Can we find a holomorphic function $g$ on an open disk such that $\sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$?

Let $ f : \mathbb{C} \to \mathbb{C}$ be a continuous function with $ f(0)=0$ .

Let $ \{a_i\}_{i\in \mathbb{N}}$ be a set of scalars in $ \mathbb{C}$ such that $ $ \exists C > 0 : \forall i\in \mathbb{N} : |a_i| \leq C $ $ Can we always find a holomorphic function $ g$ on $ B(0,C+1)$ (the open disk of radius $ C+1$ ) such that $ g(0)=0$ and $ \sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$ ?

Evaluate $\prod_{k=1}^{n} \sum_{i\ =1}^{k} a_i$ in $\mathcal{O}(n)$


Write a Matlab program, which takes the vector $ (a_1, \ldots, a_n)$ and outputs $ \prod_{k=1}^{n} \sum_{i\ =1}^{k} a_i$ . You are only allowed to define two variables and have to solve the problem in $ \mathcal{O}(n)$ .

I am pretty sure this is closely related to Horner’s method, but I haven’t been able to simplify the expression for small $ n$ into a form that looked like Horner.

Any help is appreciated 🙂

Algorithm to solve $L_1$ optimization of $\sum_i ||\mathbf{A_i x} – \mathbf{b_i}||_1$

Is there is an efficient algorithm to solve the following optimization:

$ \mathbf{x}^* = \arg\min_\mathbf{x}\sum_i ||\mathbf{A_i x} – \mathbf{b_i}||_1$

where $ || \mathbf{y} ||_1$ is the $ L_1$ norm (i.e. $ \sum_j |y_j|$ ) and $ \mathbf{A_i}$ are big matrices which we don’t have their explicit representations, only access to operations $ \mathbf{A_i y}$ and $ \mathbf{A_i^T z}$ ?

I can solve the problem for small matrices $ \mathbf{A_i}$ using linear programming using the same trick from here, but if the matrices are large, there is no obvious way (at least for me) to solve it using linear programming.

Any ideas and suggestions are welcome.