Periodic Entire Functions

A uniqueness theorem for harmonic functions

Extending a result of N. Wiener, it is shown that functions on the circle with positive Fourier coefficients that are pth power integrable near 0, 1 < P < 2, have Fourier coefficients in lp . The following result was proved (but never published) by Norbert Wiener in the early 1950's. (See [1, pp. 242, 250] and [3].) WIENER'S THEOREM. // YLcneint is the Fourier series of a function f € L1(—7T, 7t) with cn > 0 for all n, and f restricted to a neighborhood (—6,6) of the origin belong to L2(—6,6), then f belongs to L2(—tt,tt). A question which immediately arises in connection with this result is the following: does the theorem remain true if one replaces L2(—6,6) and L2(—tt,tt) in its statement respectively by Lp(—6,6) and Lp(—-k,tt), with 1 < p < oo? In 1969 Stephen Wainger showed, by ingenious counterexamples, that the answer is negative for 1 < p < 2 [4]. If p is an even integer or oo it is very easy to see that the answer is "yes." For every other exponent between 2 and oo it is "no," as was shown in 1975 by Harold S. Shapiro [3]. These negative results have been extended to compact abelian groups [2]. However, the conclusion of Wiener's theorem can be stated equivalently as "then J2 cn < °°-" This suggested the following theorem. THEOREM. If¿~2cneint is the Fourier series of a function f S Lx(—tt, tt) with cn > 0 for all n, and f restricted to a neighborhood (—6,6) of the origin belongs to Lp(—6,6) with 1 < p < 2, then ^cn < °°> where p' = p/(p — 1). PROOF. (See [3, p. 12].) Let h(t) be the 27r-periodic function which for |t| < -k is defined by M-KIA M Si, I 0, 6 < \t\ < it. Received by the editors March 2, 1987. 1980 Mathematics Subject Classification. Primary 42A32, 42A16; Secondary 43A15. 1 The research presented here was supported in part by a grant from the University Research Council of DePaul University. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

/pdf/fourier-series-with-positive-coefficients-53y3iow804.pdf

Fourier series with positive coefficients

In fact, (1.5) holds only if fo(z) is [7, p. 59] a constant multiple of (z-50)-1 sin a(zo). The form of the extremal function in the case p = 2 suggests that for functions which are real for real z we should hope to get better estimates at nonreal points. It is equally clear that for functions which do not vanish in the upper half plane inequality (1.2) can be refined for points in the lower half plane. We prove

/pdf/some-inequalities-concerning-functions-of-exponential-type-4ejw27f8uu.pdf

Some inequalities concerning functions of exponential type

Let G be a homogeneous group with the graded Lie algebra or a noncompact semisimple Lie group with finite center. We define the Fourier transform f of f as a family of operators f (X) = frG (x) (x) dx (7r E G), and we say that f is positive if all f(7r) are positive. Then, we construct an integrable function f on G with positive f and the restriction of f to any ball centered at the origin of G is square-integrable, however, f is not square-integrable on G.

/pdf/functions-on-noncompact-lie-groups-with-positive-fourier-w25xlw8e5p.pdf

Periodic Entire Functions

Citations

A uniqueness theorem for harmonic functions

Fourier series with positive coefficients

Some inequalities concerning functions of exponential type

Functions on noncompact lie groups with positive fourier transforms

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