Let $A$ be Hermitian positive definite and let $f_m$ denote the Lanczos approximation to $f(A)b$. We prove that if $f(z)$ or $f(z) / z$ is Stieltjes, then the $A^\alpha$-norm error of the Lanczos approximation is within a factor $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ of the the best possible Krylov Subspace Method, where $\kappa(A)$ is the condition number of $A$ and $E = \max\{\alpha,1-\alpha\}$. Our result strengthens and generalizes the upper bound of [Schweitzer; SIMAX, 46.3 (2025)]. Moreover, we prove that the constant $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ is optimal.
The need to evaluate expressions of the form $f(A)\mathbf{b}$, where $A$ is a square matrix, $f$ is a function, and $\mathbf{b}$ is a vector, arises in several areas of applied mathematics. When the matrix $A$ is very large, it is usually not attractive to evaluate $f(A)$. Instead, $f(A)\mathbf{b}$ often is approximated by computing an estimate in a Krylov subspace that depends on $A$ and $\mathbf{b}$, and only requires that $f$ be evaluated at a small matrix. This paper explores the application of several variants of randomized block Krylov methods to the approximation of $f(A)\mathbf{b}$. Computed examples suggest that block Krylov methods with an initial block vector that contains $\mathbf{b}$ as well as a few randomly generated vectors may require less computing time and reduce the number of Krylov steps than standard Krylov methods.
J. Kane, Lucas Onisk, Lothar Reichel et al.· 0 citations
The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma$ is the standard Gaussian in $\mathbb R^n$, $K \subset \mathbb R^n$ is an origin-symmetric convex set, and $s, t \in \mathbb R$ then $\gamma\left(e^{\frac{s + t}{2}} K\right) \ge \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$. Herscovici, Livshyts, Rotem and Volberg proved a stability version of this result, showing that if one has equality up to a factor $(1 + \epsilon)$ in the (B)-inequality for $K$ then the inradius of $K$ must be either ``very large''or ``very small,''where the bounds depend on $\epsilon$ and on $n$. We give a new stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers of the (B)-inequality. In particular, our results imply that if $\gamma\left(e^{\frac{s + t}{2}} K\right) \le (1 + \epsilon) \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$, then every principal component of the covariance matrix of the probability measure obtained by restricting the Gaussian to $K$ must either be at least $1 - O(\epsilon)$ or at most $O(\epsilon)$, which is sharp. Our method extends immediately to yield stability estimates for generalizations of the (B)-inequality, namely the ``strong''and ``functional''(B)-inequalities, which reduce to spectral questions about $1$-log-concave measures on $\mathbb R^n$.
We continue our studies on parametric norm forms $F_t({\bf x})$, with ${\bf x}=(x_0,x_1,\ldots,x_{d-1})$ lying in some parametric linear subvariety $W_t$ and integers $t$ sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions $\bf x$ of $F_t({\bf x})=1$. Here we modify our techniques to treat $F_t({\bf x})=q$ for an arbitrary integer $q$. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all $\bf x$ are polynomially bounded in terms of $|q|$ and $t$. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with $x_2=\cdots=x_{d-1}=0$. We present an example with eight variables.
We study the simultaneous approximation of constant-degree polynomials over convex sets. For any family of $m$ degree-$d$ polynomials and any convex set ${H} \subseteq \mathbb{R}_{\ge0}^n$, we construct an $\epsilon$-Cover of the joint value set $\{(f_1(x), \dots, f_m(x)) : x \in {H}\}$ in the $\ell_\infty$-norm. This cover is of size $n^{O(\log(mn)/\epsilon^2)}$, provided the polynomials have constant range over the smallest $\ell_1$-ball inscribing ${H}$. Our approach extends classical net-based sparsifications for linear functions (e.g., Lipton, Markakis, and Mehta [2003]) to arbitrary families of constant-degree polynomials over general convex sets. We use a two-step scheme: first, we construct a quasi-polynomial pre-cover of the family on the smallest $\ell_1$-ball containing ${H}$ by using a concentration argument and leveraging a connection between Bernstein approximation and multinomial distributions; we then compress the pre-cover to ${H}$ by using a recursive degree reduction and feasibility programs anchored at points of the pre-cover. The existence of these covers immediately yields a unified framework for Quasi-Polynomial Time Approximation Schemes (QPTAS) across a wide range of a problems, including fixed-degree polynomial minimization over polyhedral sets, Constraint Satisfaction Problems (CSPs), Free Games, variational inequalities with polynomial operators (which implies guarantees for local Nash equilibria in polynomial games), and additive approximation for normalized densest $k$-subhypergraph on $O(1)$-uniform hypergraphs.
Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.· arXiv.org· 1 citation
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.
We establish the bound for the second-order Hankel determinant $H_{2,2}(F) = A_2 A_4 - A_3^2$ associated with the class $\mathcal{C}_{B}^{\beta}(\mathbb{B})$ of normalized $\beta$-spirallike quasi-convex mappings of type $B$ on the open unit ball $\mathbb{B}$ of a complex Banach space. By utilizing a generalized framework based on a directional slice homogeneous polynomial expansion, we eliminate the standard, restrictive assumption that the mapping is of the form $F(x) = g(x)x$. Under these weaker operational conditions, we parameterize the targeted scalar invariants $A_n$ via the classical Carath\'{e}odory functional parameters. A rigorous optimization analysis proves that the established upper bound is strictly sharp for the classical non-spirallike case $\beta = 0$, yielding a maximal value of $1/8$. This sharp bound is verified by constructing explicit multi-dimensional extremal mappings that lift the corresponding single-variable convex profile. Finally, an unresolved open question regarding the exact variational behavior for $\beta \neq 0$ is formulated.
M. B. Ahamed, Nabadwip Sarkar, Pradip Das· 0 citations
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