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TheJacobiPolynomialsOrthogonalityMeasureIsTheSpectralDensityOfAFamilyOfGaussianProcesses.tm
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<TeXmacs|2.1.4>
<style|<tuple|generic|boring-white|framed-theorems>>
<\body>
<doc-data|<doc-title|Fourier Transform of the Jacobi Weight
Function>|<doc-author|<author-data|<author-name|Stephen
Crowley>|<\author-affiliation>
<date|>
</author-affiliation>>>|<doc-date|>>
<\theorem>
For <math|\<alpha\>,\<beta\>\<gtr\>-1>, the Fourier transform of the
Jacobi weight function
<\equation*>
w<around|(|x|)>=<around|(|1-x|)><rsup|\<alpha\>>*<around|(|1+x|)><rsup|\<beta\>>*<space|1em><text|on
><around|[|-1,1|]>
</equation*>
is given by
<\equation*>
<wide|w|^><around|(|t|)>=2<rsup|\<alpha\>+\<beta\>+1>*\<Gamma\>*<around|(|\<alpha\>+1|)>*\<Gamma\>*<around|(|\<beta\>+1|)>*<frac|J<rsub|\<alpha\>+\<beta\>+1>*<around|(|2*t|)>|<around|(|2*t|)><rsup|\<alpha\>+\<beta\>+1>>*e<rsup|i*t>
</equation*>
where <math|J<rsub|\<nu\>>> denotes the Bessel function of the first kind
of order <math|\<nu\>>.
</theorem>
<\proof>
<with|font-series|bold|1. Initial Setup and Conditions:><next-line>The
conditions <math|\<alpha\>,\<beta\>\<gtr\>-1> ensure:
<\itemize>
<item>The weight function is integrable on <math|<around|[|-1,1|]>>
<item>The Beta function <math|B<around|(|\<alpha\>+1,\<beta\>+1|)>> is
well-defined
<item>The resulting Bessel function expression converges
</itemize>
We need to compute the Fourier transform:
<\equation>
<wide|w|^><around|(|t|)>=<big|int><rsub|-1><rsup|1><around|(|1-x|)><rsup|\<alpha\>>*<around|(|1+x|)><rsup|\<beta\>>*e<rsup|-i*x*t>*<space|0.17em>d*x
</equation>
<with|font-series|bold|2. Change of Variables:><next-line>Let
<\equation>
u=<frac|1+x|2>
</equation>
then:
<\equation>
<tabular|<tformat|<table|<row|<cell|x>|<cell|=2*u-1>>|<row|<cell|d*x>|<cell|=2*d*u>>|<row|<cell|<text|when
>x>|<cell|=-1,u=0>>|<row|<cell|<text|when >x>|<cell|=1,u=1>>>>>
</equation>
The integral becomes:
<\equation>
<wide|w|^><around|(|t|)>=2<rsup|1+\<alpha\>+\<beta\>>*<big|int><rsub|0><rsup|1><around|(|1-u|)><rsup|\<alpha\>>*u<rsup|\<beta\>>*e<rsup|-i*<around|(|2*u-1|)>*t>*<space|0.17em>d*u
</equation>
<with|font-series|bold|3. Exponential Splitting:>
<\equation>
e<rsup|-i*<around|(|2*u-1|)>*t>=e<rsup|-i*2*u*t>*e<rsup|i*t>
</equation>
<with|font-series|bold|4. Connection to Hypergeometric
Functions:><next-line>The integral now takes the form:
<\equation>
2<rsup|1+\<alpha\>+\<beta\>>*e<rsup|i*t>*<big|int><rsub|0><rsup|1><around|(|1-u|)><rsup|\<alpha\>>*u<rsup|\<beta\>>*e<rsup|-i*2*u*t>*<space|0.17em>d*u
</equation>
This integral relates to the generalized hypergeometric function
<math|<rsub|1>F<rsub|1>> through:
<\equation>
<big|int><rsub|0><rsup|1>u<rsup|\<beta\>>*<around|(|1-u|)><rsup|\<alpha\>>*e<rsup|-i*2*u*t>*<space|0.17em>d*u=B<around|(|\<alpha\>+1,\<beta\>+1|)><rsub|1>*F<rsub|1><around|(|\<beta\>+1;\<alpha\>+\<beta\>+2;-2*i*t|)>
</equation>
<with|font-series|bold|5. Transformation to Bessel
Functions:><next-line>The hypergeometric function transforms to Bessel
form through three key steps:
First, applying the Kummer transformation:
<\equation>
<rsub|1>F<rsub|1><around|(|a;b;z|)>=e<rsup|z><rsub|1>*F<rsub|1><around|(|b-a;b;-z|)>
</equation>
Second, using the limiting relation between confluent hypergeometric and
Bessel functions:
<\equation>
J<rsub|\<nu\>><around|(|z|)>=<frac|<around|(|z/2|)><rsup|\<nu\>>|\<Gamma\>*<around|(|\<nu\>+1|)>><rsub|0>*F<rsub|1>(;\<nu\>+1;-z<rsup|2>/4)
</equation>
Finally, through Hankel's contour integral representation:
<\equation>
J<rsub|\<nu\>><around|(|z|)>=<frac|z<rsup|\<nu\>>|2<rsup|\<nu\>>*\<Gamma\>*<around|(|\<nu\>+1|)>><rsub|0>*F<rsub|1><around*|(|;\<nu\>+1;-<frac|z<rsup|2>|4>|)>
</equation>
These transformations yield:
<\equation>
<big|int><rsub|0><rsup|1><around|(|1-u|)><rsup|\<alpha\>>*u<rsup|\<beta\>>*e<rsup|-i*2*u*t>*<space|0.17em>d*u=B<around|(|\<alpha\>+1,\<beta\>+1|)>*<frac|J<rsub|\<alpha\>+\<beta\>+1>*<around|(|2*t|)>|<around|(|2*t|)><rsup|\<alpha\>+\<beta\>+1>>
</equation>
<with|font-series|bold|6. Final Result:><next-line>Combining all terms:
<\equation>
<wide|w|^><around|(|t|)>=2<rsup|1+\<alpha\>+\<beta\>>*B<around|(|\<alpha\>+1,\<beta\>+1|)>*<frac|J<rsub|\<alpha\>+\<beta\>+1>*<around|(|2*t|)>|<around|(|2*t|)><rsup|\<alpha\>+\<beta\>+1>>*e<rsup|i*t>
</equation>
Using the Beta function relation <math|B<around|(|a,b|)>=<frac|\<Gamma\><around|(|a|)>*\<Gamma\><around|(|b|)>|\<Gamma\>*<around|(|a+b|)>>>,
we obtain our final result:
<\equation>
<wide|w|^><around|(|t|)>=2<rsup|\<alpha\>+\<beta\>+1>*\<Gamma\>*<around|(|\<alpha\>+1|)>*\<Gamma\>*<around|(|\<beta\>+1|)>*<frac|J<rsub|\<alpha\>+\<beta\>+1>*<around|(|2*t|)>|<around|(|2*t|)><rsup|\<alpha\>+\<beta\>+1>>*e<rsup|i*t>
</equation>
The <math|e<rsup|i*t>> term carries the essential phase information of
the Fourier transform, completing the proof.
</proof>
</body>
<\initial>
<\collection>
<associate|magnification|1.2>
<associate|page-height|auto>
<associate|page-medium|paper>
<associate|page-type|letter>
<associate|page-width|auto>
</collection>
</initial>