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Riesz-Thorin Theorem
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin. This theorem bounds the norms of linear maps acting between Lp spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L2 which is a Hilbert space, or to L1 and L∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Sublinear Function
A sublinear function (or functional, as is more often used in functional analysis), in linear algebra and related areas of mathematics, is a function f : V → F displaystyle f:Vrightarrow mathbf F on a vector space V over F, an ordered field (e.g
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Young's Convolution Inequality
In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,[1] named after William Henry Young.Contents1 Statement1.1 Euclidean Space 1.2 Generalizations2 Applications 3 Proof3.1 Proof by Hölder's inequality4 Sharp constant 5 Notes 6 External linksStatement[edit] Euclidean Space[edit] In real analysis, the following result is called Young's convolution inequality:[2] Suppose f is in Lp(Rd) and g is in Lq(Rd) and 1 p + 1 q = 1 r + 1 displaystyle frac 1 p + frac 1 q = frac 1 r +1 with 1 ≤ p, q, r ≤ ∞
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Hilbert Transform
In mathematics and in signal processing, the Hilbert transform
Hilbert transform
is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function 1 / ( π t ) displaystyle 1/(pi t) : H ( u ) ( t ) = 1 &
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Cauchy Principal Value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.Contents1 Formulation 2 Distribution theory2.1 Well-definedness as a distribution 2.2 More general definitions3 Examples 4 Nomenclature 5 See also 6 ReferencesFormulation[edit] Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:1) For a singularity at the finite number b: lim ε → 0 + [ ∫ a b − ε f ( x ) d x + ∫ b
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Schwartz Space
In mathematics, Schwartz space
Schwartz space
is the function space of all functions whose derivatives are rapidly decreasing (defined rigorously below). This space has the important property that the Fourier transform
Fourier transform
is an automorphism on this space. This property enables one, by duality, to define the Fourier transform
Fourier transform
for elements in the dual space of S, that is, for tempered distributions
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Cauchy–Schwarz Inequality
In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.[1] The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved by Viktor Bunyakovsky (1859)
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Self-adjoint Operator
In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product ⟨ ⋅ , ⋅ ⟩ displaystyle langle cdot ,cdot rangle is a linear map A (from V to itself) that is its own adjoint: ⟨ A v , w ⟩ = ⟨ v , A w ⟩ displaystyle langle Av,wrangle =langle v,Awrangle . If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is Hermitian, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers
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Hardy–Littlewood Maximal Operator
In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis
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Calderón–Zygmund Theory
In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function  f  : Rd → C, where Rd denotes Euclidean space and C denotes the complex numbers, the lemma gives a precise way of partitioning Rd into two sets: one where  f  is essentially small; the other a countable collection of cubes where  f  is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of  f , wherein  f  is written as the sum of "good" and "bad" functions, using the above sets.Contents1 Covering lemma 2 Calderón–Zygmund decomposition 3 See also 4 ReferencesCovering lemma[edit]Let  f  : Rd → C be integrable and α be a positive constant
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Hardy-Littlewood Maximal Operator
In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis
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Lorentz Space
In mathematical analysis, Lorentz spaces, introduced by George Lorentz in the 1950s,[1][2] are generalisations of the more familiar L p displaystyle L^ p spaces. The Lorentz spaces are denoted by L p , q displaystyle L^ p,q . Like the L p displaystyle L^ p spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the L p displaystyle L^ p norm does
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Marcel Riesz
Marcel Riesz (Hungarian: Riesz Marcell [ˈriːs ˈmɒrt͡sɛll]; 16 November 1886 – 4 September 1969) was a Hungarian-born mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund
Lund
(Sweden).Contents1 Biography 2 Mathematical work2.1 Classical analysis 2.2 Functional-analytic methods 2.3 Potential theory, PDE, and Clifford algebras 2.4 Students3 Publications 4 References 5 External linksBiography[edit] Marcel Riesz was born in Győr, Hungary
Hungary
(Austria-Hungary); he was the younger brother of the mathematician Frigyes Riesz. He obtained his PhD at Eötvös Loránd University
Eötvös Loránd University
under the supervision of Lipót Fejér
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Sequence Space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence
Sequence
spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important sequence spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers
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Schauder Basis
In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder
Juliusz Schauder
in 1927,[1][2] although such bases were discussed earlier. For example, the Haar basis was given in 1909, and G
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