Introducing the Polylogarithmic Hardy Space

Clustering on Sliding Windows in Polylogarithmic Space

In PODS 2003, Babcock, Datar, Motwani and O’Callaghan [4] gave the first streaming solution for the k-median problem on sliding windows using O( k τ4W 2τ log2 W ) space, with a O(2O(1/τ)) approximation factor, where W is the window size and τ ∈ (0, 2 ) is a user-specified parameter. They left as an open question whether it is possible to improve this to polylogarithmic space. Despite much progr...

Constrained Hardy space approximation

We consider the problem of minimizing the distance ‖f − φ‖Lp(K), where K is a subset of the complex unit circle ∂D and φ ∈ C(K), subject to the constraint that f lies in the Hardy space H(D) and |f | ≤ g for some positive function g. This problem occurs in the context of filter design for causal LTI systems. We show that the optimization problem has a unique solution, which satisfies an extrema...

The Hardy Space of a Slit Domain

Linear-Space Substring Range Counting over Polylogarithmic Alphabets

Bille and Gørtz (2011) recently introduced the problem of substring range counting, for which we are asked to store compactly a string S of n characters with integer labels in [0, u], such that later, given an interval [a, b] and a pattern P of length m, we can quickly count the occurrences of P whose first characters’ labels are in [a, b]. They showed how to store S in O(n logn/ log logn) spac...

Spectrum and essential spectrum of linear combinations of composition operators on the Hardy space H2

Let -----. For an analytic self-map ---  of --- , Let --- be the composition operator with composite map ---  so that ----. Let ---  be a bounded analytic function on --- . The weighted composition operator ---  is defined by --- . Suppose that ---  is the Hardy space, consisting of all analytic functions defined on --- , whose Maclaurin cofficients are square summable. .....