# Questions tagged [hardy-spaces]

For questions about Hardy spaces. Use the other related tag like (tag: complex-analysis) or (operator-theory).

202 questions
Filter by
Sorted by
Tagged with
2 votes
1 answer
24 views

• 2,494
0 votes
0 answers
24 views

• 337
1 vote
0 answers
30 views

• 335
-1 votes
1 answer
39 views

### Every simple functions in $L_p(\mathbb{T})$ is a trigonometric polynomial.

I know for a finite measure space $(\Omega, \mathcal{A}, \mu)$, Simple functions are dense in $L_p(\mu)$. Also it is true that, if we consider normalized Lebesgue measure on the unit circle then ...
• 113
35 votes
1 answer
3k views

### Major error in classic "Banach Spaces of Analytic Functions"

"Banach Spaces of Analytic Functions" by Kenneth Hoffman is an excellent introduction to $H^{p}$ spaces and is considered a classic mathematical analysis textbook. I was therefore surprised ...
• 1,886
1 vote
1 answer
51 views

2 votes
1 answer
79 views

### Functions in $A(\mathbb{C})$ that vanish at zero [closed]

Let $A\subset L^{\infty}(-\pi,\pi)$ be the closed subspace spanned by $\{e^{inx}\}_{n\geq0}$, i.e., all continuous functions on the unit circle that are uniform limits of trigonometric polynomials of ...
• 1,886
0 votes
0 answers
56 views

### Blaschke product: What does extending continuously mean here?

I am studying Factorisation of $H^p$ functions from Hoffman's Banach Spaces of Analytic Functions. The author is talking about extending the Blaschke product continuously to the accumulation points of ...
• 3,985
0 votes
0 answers
75 views

### Rudin RCA Chapter 17 Exercise 3

I am going through Rudin's RCA (3rd Ed), and it seems to me that Chapter 17 ($H^p$-Spaces), Exercise 3 (the initial part) has an incorrect equivalency statement. Namely, it states the following: ...
1 vote
1 answer
145 views

### Blaschke product having poles proof

I am studying Banach Spaces of Analytic Function by Hoffman. Hoffman proves the following theorem: The Blaschke product whose zeroes are \begin{align*} \alpha_{1} , \alpha_2 , \ldots \end{align*} ...
• 3,985
1 vote
0 answers
48 views

• 311
2 votes
0 answers
55 views

### Prove that, $H^\infty+C(\Bbb{T})$ is not a $C^*$-algebra

Here $H^\infty:=\left\{f\in L^\infty(\Bbb{T}):\ \frac{1}{2\pi}\int\limits_0^{2\pi} f\chi_{-n}\ dt=0\ \forall n<0\right\}$ i.e. all those $L^\infty$ functions whose negative fourier coefficients are ...
• 3,104
0 votes
0 answers
27 views

2 votes
0 answers
70 views

### Approximations of BMO functions

I couldn't think of a good title. There was a small detail when I was trying to prove something about the norm of BMO functions. This small detail is not directly related to the problem itself, but ...
• 95
2 votes
1 answer
82 views

### Let $f\in L^2(\Bbb{T}),\ f\ne0$ and $M=\overline{f\wp_+}$. Prove that, $M$ is simply invariant for $\chi_1$ iff $f\notin \chi_N M$ for some $N>0$

Here we denote $\chi_n$ to be the functions on $\Bbb{T}$ defined as $\chi_n(z)=z^n\ \forall z\in\Bbb{T}$ for $n\in\Bbb{Z}$. We define $\wp_+=\{\sum\limits_{n=0}^N a_n\chi_n|\ a_n\in\Bbb{C},N\ge0\}$ be ...
• 3,104
0 votes
0 answers
75 views

### $\int_{-\pi}^{\pi}{\frac{d\theta}{[(1-r)^2 + 2\pi^{-2}\theta^2]^p}} < \frac{1}{(1-r)^{2p-1}}\int_{-\infty}^{\infty}{\frac{dt}{[1+2\pi^{-2}t^2]^p}}$

I found this inequality from Duren's book Theory of Hp-spaces(2000) page 66. According to the book inequality should be true for $\frac{1}{2}\leq r < 1$ and $p>\frac{1}{2}$. I have tried to ...
1 vote
0 answers
31 views

• 957
0 votes
1 answer
147 views

0 votes
0 answers
129 views

### Why does the Laplace transform of a function in L2 belong in the Hardy space?

Does anyone have a proof / mathematical explanation to why the Laplace transform of a function of $\mathbb{L}^2(0,\infty )$ belongs to the Hardy space? Any guidance would be fantastic! I cant find ...
0 votes
0 answers
60 views

### a generalized Jensen's equality for Nevanlinna class

Define the Nevanlinna class of holomorphic functions on unit disk by $\displaystyle N(\mathbb{D})=\left\{f\in\text{Hol}(\mathbb{D}):\sup_{0<r<1}\int_\mathbb{T}\log^+|f_r|dm<\infty\right\}$ ...
• 2,172
5 votes
0 answers
101 views

### Showing $\arcsin(z)-i\operatorname{Log}(-iz)-i\log 2 \in H^2(\mathbb{H})$

In order to prove a result about Hilbert transforms, I need to show the complex function $F(z) = \arcsin(z)-i\operatorname{Log}(-iz)-i\log 2$ lies in $H^2(\mathbb{H})$, the Hardy Space for the upper ...
• 51
3 votes
1 answer
90 views

### containment of $H^1$ spaces into $L^1$ spaces

Let $\mathbb{D}$ denote the unit disk and $\mathbb{T}$ the unit circle , if $f\in H^1(\mathbb{D})$ is outer , then necessarily $\log|f|\in L^1(\mathbb{T})$ ? We know from theory of Hardy spaces that ...
• 2,172