4,962 research outputs found
Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace
We consider the problem of efficient integration of an n-variate polynomial
with respect to the Gaussian measure in R^n and related problems of complex
integration and optimization of a polynomial on the unit sphere. We identify a
class of n-variate polynomials f for which the integral of any positive integer
power f^p over the whole space is well-approximated by a properly scaled
integral over a random subspace of dimension O(log n). Consequently, the
maximum of f on the unit sphere is well-approximated by a properly scaled
maximum on the unit sphere in a random subspace of dimension O(log n). We
discuss connections with problems of combinatorial counting and applications to
efficient approximation of a hafnian of a positive matrix.Comment: 15 page
Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent
The verification of many algorithms for calculating transcendental functions
is based on polynomial approximations to these functions, often Taylor series
approximations. However, computing and verifying approximations to the
arctangent function are very challenging problems, in large part because the
Taylor series converges very slowly to arctangent-a 57th-degree polynomial is
needed to get three decimal places for arctan(0.95). Medina proposed a series
of polynomials that approximate arctangent with far faster convergence-a
7th-degree polynomial is all that is needed to get three decimal places for
arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness
and convergence rate of this sequence of polynomials. The proof is particularly
beautiful, in that it uses many results from real analysis. Some of these
necessary results were proven in prior work, but some were proven as part of
this effort.Comment: In Proceedings ACL2 2014, arXiv:1406.123
O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing
A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument is an element of [-1, 1]. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is O(1). The proposed algorithm also immediately yields an O(1) algorithm for computing an arbitrary Gauss-Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss-Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the O(1) complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss-Legendre quadrature rules
Ultraspherical multipliers revisited
Sufficient ultraspherical multiplier criteria are refined in such a way that
they are comparable with necessary multiplier conditions. Also new necessary
conditions for Jacobi multipliers are deduced which, in particular, imply known
Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an
essential way
Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots
We study a recent result of Bourgain, Clozel and Kahane, a version of which
states that a sufficiently nice function
that coincides with its Fourier transform and vanishes at the origin has a root
in the interval , where the optimal satisfies . A similar result holds in higher dimensions. We improve the
one-dimensional result to , and the lower bound in
higher dimensions. We also prove that extremizers exist, and have infinitely
many double roots. With this purpose in mind, we establish a new structure
statement about Hermite polynomials which relates their pointwise evaluation to
linear flows on the torus, and applies to other families of orthogonal
polynomials as well.Comment: 26 pages, 4 figure
Oscillation of Fourier transform and Markov-Bernstein inequalities
Under certain conditions on an integrable function f having a real-valued
Fourier transform Tf=F, we obtain a certain estimate for the oscillation of F
in the interval [-C||f'||/||f||,C||f'||/||f||] with C>0 an absolute constant.
Given q>0 and an integrable positive definite function f, satisfying some
natural conditions, the above estimate allows us to construct a finite linear
combination P of translates f(x+kq)(with k running the integers) such that
||P'||>c||P||/q, where c>0 is another absolute constant. In particular, our
construction proves sharpness of an inequality of H. N. Mhaskar for Gaussian
networks
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