The great mathematician G. H. Hardy told us that “Beauty is the first test: there is no permanent place in the world for ugly mathematics” (see [24, p. 85]). It is clear why Hardy loved complex analysis: it is a very beautiful part of classical mathematics. The theory of Hilbert spaces and of operatorson the misalmostas classical and is perhaps as beautiful as complex analysis. The study of the Hardy Hilbert space and of operators on that space, combines these two subjects. The interplay produces a number of extraordinarily elegant results. For example, very elementary concepts from Hilbert space provide simple proofs of the Poisson integral (Theorem 1. 1. 21 below) and Cauchy integral (Theorem 1. 1. 19) formulas. The fundamental theorem about zeros of functions in the Hardy–Hilbert space (Corollary 2. 4. 10) is the central ingredient of a beautiful proof that every continuous function on [0,1] can be uniformly approximated by polynomials with prime exponents (Corollary 2. 5. 3).
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