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We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-\pi\lambda x^2}$ by entire functions of exponential type.
Metric spaces and completely monotone functions,
I. J. Schoenberg, · 1938
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Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen II,
B. Sz.-Nagy, · 1939
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Théorie des Distributions
L. Schwartz, · 1951
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On one-sided approximation by trigonometric polynomials,
T. Ganelius, · 1956
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Trigonometric Series
A. Zygmund, · 1959
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Linear Partial Differential Operators
L. Hörmander, · 1964
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Inequalities
G. H. Hardy, G. Polya, and J. E. Littlewod, · 1967
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Singular Integrals and Differentiability Properties of Functions
E. Stein, · 1970
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Introduction to Fourier Analysis on Euclidean Spaces
E. Stein and G. Weiss, · 1971
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Topics in Approximation Theory
H. Shapiro, · 1971
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Hilbert’s inequality,
H. L. Montgomery and R. C. Vaughan, · 1974
Cited alongside, same era.
The analytic principle of the large sieve,
H. L. Montgomery, · 1978
Cited alongside, same era.
A class of extremal functions for the Fourier transform,
S. W. Graham and J. D. Vaaler, · 1981
Cited alongside, same era.
Extremal functions for the Fourier transform and the large sieve,
S. W. Graham and J. D. Vaaler, · 1981
Cited alongside, same era.
Elliptic Functions,
K. Chandrasekharan, · 1985
Cited alongside, same era.
Some extremal functions in Fourier analysis,
J. D. Vaaler, · 1985
Cited alongside, same era.
The Theory of the Riemann Zeta-function
E.C. Titchmarsh, · 1986
The Beurling-Selberg extremal functions for a ball in the Euclidean space,
J. Holt and J. D. Vaaler, · 1996
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Some trigonometric extremal functions and the Erdös-Turán type inequalities,
X. J. Li and J. D. Vaaler, · 1999
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Note on a Diophantine inequality in several variables,
J. T. Barton, H. L. Montgomery, and J. D. Vaaler, · 2000
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Classical and Modern Fourier Analysis,
L. Grafakos, · 2004
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Entire approximations to the truncated powers,
F. Littmann, · 2005
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One-sided approximation by entire functions,
F. Littmann, · 2006
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Cited alongside, same era.
Remarks on sieves, Proc. 1972 Number Theory conf. (Univ. of Colorado, Boulder), pp 205–216;
A. Selberg, · 1989
Cited alongside, same era.
Lectures on Sieves, Atle Selberg: Collected Papers
A. Selberg, · 1991
Cited alongside, same era.
Refinements of the Erdös-Turán inequality,
J. D. Vaaler, · 1993
Cited alongside, same era.
Some extremal functions in Fourier analysis, II,
E. Carneiro and J. D. Vaaler,
Cited in the paper.
Some extremal functions in Fourier analysis, III,
E. Carneiro and J. D. Vaaler,
Cited in the paper.
Entire majorants via Euler-Maclaurin summation,
F. Littmann, · 2006
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Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials,
E. Carneiro, · 2008
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Bounding | ζ ( 1 2 + i t ) | |\zeta(\tfrac{1}{2}+it)| on the Riemann hypothesis,
V. Chandee and K. Soundararajan, · 2008
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Best approximating entire functions to | x | α |x|^{\alpha} in L 2 L^{2} ,
M. I. Ganzburg and D. S. Lubinsky, · 2008
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