Segal-Bargmann transform and Fock space

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Problem 465 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.5.1 "Geometry of Hilbert Space"

(a)(b)

Apply the orthogonalization process to the sequence of monomials $1,z,z^2,\ldots$ in the pre-Hilbert spaces of polynomials with the following scalar products:

(a)

$\displaystyle \left\langle P, Q \right\rangle =\iint_{\left|z\right|\le R}P(z)\overline{Q(z)}\, dx\, dy$ ;

(b)

$\displaystyle \left\langle P, Q \right\rangle =\iint_{\mathbb {C}}P(z)\overline{Q(z)}e^{-\left|z\right|^2}\, dx\, dy$ .

Problem 466 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.5.1 "Geometry of Hilbert Space"

Express the linear functional $F_x(f)=f(x)$ in the spaces of Problem 465 in the form of a scalar product.

Problem 467 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.5.1 "Geometry of Hilbert Space"

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References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.