Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. A general co...

We study spherical quadrilaterals whose angles are odd multiples of π/2, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, ...

The modulus and phase of the reduced logarithmic derivative of the Hankel function zH^'iz)/Hi'\z) for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.

We present a detailed computation of Ramanujan’s most famous singular modulus, k210, based on the Kronecker Limit Formula.

Let X be a completely regular Hausdorff space, E a Banach lattice, and μ an E-valued countably additive, regular Borel measure on X. Some results about the countable additivity and regularity of the modulus |μ| are proved. Also in special cases, it is proved that L1(μ) = L1(|μ|).

We present three new approximation properties of the Balázs–Szabados operators. Firstly, we prove that, in certain cases, these operators approximate some super-exponential functions on compact intervals. Next, provide a estimate error using suitable modulus continuity. Finally, characterize which can be uniformly approximated weighted norm polynomial weight spaces.

In this paper, using the Steklov function, we introduce the generalized continuity modulus and denethe class of functions Wr;kp;' in the space Lp. For this class, we prove an analog of the estimates in [1]in the space Lp.

In this article we introduce the difference sequence spacesW0[ f ,∆m],W1[ f ,∆m], W∞[ f ,∆m] and S[ f ,∆m], defined by a modulus function f . We obtain a relation between W1[ f ,∆m]∩ l∞[ f ,∆m] and S[ f ,∆m]∩ l∞[ f ,∆m] and prove some inclusion results.