Nonharmonic Fourier series and spectral theory
Author:
Harold E. Benzinger
Journal:
Trans. Amer. Math. Soc. 299 (1987), 245259
MSC:
Primary 42A65; Secondary 34B25, 42A20, 47B38
MathSciNet review:
869410
Fulltext 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. Halfrange 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 wellbounded operators.
 [1]
Harold
E. Benzinger, Functions of wellbounded
operators, Proc. Amer. Math. Soc.
92 (1984), no. 1,
75–80. MR
749895 (86b:47030), http://dx.doi.org/10.1090/S00029939198407498953
 [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
(84b:47038), http://dx.doi.org/10.1090/S00029947198306827134
 [3]
H.
R. Dowson, Spectral theory of linear operators, London
Mathematical Society Monographs, vol. 12, Academic Press Inc.
[Harcourt Brace Jovanovich Publishers], London, 1978. MR 511427
(80c:47022)
 [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 (4,97e), http://dx.doi.org/10.1090/S000299041942077974
 [5]
R.
J. Duffin and A.
C. Schaeffer, A class of nonharmonic Fourier
series, Trans. Amer. Math. Soc. 72 (1952), 341–366. MR 0047179
(13,839a), http://dx.doi.org/10.1090/S00029947195200471796
 [6]
M.
Ĭ. Kadec′, The exact value of the PaleyWiener
constant, Dokl. Akad. Nauk SSSR 155 (1964),
1253–1254 (Russian). MR 0162088
(28 #5289)
 [7]
Norman
Levinson, Gap and Density Theorems, American Mathematical
Society Colloquium Publications, v. 26, American Mathematical Society, New
York, 1940. MR
0003208 (2,180d)
 [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 (98a:01023)
 [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
(47 #702)
 [10]
Harry
Pollard, The mean convergence of nonharmonic
series, Bull. Amer. Math. Soc. 50 (1944), 583–586. MR 0010637
(6,48f), http://dx.doi.org/10.1090/S000299041944081962
 [11]
D. J. Ralph, Semigroups of wellbounded operators and multipliers, Thesis, Univ. of Edinburgh, 1977.
 [12]
J.
R. Ringrose, On wellbounded operators. II, Proc. London Math.
Soc. (3) 13 (1963), 613–638. MR 0155185
(27 #5124)
 [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, 1980. MR 591684
(81m:42027)
 [14]
Raymond
M. Redheffer, Completeness of sets of complex exponentials,
Advances in Math. 24 (1977), no. 1, 1–62. MR 0447542
(56 #5852)
 [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
(84c:42047), http://dx.doi.org/10.1090/S00029947198306900428
 [1]
 H. E. Benzinger, Functions of wellbounded operators, Proc. Amer. Math. Soc. 92 (1984), 7580. MR 749895 (86b:47030)
 [2]
 H. E. Benzinger, E. Berkson and T. A. Gillespie, Spectral families of projections, semigroups and differential operators, Trans. Amer. Math. Soc. 275 (1983), 431475. MR 682713 (84b:47038)
 [3]
 H. R. Dowson, Spectral theory of linear operators, London Math. Soc. Mono., No. 12, Academic Press, New York, 1978. MR 511427 (80c:47022)
 [4]
 R. J. Duffin and J. J. Eachus, Some notes on an expansion theorem of Paley and Wiener, Bull. Amer. Math. Soc. 48 (1942), 850855. MR 0007173 (4:97e)
 [5]
 R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341366. MR 0047179 (13:839a)
 [6]
 M. I. Kadec, The exact value of the PaleyWiener constant, Soviet Math. Dokl. 5 (1964), 559561. MR 0162088 (28:5289)
 [7]
 N. Levinson, Gap and density theorems, Amer. Math. Soc. Colloq. Publ., Vol. 26, Amer. Math. Soc., Providence, R. I., 1940. MR 0003208 (2:180d)
 [8]
 R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., Vol. 19, Amer. Math. Soc., Providence, R. I., 1934. MR 1451142 (98a:01023)
 [9]
 S. K. Pichorides, On the best value of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165179. MR 0312140 (47:702)
 [10]
 H. Pollard, The mean convergence of nonharmonic series, Bull. Amer. Math. Soc. 50 (1944), 583586. MR 0010637 (6:48f)
 [11]
 D. J. Ralph, Semigroups of wellbounded operators and multipliers, Thesis, Univ. of Edinburgh, 1977.
 [12]
 J. R. Ringrose, On wellbounded operators. II, Proc. London Math. Soc. (3) 13 (1963), 613638. MR 0155185 (27:5124)
 [13]
 R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, 1980. MR 591684 (81m:42027)
 [14]
 R. M. Redheffer, Completeness of sets of complex exponentials, Adv. in Math. 24 (1977), 162. MR 0447542 (56:5852)
 [15]
 R. M. Redheffer and R. M. Young, Completeness and basis properties of complex exponentials, Trans. Amer. Math. Soc. 277 (1983), 93111. MR 690042 (84c:42047)
Similar Articles
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:
http://dx.doi.org/10.1090/S00029947198708694100
PII:
S 00029947(1987)08694100
Article copyright:
© Copyright 1987 American Mathematical Society
