Nonharmonic Fourier series and spectral theory

Author:
Harold E. Benzinger

Journal:
Trans. Amer. Math. Soc. **299** (1987), 245-259

MSC:
Primary 42A65; Secondary 34B25, 42A20, 47B38

MathSciNet review:
869410

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of using functions to form biorthogonal expansions in the spaces , for various values of . The work of Paley and Wiener and of Levinson considered conditions of the form which insure that is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for . In this paper, rather than imposing an explicit growth condition, we assume that is a multiplier sequence on . Conditions are given insuring that inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, and are shown to be the eigenvalues and eigenfunctions of an unbounded operator which is closely related to a differential operator, generates a strongly continuous group and generates a strongly continuous semigroup. Half-range expansions, involving or on 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.

**[1]**Harold E. Benzinger,*Functions of well-bounded operators*, Proc. Amer. Math. Soc.**92**(1984), no. 1, 75–80. MR**749895**, 10.1090/S0002-9939-1984-0749895-3**[2]**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**, 10.1090/S0002-9947-1983-0682713-4**[3]**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****[4]**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**0007173**, 10.1090/S0002-9904-1942-07797-4**[5]**R. J. Duffin and A. C. Schaeffer,*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc.**72**(1952), 341–366. MR**0047179**, 10.1090/S0002-9947-1952-0047179-6**[6]**M. Ĭ. Kadec′,*The exact value of the Paley-Wiener constant*, Dokl. Akad. Nauk SSSR**155**(1964), 1253–1254 (Russian). MR**0162088****[7]**Norman Levinson,*Gap and Density Theorems*, American Mathematical Society Colloquium Publications, v. 26, American Mathematical Society, New York, 1940. MR**0003208****[8]**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****[9]**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). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, II. MR**0312140****[10]**Harry Pollard,*The mean convergence of non-harmonic series*, Bull. Amer. Math. Soc.**50**(1944), 583–586. MR**0010637**, 10.1090/S0002-9904-1944-08196-2**[11]**D. J. Ralph,*Semigroups of well-bounded operators and multipliers*, Thesis, Univ. of Edinburgh, 1977.**[12]**J. R. Ringrose,*On well-bounded operators. II*, Proc. London Math. Soc. (3)**13**(1963), 613–638. MR**0155185****[13]**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****[14]**Raymond M. Redheffer,*Completeness of sets of complex exponentials*, Advances in Math.**24**(1977), no. 1, 1–62. MR**0447542****[15]**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**, 10.1090/S0002-9947-1983-0690042-8

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
42A65,
34B25,
42A20,
47B38

Retrieve articles in all journals with MSC: 42A65, 34B25, 42A20, 47B38

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869410-0

Article copyright:
© Copyright 1987
American Mathematical Society