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harmonic evaluation – The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures Answer

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harmonic evaluation – The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures

Let $Cleft(mathbb{R}/mathbb{Z}privilege)$ denote the Banach house of steady, $1$-periodic complex-valued features on the unit interval, let $Mleft(mathbb{R}/mathbb{Z}privilege)$ denote its twin, the house of finite omplex Borel measures on $mathbb{R}/mathbb{Z}$, and let $mathcal{A}left(mathbb{D}privilege)$ denote the vector house house of all holomorphic features $f:mathbb{D}rightarrowmathbb{C}$ on the launch unit disk. We equip $mathcal{A}left(mathbb{D}privilege)$ with the topology of compact convergence on $mathbb{D}$—that’s, uniform convergence on each compact subset of $mathbb{D}$. Next, letting $omega$ breathe a constructive actual quantity, outline the organize $omega$ Cauchy remodel $mathscr{C}_{omega}$ because the linear operator $mathscr{C}_{omega}:Mleft(mathbb{R}/mathbb{Z}privilege)rightarrowmathcal{A}left(mathbb{D}privilege)$ given by: $$mathscr{C}_{omega}left{ dmuright} left(zright)overset{textrm{def}}{=}int_{0}^{1}frac{dmuleft(tright)}{left(1-e^{-2pi it}zright)^{omega}},textrm{ }forallleft|zright|<1,textrm{ }forallmuin Mleft(mathbb{R}/mathbb{Z}privilege)$$ Finally, for any $muin Mleft(mathbb{R}/mathbb{Z}privilege)$, let $hat{mu}:mathbb{Z}rightarrowmathbb{C}$ denote the Fourier coefficients of $mu$/ Fourier-Stieltjes remodel of $mu$: $$hat{mu}left(nright)overset{textrm{def}}{=}int_{0}^{1}e^{-2pi int}dmuleft(tright),textrm{ }forall ninmathbb{Z}$$

I’ve been doing fairly a little bit of studying about Cauchy transforms (fractional or in any other case), however I have not been capable of finding mighty relating to the conduct of the remodel with respect to the Fourier coefficients of sequences of parts in $Mleft(mathbb{R}/mathbb{Z}privilege)$. Specifically, let $left{ mu_{m}privilege} _{mgeq1}$ breathe a sequence in $Mleft(mathbb{R}/mathbb{Z}privilege)$, and let:$$f_{m}left(zright)overset{textrm{def}} {=}mathscr{C}_{omega}left{ dmu_{m}privilege} left(zright),textrm{ }forall mgeq1$$

Now, suppose that:

I. As $mrightarrowinfty$, the $f_{m}$s converge compactly over $mathbb{D}$ to a restrict $finmathcal{A}left(mathbb{D}privilege)$.

II. There is a duty $c:mathbb{Z}rightarrowmathbb{C}$ in order that: $$lim_{mrightarrowinfty}sup_{ninmathbb{Z}}left|cleft(nright)-hat{mu}_{m}left(nright)privilege|=0$$

With these hypotheses, does it then succeed that there’s a touchstone $dmu$ in order that each:

i. $c=hat{mu}$

ii. $f=mathscr{C}_{omega}left{ dmuright}$

That is to say: is the pointwise restrict of the Fourier coefficients of the $mu_{m}$s the Fourier coefficients of a touchstone $mu$, and is $f$ the Cauchy remodel of this $dmu$?

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