The present paper is concerned with the L2-approximation properties of exponential splines with n knots for functions defined on an interval ohm = (a, b). We first specify the Banach native space for these splines and show that it continuously embeds in L2(ohm). We derive sharp bounds on the approximation error rate for functions in this native space by exponential splines with n knots. To prove the optimality of these rates, we show that these native spaces are equivalent (as Banach spaces) to spaces of higher-order bounded variation on ohm. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
In 2023, Güntürk and Thao proved that the sequence (x(n))n∈N generated by random (relaxed) projections drawn from a finite innately regular collection of closed subspaces in a real Hilbert space satisfies ∑n∈N‖x(n)−x(n+1)‖γ<+∞ for all γ>0. We show that the Güntürk–Thao result is sharp in the sense that it fails for two closed subspaces that is not innately regular. We also extend their result to a finite collection of polyhedral cones. Moreover, we construct examples showing the tightness of our extension: indeed, the result fails for a line and a convex set in R2, and for a plane and a nonpolyhedral cone in R3.
This paper studies binary classification using ReLU neural networks within the statistical learning framework. Unlike existing approaches that impose sparsity constraints on hidden neurons and boundedness assumptions on output functions, we consider a hypothesis space generated by a fixed hard-tanh layer and a trainable shallow ReLU layer. Using recent advances in approximation theory and refined complexity control for this hypothesis space, we derive optimal learning rates, up to logarithmic factors, for the excess misclassification error under smoothness assumptions on the regression function and the Tsybakov noise condition. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
A polynomial p(z) of degree n is self-reciprocal if p(z) = z(n) p(1/z).The problem of finding the sharp estimate of ||p'|| in terms of ||p|| for self-reciprocal polynomials has been open for a long time and many papers have appeared in this direction. Govil, Jain and Labelle (Proceedings of the American Mathematical Society, 57(2):238-242, 1976) proved that for self-reciprocal polynomials having all the zeros either in left half-or right half-plane, the inequality n/2 parallel to p parallel to <= ||p'|| <= n/root 2 parallel to p parallel to holds. In this paper, we prove that for all such polynomials ||p'||/parallel to p parallel to <= n(root 2+1/4) + root 2-1/2 = n/root 2 - root 2-1/4 (n - 2) This sharpens the result of Govil, Jain and Labelle for all n >= 2 and also answers a conjecture of the authors (Acta Mathematica Hungarica, 172(1):146-160, 2024). Moreover, we prove that for all such polynomials of large degree n >= 2 root 2/epsilon [log 1+epsilon/root 2/log(1-epsilon(6)/36)], where 0 < epsilon <= root 2-1/4, ||p '|| / ||p|| <= (1/2 + epsilon)n.
In this paper, we explore the expressive power of deep 2D ReLU convolutional neural networks (CNNs). We introduce a novel construction of localized CNNs that selectively focus on specific elements of an input matrix, enabling precise controls over feature extractions. We then prove that any ReLU fully-connected neural network (FNN) can be represented as a 2D ReLU CNN of comparable size, demonstrating the versatility of CNNs in approximating functions traditionally handled by FNNs. Based on this representation, we extend approximation results for FNNs to 2D CNNs, covering Sobolev spaces, Korobov spaces, and variation spaces. Our analysis focuses on CNNs with zero-padding convolutions and no pooling operations, highlighting the effectiveness and flexibility of such architectures for high-dimensional approximation tasks. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
