$#C+1 = 31 : ~/encyclopedia/old_files/data/H046/H.0406370 Hardy\ANDLittlewood theorem
The Hardy–Littlewood theorem in the theory of functions of a complex variable: If $ a _ {k} \geq 0 $,
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...e Abel summation method is used in conjunction with [[Tauberian theorems | Tauberian theorems]] to demonstrate the convergence of a series.
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$#C+1 = 77 : ~/encyclopedia/old_files/data/T092/T.0902280 Tauberian theorems,
''theorems of Tauberian type''
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...tudy of [[arithmetic function]]s and yields a proof of the [[Prime number theorem]]. It was proved by [[Norbert Wiener]] and his student [[Shikao Ikehara]]
An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form $\sum_{n=1}^\infty a(n) n^{-s}$ wher
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...zero as $t \to \infty$. Established by N. Wiener [[#References|[1]]]. This theorem was generalized to include any commutative locally compact non-compact grou
This theorem is based on the regularity of the [[group algebra]] of a commutative locall
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i) The uniform convergence theorem: for $ f $
ii) The representation theorem: $ f $
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...assigns the sum in the usual sense to any convergent series (an [[Abelian theorem]]). The series
<TR><TD valign="top">[3]</TD> <TD valign="top">Jacob Korevaar (2004). "Tauberian theory. A century of developments". Grundlehren der Mathematischen Wissensc
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...nge theorem|Delange theorem]]; [[Elliott–Daboussi theorem|Elliott–Daboussi theorem]]), led E. Wirsing in 1961 [[#References|[a6]]] to the following result, wh
...the Hardy–Littlewood–Karamata Tauberian theorem (cf. [[Tauberian theorems|Tauberian theorems]]).
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...of the transformation defining the summation method. For example, Cauchy's theorem establishes that $ ( s _ {0} + \dots + s _ {n} )/( n+ 1) \rightarrow s $
...infer the properties of the transformed sequence (see [[Tauberian theorems|Tauberian theorems]]).
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...e., if $x \neq 0$, by the Tauberian theorem (cf. also [[Tauberian theorems|Tauberian theorems]]). This hull is called the (Arveson) spectrum of $x$ and is denot
...contribution of Arveson in this connection is a result that generalizes a theorem of F. Forelli [[#References|[a5]]] that relates the spectral subspaces of $
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...th.org/legacyimages/h/h046/h046420/h04642046.png" />. Pontryagin's duality
theorem states that the mapping
...org/legacyimages/h/h046/h046420/h04642090.png" />. The generalized Bochner
theorem applies [[#References|[4]]], [[#References|[6]]]: The function <img align="
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This system should be regarded as the generalization of Lie's first direct theorem to the case of generalized displacement operators (see [[#References|[3]]]
...lacement operators. This assertion is the analogue of Lie's first converse theorem [[#References|[5]]]. Analogues of Lie's second and third (direct and conver
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theorem it follows that if $ X $
...1</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> H.P. Lotz, "Tauberian theorems for operators on $L^\infty$ and similar spaces" , ''Functional Ana
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...rators, there is the following final result, the so-called equiconvergence theorem: The expansion of a given summable function into the eigenfunctions of a di
is known, then the application of Tauberian theorems enables one to find the asymptotics of $ \lambda _ {k} $.
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According to a theorem of Keldysh, under the assumptions made on the coefficients of $ {\mathcal
...iour of the eigen values, Keldysh [[#References|[3]]] used a new Tauberian theorem due to him.
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