A Brief Review of the Development of Qualitative Control by Marchenko V. M.

By Marchenko V. M.

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Ball. 2 A. Aleman, S. Richter and C. 1. There exists an absolute constant C > 0 such that for every ν ∈ 1 d|ν|(z) < ∞ there exists r0 > 0 Mc (C) and for every λ ∈ C with U|ν| (λ) = |z−λ| such that for all polynomials p and for all 0 < r r0 we have C |p(z)Cν(z)|dA(z). |p(λ)Cν(λ)| r2 B(λ,r) Here r0 depends only on |Cν(λ)|, U|ν| (λ) and U|ν| (λ, r) as r → 0. The theorem is nontrivial only at points when Cν(λ) = 0 and we will see that there is an absolute constant K0 > 0 such that for all such points any r0 satisfying √ √ U|ν| (λ, r0 + r0 ) + r0 U|ν| (λ) K0 |Cν(λ)| will work.

He can advise in the most difficult situations, but does not interfere and volunteer his advice, unless he is asked for his opinion. Although he has very logical and analytical mind and believes in thinking things through, sometimes he tends to rely on his intuition. He can spend hours concentrating on mathematical research, being disconnected from the world around him, but will drop everything in a second to help his children or grandchildren with their homework. Speeches and Reminiscences xxxi Can’t live without email, loves technology and internet communication, but on the other hand loves nature, enjoys long walks, good swim and camping away from the civilization.

Thus we think of the current paper mostly as an expository note, and we plan to take this opportunity to once more carefully explain how X. Tolsa’s theorem on analytic capacity, [9] and an adaptation of Thomson’s coloring scheme, [8] come together to prove the current result. In Section 5 we explain how the current approach can also be used to establish that every bounded point evaluation must either arise because of an atom of μ or it must be an analytic bounded point evaluation. 2. Thomson’s theorem Let μ be a positive finite compactly supported measure in the complex plane C, let 1 t < ∞ and let P t (μ) denote the closure of the polynomials in Lt (μ).

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