Mathematics Problem Archive

For a bounded plane domain $D$, denote by $N(D)$ the class of functions analytic on $D$ of bounded characteristic (i.e., all quotients of functions in...

Let $D$ be a simply-connected domain in the complex plane $\mathbb{C}$ and $A(D)$ the ring of functions $f: D\to\mathbb{C}$ analytic in $D$. Bers (no ...

Let $G$ be a domain in $\mathbb{C}$, and $H(G)$ the ring of functions analytic on $G$. It is known that, for two domains $G_1$ and $G_2$, $H(G_1)$ is ...

Let $A^p$, $p > 0$ be the Bergmann space of functions $f(z)$ analytic in $|z|< 1$ such that \[\|f\|_p=\Huge({\int\int}_{|z|<1}|f(re^{i\theta})|^p\,r\,...

We use the notation of Problem 8.6. Let $\{z_n\}^\infty_1$ be a sequence of points in $\mathbb{D}$ such that the kernel functions $k_n(z) = (1-z_nz)^{...

Suppose that $F$ is a relatively-closed subset of $\mathbb{D}$, and let \[\|f\|_F=\sup\{|f(z)|;z\in F\}\] for functions $f$ in the Bergman space $A^2$...

Determine \[\|\Lambda\|=\sup\Big|{\int\int}_{|z|<1}f(z)\phi(z)\,d\sigma_z\Big|\] over those $f\in A^1$ with $\|f\|_1\leq1$, where \[\phi(z)=\text{sgn}...

If $f$ and $1/f$ belong to the Bergman space $A^2$, does it follow that $\mathcal{P}f$ is dense in $A^2$? Here $\mathcal{P}f$ denotes the set of all p...

Let $g$ be a function in the space $D$ of functions analytic in $|z|< 1$ with finite Dirichlet integral, i.e., \[{\int\int}_{|z|<1}|h'(z)|^2\,dx\,dy<\...

Let $A$ denote the set of functions continuous on $|z|= 1$ which extend continuously to analytic functions on $|z|< 1$ (the disc algebra). Let $\tilde...

Does there exist a singular measure in Zygmund's class $A^*$ all of whose Fourier-Stieltjes coefficients are non-negative? (F. Holland)...

Which functions in $L^\infty$ on the unit circle generate positive Hankel operators? (F. Holland)...

Characterise the Hankel operators on the Hardy space $H^2$ on the circle that are of trace class. (F. Holland)...

Miles and Rudin have shown that in $\mathbb{C}^n$ a function analytic in the unit polydisc and in the Hardy class $H^1$ may not be expressible as the ...

In the ring of bounded analytic functions on the unit ball or the polydisc in $n$ variables, is the intersection of two finitely-generated ideals agai...

For which simply-connected domains $D$ (with $0 \in D$) is it true that there is a constant $K = K(D)$ such that $$ {\int\int}_D|f|^2\,dx\,dy\leq K{\i...

Let $\mu\geq1$ be a singular measure on the boundary of the unit disc $\mathbb{D}$; and let $S_\mu$ be the corresponding singular inner function \[S_\...

Let $f_1,\ldots,f_n$ and $g$ be $H^\infty$ functions on $\mathbb{D}$. If $|g(z)| \leq |f_1(z)| + \ldots + |f_n(z)|$, are there necessarily functions $...

Let $w\geq0$ be an integrable function on the circle or on the line. Then $w$ is said to belong to the class $A_2$ if \[\sup_l\Big(\frac{1}{|l|}\int_l...

Do there exist inner functions in any strictly pseudo-convex domain of $\mathbb{C}^n$, $n\geq2$? Alexandrov (no citation) and independently L{\o}w (no...

In Pompeiu's formula, $f(z)$ and $\int_\Gamma f(\zeta)/(\zeta-z)\,d\zeta$ make sense for merely continuous functions, but \[\int\int_\Omega\frac{\part...

Let the sequence $\{M_k\}^\infty_0$ of positive numbers be such that \[M_0=1\hspace{1cm}\text{ and }\hspace{1cm}\frac{M_{k+j}}{M_kM_j}\geq\begin{pmatr...

Let $\psi:S^1\to S^1$ be a direction-reversing homeomorphism, and let $A_\psi$ denote the set of functions $f:S^1\to\mathbb{C}$ such that both $f$ and...

Let $\psi:S^1\to S^1$ be a homeomorphism. When is it true that $\text{Re }A = \text{Re }A \circ \psi$? That is, when is each function $f$ in the real ...

A sequence $\{z_n\}^\infty_1$ in $|z|< 1$ is interpolating for bounded analytic (harmonic) functions if, for each bounded sequence $\{\alpha_n\}^\inft...

Let $f(z)\in H^\infty$, and let $\{z_n\}$ be a Blaschke sequence \[\sum^\infty_{n=1}(\-|z_n|)<\infty.\] [(a)] ; Does there always exist a Blaschke pro...

[(a)] ; Suppose that $f, f_1, f_2,\ldots,f_n\in H^\infty$, and \[|f|\leq|f_1|+|f_2|+\ldots+|f_n|.\] Do there necessarily exist $h_1,h_2,\ldots,h_n\in ...

For each pair of $f,g\in H^\infty$, does there necessarily exist another pair of functions $a,b\in H^\infty$ such that $$ af+gb\neq0,\hspace{1cm}|z|<1...

Let $K_1, K_2, K_3$ be disjoint closed sets in the extended complex plane, and $C_1, C_2, C_3$ constants. Let $\rho_n(f)$ be the best rational approxi...

Let $D$ be an open subset of the extended complex plane with non-empty boundary $\partial D$, and let $F$ be a relatively-closed subset of $D$. Let $f...

Let us call a closed set $E$ in $\mathbb{C}$ a weak Arakelian set if, corresponding to each function $g(z)$ continuous on $E$ and analytic in the inte...

Let $\gamma$ be a Jordan arc in $\mathbb{C}^n$, $n\geq2$ such that the projections $\gamma_j$ on the complex coordinate planes $j=1,\ldots,n$ have are...

Does the condition $\sum 1/p_n<\infty$ for positive integers $p_n$ guarantee that the sequences of powers $\{z^{p_n}\}$ fails to span $C(\gamma)$ for ...

Let $F$ be a closed subest of $\mathbb{R}^n$, $n\geq2$. Call $F$ a set of harmonic approximation if every function continuous on $F$ and harmonic in t...

Let $D$ be a planar domain. A sequence $\{z_j\}$ of points in $D$ is said to be an interpolating sequence if whenever $\{\alpha_j\}\in\ell^\infty$ the...

Let $\Gamma\subset\mathbb{C}$ be a Jordan curve of logarithmic capacity $1$, and let $\phi$ be a conformal map from the exterior of $\Gamma$ to the ex...

Let $K$ be a compact subset of $\mathbb{R}^n$, $n\geq3$. For $\phi\in\mathcal{D}$, let $D(\phi)$ be a least-diameter disc containing $\text{spt }\phi$...

Let $f$ be continuous on a compact subset $K$ of $\mathbb{C}$. If there exists a sequence $\{f_n\}^\infty_1$ of functions analytic near $K$ for which ...

Let $f_1$ and $f_2\in H^\infty(\text{unit disc}) = H^\infty$; and let the function $g\in H^\infty$ satisfy the inequality \[|g(z)|\leq|f_1(z)|+|f_2(z)...

Let $\Gamma$ be a curve of the form \[\{x+iA(x):-\infty<x<\infty\}\] with \[|A(x_1)-A(x_2)|\leq M|x_1-x_2|.\] Let $E$ be a compact subset of $\Gamma$,...

Let $K$ denote the $\frac{1}{3}$-Cantor set on $\mathbb{R}$; let $E = K\times K$, and let $\Omega=\mathbb{C}^*\setminus E$. Prove the corona theorem f...

Does the Flint Hills series $\sum_{n=1}^{\infty} 1/(n^3\sin^2 n)$ converge?...

Are there infinitely many Lehmer pairs of zeros in the sense used in the theory of the de Bruijn–Newman constant?...

For viscous incompressible flow in $\mathbb{R}^3\setminus B$ past a fixed obstacle $B$, with velocity approaching a nonzero constant vector at infinit...

Let $X$ be a centered Gaussian random vector in $\mathbb{R}^n$ with known covariance matrix, and for an orthonormal basis $B=(b_1,\ldots,b_n)$ let $N_...

For a $d\times d$ real Gaussian matrix $G_{\mathbb{R}}$ and complex Gaussian matrix $G_{\mathbb{C}}$, both normalized to entry variance $1/d$, define ...

Let ${W}$ denote a symmetric Wigner matrix with i.i.d. entries ${W_{ij}\sim \mathcal{N}(0,1)}$ . Also, given ${B\in\mathbb{R}^{n\times n}}$ symmetric,...

Is there a polynomial time algorithm that is able to find the largest clique of $G$ (with high probability) for $\omega \ll \sqrt{n}$ ? For example, f...

Does there exist $c>0$ such that it is NP-hard, given a graph $G$ and $\phi>0$, to distinguish $h_G\le\phi$ from $h_G\ge c\sqrt{\phi}$?...

Given a symmetric matrix ${M}$ with small condition number, is there a quasi-linear time (on ${n}$ and the number of non-zero entries of ${M}$ ) proce...