## Nonharmonic Fourier series and spectral theory

HTML articles powered by AMS MathViewer

- by Harold E. Benzinger PDF
- Trans. Amer. Math. Soc.
**299**(1987), 245-259 Request permission

## Abstract:

We consider the problem of using functions ${g_n}(x): = exp(i{\lambda _n}x)$ to form biorthogonal expansions in the spaces ${L^p}( - \pi , \pi )$, for various values of $p$. The work of Paley and Wiener and of Levinson considered conditions of the form $\left | {{\lambda _n} - n} \right | \leq \Delta (p)$ which insure that $\{ {g_n}\}$ is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for $p = 2$. In this paper, rather than imposing an explicit growth condition, we assume that $\{ {\lambda _n} - n\}$ is a multiplier sequence on ${L^p}( - \pi , \pi )$. Conditions are given insuring that $\{ {g_n}\}$ inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, ${\lambda _n}$ and ${g_n}$ are shown to be the eigenvalues and eigenfunctions of an unbounded operator $\Lambda$ which is closely related to a differential operator, $i\Lambda$ generates a strongly continuous group and $- {\Lambda ^2}$ generates a strongly continuous semigroup. Half-range expansions, involving ${\text {cos}}{\lambda _n}x$ or ${\text {sin}}{\lambda _n}x$ on $(0, \pi )$ are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.## References

- Harold E. Benzinger,
*Functions of well-bounded operators*, Proc. Amer. Math. Soc.**92**(1984), no. 1, 75–80. MR**749895**, DOI 10.1090/S0002-9939-1984-0749895-3 - Harold Benzinger, Earl Berkson, and T. A. Gillespie,
*Spectral families of projections, semigroups, and differential operators*, Trans. Amer. Math. Soc.**275**(1983), no. 2, 431–475. MR**682713**, DOI 10.1090/S0002-9947-1983-0682713-4 - H. R. Dowson,
*Spectral theory of linear operators*, London Mathematical Society Monographs, vol. 12, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR**511427** - R. J. Duffin and J. J. Eachus,
*Some notes on an expansion theorem of Paley and Wiener*, Bull. Amer. Math. Soc.**48**(1942), 850–855. MR**7173**, DOI 10.1090/S0002-9904-1942-07797-4 - R. J. Duffin and A. C. Schaeffer,
*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc.**72**(1952), 341–366. MR**47179**, DOI 10.1090/S0002-9947-1952-0047179-6 - M. Ĭ. Kadec′,
*The exact value of the Paley-Wiener constant*, Dokl. Akad. Nauk SSSR**155**(1964), 1253–1254 (Russian). MR**0162088** - Norman Levinson,
*Gap and Density Theorems*, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR**0003208** - Raymond E. A. C. Paley and Norbert Wiener,
*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142**, DOI 10.1090/coll/019 - S. K. Pichorides,
*On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov*, Studia Math.**44**(1972), 165–179. (errata insert). MR**312140**, DOI 10.4064/sm-44-2-165-179 - Harry Pollard,
*The mean convergence of non-harmonic series*, Bull. Amer. Math. Soc.**50**(1944), 583–586. MR**10637**, DOI 10.1090/S0002-9904-1944-08196-2
D. J. Ralph, - J. R. Ringrose,
*On well-bounded operators. II*, Proc. London Math. Soc. (3)**13**(1963), 613–638. MR**155185**, DOI 10.1112/plms/s3-13.1.613 - Robert M. Young,
*An introduction to nonharmonic Fourier series*, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**591684** - Raymond M. Redheffer,
*Completeness of sets of complex exponentials*, Advances in Math.**24**(1977), no. 1, 1–62. MR**447542**, DOI 10.1016/S0001-8708(77)80002-9 - Raymond M. Redheffer and Robert M. Young,
*Completeness and basis properties of complex exponentials*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 93–111. MR**690042**, DOI 10.1090/S0002-9947-1983-0690042-8

*Semigroups of well-bounded operators and multipliers*, Thesis, Univ. of Edinburgh, 1977.

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**299**(1987), 245-259 - MSC: Primary 42A65; Secondary 34B25, 42A20, 47B38
- DOI: https://doi.org/10.1090/S0002-9947-1987-0869410-0
- MathSciNet review: 869410