Titlebook: Explorations in Harmonic Analysis; With Applications to Steven G. Krantz Textbook 2009 Birkh?user Boston 2009 Fourier analysis.Fourier tran

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https://doi.org/10.1007/978-981-99-0872-1m of P. Fatou that a . holomorphic function on the unit disk . has radial (indeed nontangential) boundary limits almost everywhere. Hardy and Riesz wished to expand the space of holomorphic functions for which such results could be obtained.

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The Central Idea: The Hilbert Transform,intertwined in profound and influential ways. What it all comes down to is that there is only one singular integral in dimension 1, and it is the Hilbert transform. The philosophy is that all significant analytic questions reduce to a singular integral; and in the first dimension there is just one choice.

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Pseudoconvexity and Domains of Holomorphy,al geometric condition on the boundary. The second is an idea that comes strictly from function theory. The big result in the subject—the solution of the Levi problem—is that these two conditions are equivalent.

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Canonical Complex Integral Operators,lmay be discovered naturally by way of power series considerations, or partial differential equations considerations, or conformality considerations. The Poisson kernel is the real part of the Cauchy kernel. It also arises naturally as the solution operator for the Dirichlet problem. It is rather mo

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Hardy Spaces Old and New,m of P. Fatou that a . holomorphic function on the unit disk . has radial (indeed nontangential) boundary limits almost everywhere. Hardy and Riesz wished to expand the space of holomorphic functions for which such results could be obtained.

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