arXiv subjects

J. Heittokangas

Publications and source records attributed to J. Heittokangas.

3 recordsLinked to original sources

arXiv2020

On the limits of real-valued functions in sets involving $ ψ$-density, and applications

We prove new results on upper and lower limits of real-valued functions by means of $ψ$-densities introduced by P. D. Barry in 1962. This allows us to improve several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The $ψ$-densities are also used to introduce a new concept of a limit for real-valued functions. The results in this paper are of interest in real analysis as well as in the theory of meromorphic functions.

math.CV↗

arXiv2020

On Petrenko's deviations and second order differential equations

New results on the oscillation of solutions of $f''+A(z)f=0$ and on the growth of solutions of $f''+A(z)f'+B(z)f=0$ are obtained, where $A$ and $B$ are entire functions. Petrenko's magnitudes of deviation of $g$ with respect to $\infty$ play a key rôle in the results, where $g$ represents one of the coefficients $A$ or $B$. These quantities are defined by $β^-(\infty,g) = \liminf_{r\to\infty} \frac{\log M(r,g)}{T(r,g)}$ and $β^+(\infty,g) = \limsup_{r\to\infty} \frac{\log M(r,g)}{T(r,g)}$.

math.CV↗

arXiv2019

The asymptotic number of zeros of exponential sums in critical strips

Normalized exponential sums are entire functions of the form $$ f(z)=1+H_1e^{w_1z}+\cdots+H_ne^{w_nz}, $$ where $H_1,\ldots, H_n\in\C$ and $0<w_1<\ldots<w_n$. It is known that the zeros of such functions are in finitely many vertical strips $S$. The asymptotic number of the zeros in the union of all these strips was found by R. E. Langer already in 1931. In 1973, C. J. Moreno proved that there are zeros arbitrarily close to any vertical line in any strip $S$, provided that $1,w_1,\ldots,w_n$ are linearly independent over the rational numbers. In this study the asymptotic number of zeros in each individual vertical strip is found by relying on R. J. Backlund's lemma, which was originally used to study the zeros of the Riemann $ζ$-function. As a counterpart to Moreno's result, it is shown that almost every vertical line meets at most finitely many small discs around the zeros of $f$.

math.CV↗