Energy estimate for wave equations with coefficients in some Besov type class.

Tarama, Shigeo

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We compute upper and lower bounds for the approximation of hyperbolic functions at points $1 / s$ $(s = 1, 2, \dots)$ by rationals $x / y$ , such that $x, y$ satisfy a quadratic equation. For instance, all positive integers $x, y$ with $y \equiv 0 (mod 2)$ solving the Pythagorean equation $x^{2} + y^{2} = z^{2}$ satisfy $| y sinh (1 / s) - x | ≫ \frac{log log y}{log y} .$ Conversely, for every $s = 1, 2, \dots$ there are infinitely many coprime integers $x, y$ , such that $| y sinh (1 / s) - x | ≪ \frac{log log y}{log y}$ and $x^{2} + y^{2} = z^{2}$ hold simultaneously for some integer $z$ . A generalization to the approximation of $h (e^{1 / s})$ for rational...

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Let f(z), $z = r e^{i θ}$ , be analytic in the finite disc |z| < R. The growth properties of f(z) are studied using the mean values $I_{δ} (r)$ and the iterated mean values $N_{δ, k} (r)$ of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z).

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