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Full-Text Articles in Analysis
A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić
A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić
Mathematics, Physics, and Computer Science Faculty Articles and Research
For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x0, y0) such that A nontrivially intersects curves y − y0 = a(x −x0)β for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …
More Properties Of Optimal Polynomial Approximants In Hardy Spaces, Raymond Cheng, Christopher Felder
More Properties Of Optimal Polynomial Approximants In Hardy Spaces, Raymond Cheng, Christopher Felder
Mathematics & Statistics Faculty Publications
We study optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, Hp (1 < p < ∞). For fixed f ∈ Hp and n ∈ N, the OPA of degree n associated to f is the polynomial which minimizes the quantity ∥qf −1∥p over all complex polynomials q of degree less than or equal to n. We begin with some examples which illustrate, when p ≠ 2, how the Banach space geometry makes the above minimization problem interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions …