By Javad Mashreghi

The idea of Hardy areas has shut connections to many branches of arithmetic together with Fourier research, harmonic research, singular integrals, power conception and operator thought, and has came upon crucial functions in powerful regulate engineering. for every program, the power to symbolize components of those periods through sequence or crucial formulation is of extreme value. This self-contained textual content offers an advent to quite a lot of illustration theorems and offers an entire description of the illustration theorems with direct proofs for either periods of Hardy areas: Hardy areas of the open unit disc and Hardy areas of the higher part airplane. With over three hundred routines, many with accompanying tricks, this ebook is perfect for these learning complex complicated research, functionality thought or conception of Hardy areas. complex undergraduate and graduate scholars will locate the booklet effortless to stick with, with a logical development from simple idea to complicated examine.

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Thus, given ε > 0, there is ϕ ∈ C(T) such that f −ϕ p < ε. Hence |(f ∗ g)(eit ) − (f ∗ g)(eis )| ≤ | (f − ϕ) ∗ g (eit )| + | (f − ϕ) ∗ g (eis )| + |(ϕ ∗ g)(eit ) − (ϕ ∗ g)(eis )| ≤ 2 f −ϕ p g q + ωϕ (|t − s|) g q , where ωϕ (δ) = sup |ϕ(eit ) − ϕ(eis )| |t−s|≤δ is the modulus of continuity of ϕ. Since ϕ is uniformly continuous on T, ωϕ (δ) −→ 0 as δ → 0. Therefore, if |t − s| is small enough, we have |(f ∗ g)(eit ) − (f ∗ g)(eis )| ≤ 3ε g q . 1 What can we say about f ∗ g if f ∈ Lr (T) and g ∈ Ls (T) with 1 ≤ r, s ≤ ∞ and 1 1 + ≤ 1?

Then (f ∗ g)(eit ) is well-deﬁned for all eit ∈ T, f ∗ g ∈ C(T), and f ∗g ∞ ≤ f p g q. 5. Young’s inequality 19 Proof. 4, H¨ older’s inequality ensures that (f ∗ g)(eit ) is well-deﬁned for all eit ∈ T. The only new fact to prove is that f ∗ g is a continuous function on T. At least one of p or q is not inﬁnity. Without loss of generality, assume that p = ∞. 4). Thus, given ε > 0, there is ϕ ∈ C(T) such that f −ϕ p < ε. Hence |(f ∗ g)(eit ) − (f ∗ g)(eis )| ≤ | (f − ϕ) ∗ g (eit )| + | (f − ϕ) ∗ g (eis )| + |(ϕ ∗ g)(eit ) − (ϕ ∗ g)(eis )| ≤ 2 f −ϕ p g q + ωϕ (|t − s|) g q , where ωϕ (δ) = sup |ϕ(eit ) − ϕ(eis )| |t−s|≤δ is the modulus of continuity of ϕ.

Fˆ(n)|2 . 7. Parseval’s identity 53 Hence the Fourier transform L2 (T) −→ f −→ 2 (Z) fˆ is bijective and shows that the Hilbert spaces L2 (T) and isometric. 1 [Polarization identity] Let H be a complex inner product space and let x, y ∈ H. 2 2 − x−y 2 + i x + iy 2 − i x − iy 2 . Let f, g ∈ L2 (T). Show that 1 2π π −π f (eit ) g(eit ) dt = ∞ fˆ(n) gˆ(n). 1.