Antonio Córdoba

Trigonometric polynomials and lattice points

In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius R, an arc whose length is smaller than V2Rl12-l(4[m/2]+2) contains, at most, m lattice points. We use the same method to obtain sharp L4-estimates for uncompleted, Gaussian sums

Band-limited functions: ^{}-convergence

We consider the set BP(Q) (functions of LP(Tl) whose Fourier spectrum lies in (-I2, +I2) ). We prove that the prolate spheroidal wave func- tions constitute a basis of this space if and only if 4/3 < p < 4. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hubert transform.

Convergence and divergence of decreasing rearranged Fourier series

In a number of useful applications, e.g., data compression, the appropriate partial sums of the Fourier series are formed by taking into consideration the size of the coefficients rather than the size of the frequencies involved. The purpose of this paper is to show the limitations of that method of summation. We use several results from the number theory to construct counterexamples to Lp-convergence for p < 2. We also show how to obtain positive results if we combine the two points of view, i.e., cutting on frequencies and the size of coefficients at the same time. This can be considered as a kind of uncertainty principle for Fourier sums.

Principal Component Value at Risk

Value at risk (VaR) is an industrial standard for monitoring financial risk in an investment portfolio. It measures potential losses within a given confidence interval. The implementation, calculation, and interpretation of VaR contains a wealth of mathematical issues that are not fully understood. In this paper we present a methodology for an approximation to value at risk that is based on the principal components of a sensitivity-adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the variance/covariance matrix. It can be viewed as a nonlinear extension of the linear model given by the delta-normal VaR or Risk Metrics (J.P. Morgan, 1996).

Band-limited functions:

We consider the set BP(Q) (functions of LP(Tl) whose Fourier spectrum lies in [-Í2, +Í2] ). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if 4/3 < p < 4. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hubert transform.

L^p

In this paper we state some results for a maximal function and a Fourier multiplier that are connected with the Bochner-Riesz spherical summation of multiple Fourier series (see Fefferman [3], [5] ). Our purpose will be to get sharp estimates for the norm of these operators in dimension two. Proofs will appear elsewhere [2]. Let N >\ be a real number. By a rectangle of eccentricity N we mean a rectangle R such that Length of the bigger side of R __ .. Length of the smaller side of R We will define the direction of R as the direction of its bigger side. Given a locally integrable function ƒ we consider the maximal function

-convergence

The main problem in (planar) lattice point theory consists in counting lattice points under the graph of positive functions supported on [0 ,M ] and with radius of curvature comparable to M. We prove that, in some sense motivated by Feynman path integral formulation of Quantum Mechanics, for “most” functions the lattice error term in the area approximation is O(M 1/2+e ). This complements Jarnik construction of curves with an optimal O(M 2/3 ) error term.