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**An introduction to nonharmonic Fourier series.
Revised edition.**
*(English)*
Zbl 0981.42001

Orlando, FL: Academic Press. xiv, 234 p. (2001).

This is a revised first edition of the author’s celebrated book with the same title [Pure and Applied Mathematics, 93. Academic Press, Inc. New York-London (1980; Zbl 0493.42001)]. Since its publication in 1980 it has been a much used textbook on nonharmonic Fourier series, i.e., generalized Fourier expansions with respect to sets of complex exponentials \(\{e^{i\lambda_nx}\}_{n=1}^\infty\). The book does not claim to give a complete treatment of all topics, but it gives a very nice and well written account of the basic facts and several advanced results, as well as references for further reading. For many years it has been one of the favorite items on my bookshelves, and I welcome the new version warmly. In 1980 there was no reason to belive that Riesz bases and frames would become very important tools in mathematical analysis and applications like signal processing and image processing. Young’s treatment of frames was the first after the introduction of frames in 1952, and his clear presentation has a large part of the honour that researchers saw the potential in frames in the middle of the eighties. Today, a search in Zentralblatt gives hundreds of papers concerning frames and Riesz bases. The purpose of the revised edition is to bring the book up to date, mainly by extending the notes and the references considerably. Also, several new exercises have been added, for example, containing recent results which can be proved by the methods developed in the book.

The book starts with a chapter on bases in Banach spaces. After presenting the general results concerning (Schauder) bases and the associated coefficient functionals, the important Riesz bases in Hilbert spaces are discussed in detail. The stability theorem by Paley and Wiener is proved, as well as Kadec’s 1/4-theorem for families of complex exponentials. Chapter 2 concerns entire functions of exponential type. Weierstrass’ factorization theorem is proved, as well as Hadamard’s factorization theorem for an entire function of finite order. Chapter 3 is about completeness of sets of complex exponentials \(\{e^{i\lambda_nx}\}\), mainly in \(L^p(-A,A)\) and \(C(-A,A)\). Several classical results due to T. Carleman, N. Levinson and A. Ingham are proved. Exact sequences are introduced (a sequence which is complete, but fails to be complete if any element is removed) and also excess of a complete sequence (the number of elements that should be removed in order to obtain an exact sequence). It is proved that \(\{e^{\pm i(n-1/4)x}\}_{n=1}^\infty\) is exact in \(L^2(-\pi,\pi)\) but not a Riesz basis; this shows that the constant 1/4 in Kadec’s 1/4-theorem is best possible. Chapter 4 is about interpolation and bases in Hilbert spaces. Given a sequence \(\{f_n\}\) in a Hilbert space \((H,\langle \cdot, \cdot \rangle)\) and a scalar sequence \(\{c_n\}\), consider the problem of finding \(f\in H\) such that \(\{c_n\}=\{\langle f, f_n \rangle \}\). It is proved that if a solution exists, then a unique solution of minimal norm exists. Riesz-Fischer sequences (sequences for which a solution exists for all \(\{c_n\}\in \ell^2\)) are characterized, as well as Bessel sequences (sequences \(\{f_n\}\) for which \(\sum |\langle f, f_n \rangle|^2<\infty\) for all \(f\in H\)), and the results are applied to families of complex exponentials in \(L^2(-A,A)\). In section 7, the basic theory of frames and its relationship to Riesz bases is developed; and in section 8, the results are connected to families of complex exponentials. The book ends with a large collection of notes and references.

The book starts with a chapter on bases in Banach spaces. After presenting the general results concerning (Schauder) bases and the associated coefficient functionals, the important Riesz bases in Hilbert spaces are discussed in detail. The stability theorem by Paley and Wiener is proved, as well as Kadec’s 1/4-theorem for families of complex exponentials. Chapter 2 concerns entire functions of exponential type. Weierstrass’ factorization theorem is proved, as well as Hadamard’s factorization theorem for an entire function of finite order. Chapter 3 is about completeness of sets of complex exponentials \(\{e^{i\lambda_nx}\}\), mainly in \(L^p(-A,A)\) and \(C(-A,A)\). Several classical results due to T. Carleman, N. Levinson and A. Ingham are proved. Exact sequences are introduced (a sequence which is complete, but fails to be complete if any element is removed) and also excess of a complete sequence (the number of elements that should be removed in order to obtain an exact sequence). It is proved that \(\{e^{\pm i(n-1/4)x}\}_{n=1}^\infty\) is exact in \(L^2(-\pi,\pi)\) but not a Riesz basis; this shows that the constant 1/4 in Kadec’s 1/4-theorem is best possible. Chapter 4 is about interpolation and bases in Hilbert spaces. Given a sequence \(\{f_n\}\) in a Hilbert space \((H,\langle \cdot, \cdot \rangle)\) and a scalar sequence \(\{c_n\}\), consider the problem of finding \(f\in H\) such that \(\{c_n\}=\{\langle f, f_n \rangle \}\). It is proved that if a solution exists, then a unique solution of minimal norm exists. Riesz-Fischer sequences (sequences for which a solution exists for all \(\{c_n\}\in \ell^2\)) are characterized, as well as Bessel sequences (sequences \(\{f_n\}\) for which \(\sum |\langle f, f_n \rangle|^2<\infty\) for all \(f\in H\)), and the results are applied to families of complex exponentials in \(L^2(-A,A)\). In section 7, the basic theory of frames and its relationship to Riesz bases is developed; and in section 8, the results are connected to families of complex exponentials. The book ends with a large collection of notes and references.

Reviewer: Ole Christensen (Lyngby)

### MSC:

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |

30D20 | Entire functions of one complex variable (general theory) |

42Cxx | Nontrigonometric harmonic analysis |