Papers

We study a class of strongly local, regular Dirichlet forms on the standard unit interval. The aim of our work is to record some Hölder regularity properties that have not been noted in the prior literature. In particular, as our main result, we show that for every $\delta \in (0,1]$ there exists a metric measure space $(X,d,\mu)$ equipped with a strongly local, regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(\mu)$ with the property that $\delta$ is the supremum of $\alpha \in (0,1]$ for which the domain $\mathcal{F}$ of the Dirichlet form contains a non-constant $\alpha$-Hölder continuous function. To the best of our knowledge, such examples were previously known only for $\delta = 1$. In our construction, the value $\delta$ is characterized by the upper Hausdorff dimension of a certain Radon measure that is used to define $(\mathcal{E},\mathcal{F})$.

We investigate the operator system structure of finite Hankel matrices and its connection with complex polynomials on the real line. Using polynomial matrix spectral factorization, we establish a unital complete order isomorphism between the operator system of (n + 1) x (n + 1) Hankel matrices and the dual of the operator system of polynomials of degree at most 2n. We also show that the operator system of complex polynomials is minimal, while the Hankel operator system is maximal. As an application, we characterize positivity of block Hankel matrices with entries in a C*-algebra in terms of completely positive maps on the polynomial operator system.

Wasserstein spaces carry two natural geometries: a vertical one, inherited from the linear structure of signed measures, and a horizontal one, induced by couplings. We prove that a Radon-Nikodym property holds in both. In the vertical geometry, bounded subsets of $P_p(R^d)$ are dentable in an ambient Banach space which itself fails the Radon-Nikodym property; we derive variational principles, Radon-Nikodym theorems for vector measures and martingale convergence results. In the horizontal geometry, we show that a Radon-Nikodym property is inherited from the one of spaces of random variables. We introduce horizontal martingales and prove that they are exactly the laws of $L^p$ -valued martingales.

It is known that the free $p$-convex Banach lattice on a Banach space $X$ can be represented as a space of functions on the unit ball of $X^*$. In this way, it gives rise to certain related (larger) lattices. To investigate such lattices, we introduce a new tool, related to operators into $X$. This tool is then used to (i) determine whether the lattices in question possess, or fail, properties involving upper bounds of upward directed sets – namely, the Fatou property, and the related property of monotonic boundedness; (ii) investigate the regularity of embeddings between spaces in question. We find connections between the Fatou-like properties of ${\textrm{FBL}}^{(p)}[X]$ and the Radon-Nikodym property of $X$. In addition, we give an example of $X \subset Y$ such that ${\textrm{FBL}}^{(p)}[X]$ is not a regular sublattice of ${\textrm{FBL}}^{(p)}[Y]$.

In this paper, we establish a new version of one dimensional generalized discrete dual $p$-Hardy inequality with a shift. Using this generalized discrete dual p-Hardy inequality, we obtain improvements of two discrete dual $p$-Hardy inequalities. To be specific, for $p>1$ and $A\in C_c(\mathbb{N}_{0})$ satisfying $A_{0}=A_{1}=0$, we first improve the discrete dual p-Hardy inequality \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^{p}| A_{n}-A_{n-1}|^{p}\geq\frac{1}{p^{p}}\displaystyle\sum_{n=2}^{\infty}|A_{n}|^{p}, \end{align*} where the associate constant term is sharp. Subsequently, we improve its power-type weighted discrete dual p-Hardy extension \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^{\alpha}|A_{n}-A_{n-1}|^{p}\geq\Big(\frac{\alpha+1-p}{p}\Big)^{p} \displaystyle\sum_{n=2}^{\infty}\frac{|A_{n}|^{p}}{n^{p-\alpha}} \end{align*} for $p-1<\alpha\leq p$, where the associated constant term is also sharp. We also establish a discrete $p$- Birman inequality with power weights. Furthermore, we establish a multivariable dual $p$-Hardy inequality with a sharp constant. The proof proceeds by first establishing the inequality for two variables and then extending the argument to multiple variables, while preserving the sharpness of the constant.

Berger and Coburn showed that boundedness of a Toeplitz operator $T_g$ on the Fock space controls the heat transform $g^{(t)}$ for $1/4<t<1$, and conjectured the endpoint $g^{(1/4)}$ characterizes boundedness. Looi recently disproved this by constructing a bounded $T_g$ with unbounded $g^{(1/4)}$. We disprove the converse implication by showing that there exists a real-valued admissible symbol $g$ such that $g^{(1/4)}\in C_0(\mathbb C^n)$ while $T_g$ has no bounded extension. Despite this two-sided failure, we show that the forward implication has a sharp Schatten-class form, that is, $p=1$ is the exact Schatten exponent for which $T_g\in S_p$ forces $g^{(1/4)}$ to be bounded. The trace-class operators appearing in Berger and Coburn's trace formula have divergent trace norms as $t\downarrow1/4$, yet converge strongly to $2^n J$, and it follows that $g^{(1/4)}(a)=2^n\operatorname{tr}(T_gW_aJW_a^*)$ and $\|g^{(1/4)}\|_\infty\leq2^n\|T_g\|_{S_1}$ for every admissible symbol $g$ with $T_g$ trace class, where $J$ is the parity operator and $W_a$ is Weyl translation. We prove that $2^n$ is optimal, and the same bound holds for $T_g\in S_p$ with $0<p\leq1$. For $p>1$, no corresponding $S_p$ estimate is possible, even for compactly supported smooth symbols. Moreover, a Baire category argument shows there is an admissible symbol $g$ with $T_g\in S_p$ for every $p>1$ while $g^{(1/4)}$ is unbounded, though it yields no explicit symbol. We also determine the optimal constants in the Berger–Coburn estimates for $1/4<t\leq1$.

In the present work we study for a given Banach space the quotient of the family of equivalent norms modulus the collinearity relationship and prove that under a Banach-Mazur type metric this family is a complete metric space, and we also prove the completeness of the family of constructible norms for the Banach space of bounded Lipschitzian mappings between convex sets of a Banach space.

We discuss some theoretical results and their applications to specific practical problems from wireless communications. We assume that we know the noisy version of the signal, which is sparse with respect to a given system of elements (dictionary), at a finite number of points and we want to approximately recover it. This problem of recovery of a noisy signal is closely related to the problem of establishing the Lebesgue-type inequalities for the corresponding algorithms and it motivates us to prove such inequalities. Under certain conditions on a dictionary (RIP-type condition, coherence condition) we obtain different kinds of the Lebesgue-type inequalities for the OMP and its version WOMP algorithms. The most important feature of our new theoretical result is the assumption that the dictionary has the RIP-type property instead of the assumption that it is the Riesz basis, which was used in the previous results. Our approach allows us to treat redundant (overcomplete) systems, which is important in applications. We consider the setting of sparse recovery for highly-coherent dictionaries that appear in OFDM setting for wireless communication. Our main example is the oversampled Fourier dictionary and the recovery of the frequency response of a sparse channel. We develop a general approach to such problems and we prove that under certain conditions the OMP algorithm recovers all dictionary elements with large coefficients, and estimate its accuracy (in NMSE metric) in terms of signal-to-noise ratio.

We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \sup_{\theta\in\mathbb{R}}\int_{\theta}^{\theta+1}\left|P\left(\frac12+it\right)\right|^p\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ with $C_p$ independent of the number and choice of prime variables on which $P$ depends. Before the present work, the embedding was known at $p=2$ and, by taking integer powers, at the even exponents $p=2k$; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for $0<p<2$, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for $p\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\in(1,2)$, where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For $1\le p<\infty$, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon–Hedenmalm composition operators.

We prove that in $\mathbb{R}^3$ any $BV$ homeomorphism as well as the weak-* limit of $BV$ homeomorphisms cannot change the sign of the Jacobian, i.e., we have $J_f \geq 0$ almost everywhere or $J_f \leq 0$ almost everywhere.

We establish sharp localized $L^r$ bounds for compactly supported functions of bounded mean oscillation (BMO). Classical estimates obtained via the John-Nirenberg inequality bound the $L^r$ norm of such a function in terms of the diameter of its support. We show that the diameter dependence can be replaced by the Lebesgue measure of the support, the relevant quantity for thin, highly eccentric domains. The proof relies on non-increasing rearrangements and the Bennett-DeVore-Sharpley theorem, and requires no vanishing-oscillation hypothesis. We present physical and operator-theoretic applications: geometric collapse in tubular neighborhoods, a localized Cwikel-Lieb-Rozenblum bound for Schrödinger operators, and decoupling bounds for the localized action of Calderón-Zygmund singular integrals.

We establish a Sarason-type commutant lifting theorem in the framework of noncommutative Drury-Arveson space and apply it to solve the noncommutative Nevanlinna-Pick interpolation problem on the noncommutative row ball. In particular, it recovers the classical Nevanlinna-Pick interpolation theorem over the open unit disc. We also provide a Beurling-Lax-Halmos-type characterization of shift-invariant subspaces of noncommutative Drury-Arveson space using dilation theoretic technique.

In this short note, we study the small perturbations of multiplication structure of a Banach algebra $\mathcal A$, and its impact on the two Arens products on the bidual $\mathcal A^{**}$. We show that the property of being \say{Arens irregular} is stable under sufficiently small perturbation in the multiplication structure of a Banach algebra. Further we show that for any Banach function algebra which is equivalent to a uniform algebra, the Arens regularity is stable under small perturbation on the multiplication structure of the original Banach function algebra.

We study the continuity equation with reaction $\partial_t\mu + \operatorname{div}(v\mu) = w\mu$ on $[0,T]\times\mathbb{R}^d$, driven by Borel velocity and reaction fields $v$ and $w$. We provide the first version of a Lagrangian superposition principle for the equation in its natural generality, assuming only finite quadratic energy without imposing boundedness conditions on the reaction term neither any form of compactness for the support of the measure. More precisely, every solution $\mu \in \mathcal{C}([0,T];\mathscr{M}_+(\mathbb{R}^d))$ with finite quadratic energy is represented by a probability measure $\eta$ on absolutely continuous curves in the geometric cone over $\mathbb{R}^d$, concentrated on the solutions of the characteristic system $x' = v(x)$, $k' = w(x)\,k$, through the $2$-homogeneous marginal $h^2_t(\eta)=\mu_t$. We also establish the converse implication. The proof passes through a regularization of the triple $(v,w,\mu)$ which produces fields satisfying only local bounds; the core of the argument is therefore a representation theory under such local assumptions, whose key tool, of independent interest, is a two-time representation formula relating $\mu_s$ and $\mu_t$ along the flow of $v$ for arbitrary times $s,t$.

We present self-contained martingale proofs of Talagrand's convolution conjecture in the Gaussian and Boolean settings with explicit dimension-free constants. The argument is based on the analysis of the stopped density martingale and a weighted Itô isometry estimate. The Gaussian case was settled by Eldan and Lee, and Lehec; the Boolean case was proved up to a $\log\log$ factor by Chen and recently settled by Lu, Guo and Fang.

Our main result is a $3\sqrt{2\pi}$ bound for the Komlós signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $\kappa\ge0$, if a bounded open convex set supports a probability density with directional total variation at most $\kappa$ in every unit direction, then its open-set Banaszczyk transform supports another such density with the same $\kappa$, provided the translation vector $v$ satisfies $\kappa\|v\|_2\le1/3$. As a consequence, every finite set system in which each element belongs to at most $t$ sets, where $t\ge1$ is an integer, admits a two-coloring whose imbalance in each set is less than $3\sqrt{2\pi t}$. This gives the square-root dependence predicted by the Beck-Fiala conjecture. The proof was discovered by the Odin Automatic AI Research Agent.

In this paper, we establish the saturation order and a local inverse result for the uniform approximation of non-negative, bounded, and uniformly continuous functions on $\mathbb{R}$ by max-product generalized sampling operators based on suitable kernel functions. In particular, assuming that the kernel is an even centered bell-shaped function, we first show that $1/w$, $w>0$, is the uniform saturation order, with the corresponding saturation class coinciding with the class of non-negative constant functions. This means that $1/w$ is the best possible rate of convergence that the max-product generalized sampling operators can achieve when approximating non-trivial (i.e., non-constant) non-negative, bounded, and uniformly continuous functions on $\mathbb{R}$. Moreover, it is known that, for Lipschitz continuous functions on $\mathbb{R}$, the approximation order is $1/w$ as $w \to +\infty$. Here, we show that this result can be locally reversed. Specifically, we prove that if $f$ can be approximated at the rate $1/w$ on a compact interval $[a,b]\subset\mathbb{R}$, then $f$ is Lipschitz continuous on $[a,c]$ for every $c \in [a,b)$ whenever $0<a<b$, and on $[c,b]$ for every $c \in (a,b]$ whenever $ a<b<0$. Finally, under the same assumptions on the kernel, we establish a strong localization result for sequences of truncated max-product generalized sampling operators in the case of strictly positive and bounded functions defined on $[0,1]$. All these results extend previous results of Coroianu and Gal, which were established only for specific sinc-type kernels, to a broader class of kernel functions.

The weak dominance of aggregation operators, particularly between triangular norms (t-norms) and triangular conorms (t-conorms), has attracted considerable attention in aggregation operator theory. While several characterizations have been obtained for Archimedean and continuous cases, a general criterion for continuous t-conorms over continuous t-norms remains to be fully clarified. In this paper, we provide a complete characterization of a continuous t-conorm weakly dominating a continuous t-norm. We first reduce the problem for ordinal sum operators to that for their single Archimedean components, and then express the weak dominance condition entirely in terms of the additive generators of these components. Our approach covers both strict and nilpotent cases uniformly, and recovers the known results for Archimedean operators as a special case.

We extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is, for the first time, the ability to consider more than two sets, a task which has been in the mind of researchers for many years without much success in fulfilling it because of a certain theoretical obstacle related to cycles and minimizers of general functionals. The other direction is the ability to handle each set as an intersection of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to these extensions is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fejér monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.

Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition

We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.

We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1< p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dotsc,T^k)$ is disjoint hypercyclic for any $k\geq 2$. We also show that the similar typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

A Hermitian completely bounded map $\Phi:\mathscr{A}\to B(\mathcal{H})$ need not have a canonical completely positive analogue of an absolute value. We study this problem through the ordered set $\mathsf{E}(\Phi)$ of completely positive symmetric majorants of $\Phi$, which we call Wittstock envelopes. We show that every Wittstock envelope canonically determines a minimal Wittstock envelope below it. However, minimal envelopes need not be unique. We therefore study the stronger question of when a least completely positive majorant exists. We obtain a criterion, in terms of central supports of spectral projections, for the existence of least majorants below a fixed envelope. In finite dimensions, the Choi representation gives a simple characterization: a least completely positive majorant exists if and only if \(\Phi\) or \(-\Phi\) is completely positive. Finally, we use the envelope-relative modulus to define envelope-relative perspectives for Hermitian completely bounded maps. These constructions extend Okayasu's completely positive map perspectives, reduce to the usual perspective in the completely positive case, and remain dependent in general on the chosen Wittstock envelopes.

For every $n\geq2$, we consider a Gaussian zonoid of revolution $Z_n$ arising from works of Vitale and Mathis. We prove that $Z_n^*$ is also a zonoid and compute the Banach–Mazur distance from $Z_n$ to the Euclidean ball. This distance is independent of the dimension and is approximately $1.10$. Consequently, the supremal Banach–Mazur distance among zonoids whose polars are zonoids does not converge to $1$. The construction also produces a separable real Banach space $X$, not isometric to a Hilbert space, such that both $X$ and $X^*$ embed linearly isometrically into $L_1$.

Let $A^2(\mathbb D)$ be the unweighted Bergman space and write $Q_m(S)=T_{B_m(S)}$ for the map induced by the $m$th higher-order Berezin transform. Suárez asked in 2005 whether $Q_m(S)$ converges to $S$ in operator norm for every $S$ in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator $S$ such that \[ \sup_{m\geq 0}\|Q_m(S)\|=\infty. \] In particular, along a strictly increasing sequence $(m_n)$ one has \[ \|Q_{m_n}(S)-S\|\longrightarrow\infty. \] The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the $m$th matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal–Hankel theorem showing that the associated weighted Hankel transformation fails to map $\ell^2$ boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.

The main purpose of this article is to provide a fractional counterpart of the well-known Maz'ya inequality on the half-space, that is $$ \int_{\mathbb{R}^{d}_{+}}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dy\,dx\ge\mathcal{D}_{d,s,p}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)|^p}{x_d^{sp}}dx+C_{d,s,p,\tau}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)|^p}{x_{d}^{sp-\tau}\left(x_{d-1}^2+x_d^2\right)^{\tau/2}}dx, $$ where $\mathcal{D}_{d,s,p}$ stands for the sharp constant in the fractional Hardy inequality on a half-space $\mathbb{R}^{d}_{+}$. We also obtain a similar result in the setting of Sobolev–Bregman forms.

We develop a time–frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

We present a general proof of the Fong-Tsui conjecture for bounded operators on arbitrary complex Hilbert spaces. Specifically, we show that $|T|\leq|\operatorname{Re}T|$ implies that $T$ is self-adjoint. The argument combines a positive inverse of a Sylvester map with a spectral cutoff determined by the norm of the positive defect $|\operatorname{Re}T|-|T|$. A local vanishing lemma reduces the analysis to the classical squared self-adjointness criterion, while positivity of the defect yields a global norm contradiction. We formulate the argument as an abstract four-operator vanishing principle, without compactness, trace, or separability assumptions. We also establish a quantitative stability estimate: if $|T|\leq|\operatorname{Re}T|+\varepsilon I$ and $0\leq\varepsilon\leq|T|$, then $|\operatorname{Im}T|\leq6|T|^{7/8}\varepsilon^{1/8}$. The constant is independent of the dimension, and the exponent is not claimed to be optimal. Large language models (LLMs) were used to assist with proof development, algebraic calculations, numerical checks, and auditing of the arguments.

Let $\gamma\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let $\lambda_\gamma$ be the first eigenvalue of the periodic curvature Schrödinger operator $-d^2/d s^2+\kappa(s)^2$. We obtain \[ \lambda_\gamma\geq \frac{\sqrt{\pi}}{2} \left(\frac{\Gamma(7/6)}{\Gamma(5/3)}\right)^3. \] This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.

We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.