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"'''Measure algebra'''" may refer to: ...l group with the operation of convolution; see [[measure algebra (harmonic analysis)]];

$#C+1 = 37 : ~/encyclopedia/old_files/data/A011/A.0101390 Algebra of measures, ''measure algebra''

...roup $G$: If $x$ is a function on $G$, summable with respect to the [[Haar measure]], whose Fourier transform does not vanish on the group of characters $\hat This theorem is based on the regularity of the [[group algebra]] of a commutative locally compact group, and on the possibility of [[spect

$#C+1 = 49 : ~/encyclopedia/old_files/data/Q076/Q.0706560 Quasi\AAhinvariant measure A measure on a space that is equivalent to itself under "translations" of this spac

$#C+1 = 38 : ~/encyclopedia/old_files/data/C027/C.0207640 Cylindrical measure ...sure in measure theory on topological vector spaces is a finitely-additive measure $ \mu $

$#C+1 = 45 : ~/encyclopedia/old_files/data/G045/G.0405230 Group algebra of a locally compact group A topological algebra with [[Involution|involution]] formed by certain functions on the group wit

...logical group|topological group]] whose left-invariant [[Haar measure|Haar measure]] is right invariant (equivalently, is invariant under the transformation $ ...athfrak{g}$), where $\mathrm{ad}$ is the adjoint representation of the Lie algebra $\mathfrak{g}$ of $G$. Any compact, discrete or Abelian locally compact gro

...1 = 6 : ~/encyclopedia/old_files/data/H110/H.1100410 Hypergroups, harmonic analysis on locally compact ...monic analysis|Harmonic analysis]]; [[Harmonic analysis, abstract|Harmonic analysis, abstract]]; for the notion of a hypergroup, see also [[Generalized displac

...on [[#References|[a1]]] to characterize the interpolating sequences in the algebra $H ^ { \infty }$ of bounded analytic functions in the open unit disc and to ...{D} = \{ z \in \mathbf{C} : | z | < 1 \}$. Then $\mu$ is called a Carleson measure if there exists a constant $C$ such that $\mu ( S ) \leq C h$ for every sec

...and most important case of a product of spaces, see the article [[Measure|Measure]]. A more general construction is given below. Let $ I $ algebra $ S _ {i} $

...nn, H. Lebesgue, M. Plancherel, L. Fejér, and F. Riesz formulated harmonic analysis as an independent mathematical discipline. ...he problem of the natural limits of the main results of classical harmonic analysis. This problem is based on the following interpretation of an ordinary Fouri

...of locally compact groups (cf. also [[Harmonic analysis, abstract|Harmonic analysis, abstract]]). They play an important role in the duality theories of these ==Fourier–Stieltjes algebra.==

...hfrak { g } \rightarrow \mathfrak { g }$, gives an automorphism of the Lie algebra $\frak g$. The resulting linear representation $\operatorname{Ad} : G \righ ...atorname{Aut} ( \mathfrak{g} )$, the group of all automorphisms of the Lie algebra $\frak g$.

...( x ) ) ^ { 1 / p }$, where $m$ is some left-invariant [[Haar measure|Haar measure]] on $G$. Let $A _ { p } ( G )$ denote the set of all $u \in {\bf C} ^ { G ...y Abelian, $A _ { 2 } ( G )$ is precisely the [[Fourier-algebra(2)|Fourier algebra]] of $G$.

$#C+1 = 41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space ...gical vector space is that of extending a [[Pre-measure|pre-measure]] to a measure. Let $ E $

of positive measure, $ \mathop{\rm mes} E > 0 $, is a complex Borel measure on the unit circle $ \Gamma $

be the Newton potential of a measure $ \mu \geq 0 $

does not contain the support of a measure orthogonal to the polynomials. This condition holds if and only if for any ...of operators adjoint to the operators of multiplication by elements of the algebra.

...additive set functions (cf. also [[Set function|Set function]]; [[Measure|Measure]]) or, more general, concerning finite additive set functions. The pioneer ...ple way. For a fixed set $A$ from a $\sigma$-algebra $\Sigma$, a classical measure $\mu : \Sigma \rightarrow [ 0 , + \infty ]$ gives that for every set $B$ fr

algebra $ \overline {\mathcal B} \; $ if for any finite measure $ \mu $

...role in various problems of boundary properties of functions, in harmonic analysis, in the theory of power series, linear operators, random processes, and in ...g/legacyimages/h/h046/h046320/h04632085.png" /> is a non-negative singular measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or

...ences|[a3]]], W. Arveson [[#References|[a1]]] generalized and expanded the analysis just presented to cover cases when an arbitrary locally compact [[Abelian g b) $\mathcal{X}$ is a [[C*-algebra|$C ^ { * }$-algebra]] and $\{ U _ { t } \} _ { t \in G }$ is a strongly continuous representati

...algebra]]). The most important among these is the symmetric Banach measure algebra $ M (G) $ the algebra of all regular Borel measures on $ G $

..., space of]]). An early construction of a non-associative, non-commutative algebra was given by H. König [[#References|[a6]]]. The main current (2000) direct ...f. also [[Net (directed set)|Net (directed set)]]) converging to the Dirac measure in $\mathcal{D} ^ { \prime } ( \mathbf{R} ^ { n } )$ (cf. also [[Generalize

of square-summable functions with respect to the Haar measure on $ G $( the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on $ G $,

...Further, between the non-degenerate symmetric representations of the group algebra $ L _ {1} ( G) $( constructed with the left Haar measure) and the continuous unitary representations of the group $ G $

[[Category:Classical measure theory]] The term Hausdorff measures is used for a class of [[Outer measure|outer measures]] (introduced for the first time by Hausdorff in {{Cite|Ha}}

algebra $ \mathfrak B $ of subsets and a probability measure $ \mu $

...al group, then $C ^ { * } ( G )$ is isometrically isomorphic to the Banach algebra $C _ { 0 } ( \hat { G } ; \mathbf{C} )$ of all complex-valued continuous fu ...l norm and the pointwise product on $G$, $B ( G )$ is a commutative Banach algebra [[#References|[a4]]].

...abstract|Harmonic analysis, abstract]]; [[Topological algebra|Topological algebra]]). ...he homomorphisms which are usually considered are those with values in the algebra $ C ( E) $

...gebra $ A $ (cf. [[Algebra of functions|Algebra of functions]]) into the algebra $ L ( X) $ A functional calculus is one of the basic tools of general spectral analysis and the theory of Banach algebras and it enables one to use function-analyt

...in abstract harmonic analysis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]). is a locally compact space with a measure $ m $,

...><TR><TD valign="top">[7]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5</TD></TR><TR><TD valign="top" ...a3]]], Sect. 41. The basic observation is that the [[Banach algebra|Banach algebra]] of (continuous) almost-periodic functions on a (topological) group $ G

''measure of a set'' ...f a set for some mass distribution throughout the space. The notion of the measure of a set arose in the theory of functions of a real variable in connection

A [[Topological algebra|topological algebra]] $A$ over the field of complex numbers whose topology is elements being separately continuous for both factors. A Banach algebra is said to be commutative if

...d out an analogue of the parametrization (in terms of zeros and a singular measure on the circle) of an inner function for the matrix-valued case. It turns ou ...ions to more abstract [[Harmonic analysis|harmonic analysis]] and function-algebra settings. A more delicate Beurling–Lax representation theorem has been sh

such that the [[Lie algebra|Lie algebra]] $ \mathfrak g $ is the Lie algebra of $ H $

...then given $f \in L ^ { 1 } ( \mu )$ one can find a set of full [[Measure|measure]] $X _ { f }$ such that for $x$ in this set the averages ==Wiener–Wintner return-time theorem and the Conze–Lesigne algebra.==

of terms of the [[Harmonic series|harmonic series]] In mathematical analysis both convergent and divergent series are used. For the latter various metho

...s a role in very diverse branches of mathematics and physics, above all in analysis and its applications. ...n algebra; in analytic geometry it includes coordinate transformations; in analysis it includes differential and integral transforms and the Fourier integral.

...dy of partial differential equations, the calculus of variations, harmonic analysis, and fractals. ...References|[a9]]] discusses applications of some of the ideas of geometric measure theory in the theory of Sobolev spaces and functions of bounded variation.

there exists a non-trivial measure $ \mu $ Such a measure is called a [[Haar measure|Haar measure]]. It is unique up to multiplication by a constant.

$#C+1 = 165 : ~/encyclopedia/old_files/data/W097/W.0907760 White noise analysis ...developed into a viable framework for stochastic and infinite-dimensional analysis [[#References|[a4]]]–[[#References|[a6]]], with a growing number of appli

...ise. As in the case of, e.g., Banach algebras (cf. [[Banach algebra|Banach algebra]]), where the Gel'fand representation provides an answer, one asks whether ...in measure theory and the [[Poisson formula|Poisson formula]] for bounded harmonic functions on an open disc are special cases of the spectral theorem. The Fr

...erator. In particular, the set of all nuclear operators is an ideal in the algebra $ L ( E) $; Suppose that the algebra $ S ( E) $

.... Even though aviation is more than 50 years old, an instrument that would measure the disturbance of the attack angle of the aircraft wing or the altitude of ...automatic control. In particular, one may mention the methods of frequency analysis; methods based on the first approximation to [[Lyapunov stability theory|Ly