Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The full Bochner theorem on real reductive groups   The major results of Barker $[3.],$ leading to the spherical Bochner theorem
and its (spherical) extension, were made possible through the spherical
transform theory of Trombi-Varadarajan $[14.]$ and were greatly controlled by
the non-availability of the full (non-spherical) Harish-Chandra Fourier
transform theory on a general connected semisimple Lie group, $G.$ Sequel to
the recently announced results of Oyadare $[13.],$ where the full image of the
Schwartz-type algebras, $\mathcal{C}^{p}(G),$ under the full Fourier transform
is computed to be
$\mathcal{C}^{p}(\widehat{G}):=\{(\widehat{\xi_{1}})^{-1}\cdot h\cdot
(\widehat{\xi_{1}})^{-1}:h\in\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})\}$
with $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})$ given as the
Trombi-Varadarajan image of $\mathcal{C}^{p}(G//K),$ the present paper now
gives the full Bochner theorem for $G$ by lifting the results of $[3.]$ to full
non-spherical status. An extension of the full Bochner theorem to all of
$\mathcal{C}^{p}(G),$ $1\leq p\leq2,$ is established. It is also conjectured
that every positive-definite distribution $T$ on $G$ which corresponds to a
Bochner measure $\mu$ on ${\mathfrak{F}}^{\epsilon}$ extends uniquely to an
element of $\mathcal{C}^{p}(G)'$ if and only if $T$ can be expressed as a
finite sum of derivatives of a class of functions exclusively parameterized by
members of ${\mathfrak{F}}^{\epsilon}$ and $supp\;
(\mu)\subset{\mathfrak{F}}^{\epsilon},$ with $\epsilon=(\frac{2}{p})-1$ for all
$1\leq p\leq2.$ This gives the non-spherical abstract version of the extension
theorem for any positive-definite distribution on $G.$ Our results confirm the
one-to-one correspondence between tempered invariant positive-definite
distributions and the Bochner measures of the case $SU(1,1)/\{\pm1\}$ (as
computed in Barker $[5.]$) for all $G.$

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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg