On the distribution of the fourier spectrum of Boolean functions

In this paper we obtain a general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1} ^N . Roughly speaking, fixingk∈ℤ₊ and assuming thatf is not essentially determined by a bounded number (depending onk) of variables, we have that . The example of the majority function shows that this result is basically optimal.     

From characteristic function to distribution function via fourier analysis

In this paper characteristic functions of probability distributions are considered. The numerical calculation of the distribution function when the characteristic function is known is discussed and two different methods are presented.     

Absolute convergence of fourier series of almost-periodic functions

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.     

Approximate convexity of Takagi type functions

Given a bounded function Φ:R→R, we define the Takagi type function TΦ:R→R by

     

Absolute convergence of fourier series of superpositions of functions

In this paper we establish conditions for the absolute convergence of series of Fourier coefficients with respect to a generalized Haar system of a superposition of two functions.     

On the minimal fourier degree of symmetric Boolean functions

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1} ^k → {0, 1} there exists a set; \(\not 0 \ne S \subset [k]\) such that ¦S¦ = O(Γ(k)+√k, and \(\hat f(S) \ne 0\) where Γ(m)≤m ^0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10]. Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that ¦S¦=O(k=log k). We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.     

Calculation of fourier coefficients of discrete functions using cubic splines

Formulas are given for the calculation of the finite Fourier transform of a B-spline. These formulas are useful for the computation of Fourier coefficients of a function which is given at a discrete set of arbitrary points.     

Strong summability of fourier series on classes of periodic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 354–360, March, 1989.     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

Approximate portfolio analysis

This paper presents a portfolio selection model based on the idea of approximation. The model describes a portfolio by its decumulative distribution curve and a preference structure by a family of convex indifference curves. It prescribes the optimal portfolio as the one whose decumulative curve has the highest tangent indifference curve. The model extends the mean–variance model in the sense that it does not restrict the return distributions of assets to be normal. While under the assumption of normality, the model simplifies to the mean–variance model. The model has a measure of risk attitudes that resembles the Arrow–Pratt measure while combining both wealth and probability attitudes. Using this measure, we show that the smaller the curvature of a value function and the larger the curvature of a weighting function, the more risk averse an agent.     

Recovering fourier coefficients of some functions and factorization of integer numbers

It is shown that if a function determined on the segment [−1, 1] has a sufficiently good approximation by partial sums of its expansion over Legendre polynomial, then, given the function’s Fourier coefficients c _n for some subset of n ∈ [n ₁, n ₂], one can approximately recover them for all n ∈ [n ₁, n ₂]. A new approach to factorization of integer numbers is given as an application.