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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