By Kehe Zhu (auth.)

ISBN-10: 1441988009

ISBN-13: 9781441988003

Several typical Lp areas of analytic capabilities were extensively studied some time past few a long time, together with Hardy areas, Bergman areas, and Fock areas. The phrases “Hardy areas” and “Bergman areas” are by means of now typical and good validated. however the time period “Fock areas” is a distinct story.

Numerous very good books now exist with reference to Hardy areas. a number of books approximately Bergman areas, together with a number of the author’s, have additionally seemed some time past few many years. yet there was no publication out there in regards to the Fock areas. the aim of this e-book is to fill that void, in particular whilst many leads to the topic are whole by means of now. This ebook offers vital effects and strategies summarized in a single position, in order that new comers, specifically graduate scholars, have a handy connection with the subject.

This ebook includes proofs which are new and easier than the prevailing ones within the literature. particularly, the e-book avoids using the Heisenberg crew, the Fourier rework, and the warmth equation. this assists in keeping the must haves to a minimal. a customary graduate direction in every one of genuine research, advanced research, and useful research could be enough education for the reader.

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Numerous ordinary Lp areas of analytic services were largely studied some time past few a long time, together with Hardy areas, Bergman areas, and Fock areas. The phrases “Hardy areas” and “Bergman areas” are by means of now regular and good verified. however the time period “Fock areas” is a special tale. various first-class books now exist as regards to Hardy areas.

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**Additional resources for Analysis on Fock Spaces**

**Sample text**

1 of Boas’ book [38] for a detailed proof of this result. The section about lattices in the complex plane is completely elementary. Whenever we really use lattices later on, we restrict our attention to square lattices, although many arguments can easily be adapted to arbitrary lattices, even to sequences that behave like lattices. 15 looks peculiar to the reader, but it is critical for the study of Hankel operators in Chap. 8. Pseudodifferential operators constitute an important subject by itself, and there is extensive literature about them.

In this brief section, we give its definition and produce a unitary representation based on pseudodifferential operators. The Heisenberg group H is the set C × R (or R2 × R) with the following group operation: (z, s) ⊕ (w,t) = (z + w, s + t − Im (zw)), where z and w are complex and s and t are real. More generally, if n is any positive integer, the Heisenberg group Hn is the set Cn × R with the group operation (z, s) ⊕ (w,t) = (z + w, s + t − Im ( z, w )), where z = (z1 , · · · , zn ), w = (w1 , · · · , wn ), and z, w = z1 w1 + · · · + zn wn .

Therefore, sup{| f (z)| : f ∞,α ∞,α . that | f (z)| ≤ eα |z| 2 /2 ≤ 1} ≤ eα |z| 2 /2 . for all f 38 2 Fock Spaces On the other hand, the function f (w) = kz (w) is a unit vector in Fα∞ and kz (z) = 2 eα |z| /2 . Thus, we actually have sup{| f (z)| : f ≤ 1} = eα |z| 2 /2 ∞,α . This proves the case for p = ∞. 7 consist of more than constant multiples of reproducing kernels. For example, if f is any polynomial normalized so that f ∞,α = 1, then 1 = sup | f (z)|e−α |z| = | f (z0 )|e−α |z0 | 2 /2 2 /2 z∈C for some z0 ∈ C because in this case we have lim f (z)e−α |z| 2 /2 = 0.

### Analysis on Fock Spaces by Kehe Zhu (auth.)

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