Weierstrass Transforms of Positive Functions

On Positive Functions with Positive Fourier Transforms

Using the basis of Hermite–Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive constraints for a function and its Fourier transform to be both real and positive. We propose a constructive method based on the algebra of Hermite polynomials. Applications are extended to the 2-dimensional case (i.e. Fourier–Bessel transforms and the algebra of Laguerre poly...

The Fekete-Szego Coefficient Functional for Transforms of Analytic Functions

Piecewise Harmonic Subharmonic Functions and Positive Cauchy Transforms

We give a local characterization of the class of functions having positive distributional derivative with respect to z̄ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild nondegeneracy assumptions. As a consequence, we show that any subharmonic piecewise harmonic function satisfying these conditions locally coincides with the maximum of finitely man...

Rational values of Weierstrass zeta functions

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T ), for an effective constant c > 0 depending on k and on Λ. Using different methods, we a...

On Relations of Hyperelliptic Weierstrass al Functions

We study relations of the Weierstrass’s hyperelliptic al-functions over a non-degenerated hyperelliptic curve y = f(x) of arbitrary genus g as solutions of sine-Gordon equation using Weierstrass’s local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gordon equation had already obtained, our derivations of them is simple; they need only...