations, such as the Nash embedding theorem or his proof of Hilbert's nineteenth problem, work which he did in his
A_Beautiful_Mind
American mathematician
Algebra: Basics, Practice, and Theory: Published by Elsevier Science in 1988. Theorems and Counterexamples in Mathematics (with John M. H. Olmsted): Published
Bernard_Russell_Gelbaum
Polish mathematician
operators". He is also well known for Mikusinski's cube, the Antosik–Mikusinski theorem, and Mikusinski convolution algebra. Mikusiński died in Katowice in 1987
Jan_Mikusiński
Theorem in probability theory
(6). For convenience (see the proof below using the optional stopping theorem) and to specify the relation of the sequence (Xn)n∈ N {\displaystyle \mathbb
Wald's_equation
Generalized function whose value is zero everywhere except at zero
Joseph Fourier. Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur (1822) in the form: f
Dirac_delta_function
Italian-born American mathematician (1923–2023)
of a theorem by K. Jörgens" as that which he was "most proud of". At the 1954 International Congress of Mathematicians, Calabi announced a theorem on how
Eugenio_Calabi
Italian mathematician (1924–1980)
Notably he proved the Andreotti–Frankel theorem, the Andreotti–Grauert theorem, the Andreotti–Vesentini theorem and introduced, jointly with François Norguet
Aldo_Andreotti
Whose values lie in an infinite-dimensional vector space
for example, X {\displaystyle X} is a Hilbert space); see Radon–Nikodym theorem A curve is a continuous map of the unit interval (or more generally, of
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Type of integration
He also proved the change of variables theorem for multiple Bochner integrals and Fubini's theorem for Bochner integrals using Daniell integration. The
Daniell_integral
Chinese mathematician
Xiangyu; Zhu, Langfeng (2011). "On the Ohsawa–Takegoshi extension theorem and the twisted Bochner–Kodaira identity". Comptes Rendus Mathematique. 349 (13–14):
Xiangyu_Zhou
Baltic German mathematician
for the idea and method for his classic 1904 proof of the Well-ordering theorem from an "Axiom of choice", which has become an integral part of modern
Erhard_Schmidt
Tensor in differential geometry
geometric and topological consequences, as in Myers's theorem and related comparison theorems. In dimension three, the Ricci tensor determines the full
Ricci_curvature
American mathematician (1926–2003)
crucial role. As a result of Eells and Sampson's (subsequential) convergence theorem, they were able to prove the existence of harmonic maps in any homotopy
Joseph_H._Sampson
bound on the second largest eigenvalue λ 2 {\displaystyle \lambda _{2}} . Theorem (Cheeger bound): 1 − 2 Φ ≤ λ 2 ≤ 1 − Φ 2 2 . {\displaystyle 1-2\Phi \leq
Cheeger_bound
Japanese mathematician
is positive. Later, Kobayashi and Takushiro Ochiai proved some rigidity theorems for Kähler manifolds. In particular, if M is a closed Kähler manifold and
Shoshichi_Kobayashi
English polymath (1642–1727)
generalised the binomial theorem to any real number, introduced the Puiseux series, was the first to state Bézout's theorem, classified most of the cubic
Isaac_Newton
Polish American mathematician
University in 1947 (at the age of 22) under the supervision of Salomon Bochner. He was on the faculty of Caltech from 1948 to 1956, before becoming a
Samuel_Karlin
German mathematician
solved a special integral equation of Abelian type. The Herglotz–Noether theorem stated by Herglotz (1909) and independently by Fritz Noether (1909), was
Gustav_Herglotz
}}_{j+1}^{\alpha _{j-1}+1}\land \cdots \land d{\bar {\zeta }}_{n}^{\alpha _{n}+1}} Theorem 1 (Andreotti and Norguet). For every function f ∈ A(D), every point z ∈
Andreotti–Norguet_formula
Swedish mathematician (born 1928)
theorem (1962), and establishing the almost everywhere convergence of Fourier series for square-integrable functions (now known as Carleson's theorem)
Lennart_Carleson
intersections of k faces have codimension at least k. Andreotti–Norguet formula Bochner–Martinelli formula Bergmann, Stefan (1936), "Über eine Integraldarstellung
Bergman–Weil_formula
the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also, again combined
Kähler_identities
Swedish mathematician
sufficient condition for quasi-analyticity, now called the Denjoy–Carleman theorem. As a corollary, he obtained a sufficient condition for the determinacy
Torsten_Carleman
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BOCHNERS THEOREM
BOCHNERS THEOREM
BOCHNERS THEOREM
BOCHNERS THEOREM
BOCHNERS THEOREM
BOCHNERS THEOREM
BOCHNERS THEOREM
BOCHNERS THEOREM
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