A polynomial p(z) of degree n is self-reciprocal if p(z)≡znp(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 n2p≤p′≤n2p holds. In this paper, we prove that for all such polynomials p′p≤n2+14+2−12=n2−2−14(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≥22ϵlog1+ϵ2log(1−ϵ6/36), where 0<ϵ≤2−14, p′p≤12+ϵn.
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.
In this paper we investigate the monotonicity properties of the ratio of Ferrers functions of the first kind and we use these properties to derive functional bounds for this ratio, which are shown to be sharp and asymptotically accurate for large values of the parameters. By using these bounds, we establish the convexity of the Ferrers function of the first kind and we obtain some Tur & aacute;n type inequalities with respect to both of the parameters, which are sharp at the endpoints of the argument. Finally, we propose some open problems concerning the log-convexity and log-concavity of Ferrers functions of the first kind. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Functional linear model is a powerful tool for analyzing complex relationships involving functional data, which extends the traditional linear model to accommodate infinite-dimensional data. In this paper, we study an online learning algorithm with a robust loss function L sigma for functional linear model in a reproducing kernel Hilbert space (RKHS). The proposed loss function L sigma, constructed using a windowing function W and a scale parameter sigma, can generalize a wide range of commonly used robust losses for regression through appropriate selection of these parameters. Under mild conditions on the slope function regularity, noise structure and the capacity of the hypothesis space, fast convergence rates of both prediction and estimation problems are achieved. Our theoretical analysis demonstrates that with a properly chosen scale parameter sigma, the final iterate of our online algorithm achieves optimal capacity-independent convergence rates for the prediction problem in the mini-max sense. Moreover, optimal capacity-dependent rates for strong convergence in an RKHS are attained if additional information on the underlying function space is available. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Chebyshev spectral methods are fundamental in numerical analysis and scientific computing. However, existing studies primarily focus on asymptotic error estimates for polynomial approximations of certain singular functions and lack sharp pointwise error estimates for functions with interior or endpoint singularities in fractional spaces. In this work, we introduce a novel analytical framework to rigorously derive explicit and sharp pointwise error estimates for Chebyshev polynomial approximations of functions exhibiting interior or endpoint singularities within fractional spaces. Our analysis provides three key contributions: (i) new definitions of fractional space that yield more precise theoretical results, (ii) explicit and sharp pointwise error estimates for Chebyshev approximations of functions with interior or endpoint singularities, and (iii) an extension to Chebyshev spectral differentiation, resulting in explicit and improved upper error bounds. Numerical experiments are presented to support the theoretical results. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We construct new polynomial interpolation schemes of Hermite types in Rn and on the unit sphere in Rn+1. The interpolation conditions are the differential operators induced by harmonic polynomials in Rn. We also give a divisibility criterion for the polynomial parallel to x-a parallel to 2 and for the affine polynomial defining the tangent hyperplane of the unit sphere, and use it to prove the regularity of the Hermite interpolation schemes. Moreover, we establish formulas for the interpolation polynomials in the form of recurrence relations and a decomposition formula for multivariate polynomials. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm ‖f(k)‖ of the kth derivative of a function f:R→R by the values of the uniform norm of f and uniform norms of several its higher derivatives (‖f(r)‖ and ‖f(r−1)‖, or ‖f(r)‖ and ‖f(r−2)‖, or ‖f(r)‖, ‖f(r−1)‖ and ‖f(r−2)‖) using standard techniques can be obtained from the known solutions to the Kolmogorov problem about existence of a function with given norms of its derivatives.
We first prove the best constant C in a Landau type inequality parallel to f 'parallel to 5 infinity <= C parallel to f parallel to 3 infinity parallel to f ''parallel to infinity parallel to f '''parallel to infinity, and with similar approach, we prove the best constants in a more general form parallel to f 'parallel to 2+eta infinity <= C eta parallel to f parallel to 1+eta infinity parallel to f ''parallel to 1-eta infinity parallel to f '''parallel to eta infinity for all 0 <= eta <= 1. Here we have the direct expression for each of the best constants C eta = where theta eta = theta eta eta(1-theta eta)2+eta 2(1/4-theta eta 2/3)1+eta, root 3 + 3 eta - -3 eta 2-6 eta + 9 8 +4 eta . A Landau type inequality and another special case of Landau-Kolmogorov inequality (Shilov's result) can also be deduced from this general form, with the best constant. (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/).
The local Heun solution is the unique solution to Heun's equation that is analytic in the unit disk centered at 0 is an element of C and taking the value 1 at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In two particular cases when the total number of free parameters of the equation is reduced from 6 to 5, the obtained formula is used to derive an explicit estimate on the coefficients. Moreover, asymptotic behavior of the estimate for large indices is determined. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study strictly positive definite functions of finite orders on real inner product spaces. Via exploring new connections to theories of Chebyshev ranks and Schur polynomials, we obtain quantifiable conditions for such functions. To assist field scientists in selecting an ideal multivariate polynomial interpolant, we propose and study the notions of "interpolating rank" and "minimal seminorm interpolation." We quantify both Chebyshev ranks and interpolating ranks in terms of the number of common zeros of a certain finite collection of Schur polynomials. As a byproduct, we develop a mechanism to construct positive-rank Chebyshev systems of which examples are scarce in the literature. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The asymptotic behavior of the Mellin transform of the associated B-splines B & lowast;N(t) := t-N BN (t) with special knots in terms of theta-like functions is found. The proof is based on polynomial interpolation of power functions and properties of certain theta-like functions. Pointwise asymptotics of B & lowast;N and BN (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Known or essentially known results about duals of interpolation spaces are presented, taking a point of view sometimes slightly different from the usual one. Particular emphasis is placed on Alberto Calder & oacute;n's theorem describing the duals of complex interpolation spaces. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. MSC: primary 46B70; secondary 46B10
Let alpha, beta E (0, oc) and U alpha,beta be the dyadic maximal operator. In this article, the authors find some sufficient conditions on the indices alpha, beta to guarantee the boundedness of the operators U alpha,beta on weighted Lebesgue spaces. These conditions in special weight case turn out to be necessary and hence are sharp in some sense. Moreover, the authors show that a weight w belongs to the Muckenhoupt weight class Ap, p E (1, oc), if and only if the operator U alpha,beta is bounded on weighted Lp spaces. Furthermore, the authors establish the weighted vector-valued dyadic maximal inequality and the weighted weak-type (1, 1) estimate for the operators U alpha,beta. As applications, the authors obtain the convergence of Fej & eacute;r means of Walsh-Fourier series of martingales in both pointwise and Musielak-Orlicz spaces. These main results in Musielak-Orlicz space case remedy a missing necessary assumption of Theorems 3.2 and 3.4 in Weisz et al. (2023). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The focus of this work is on the properties of the q-Durrmeyer operators Mn,q, n E N, and M infinity,q introduced, for q E (0, 1), by V. Gupta and H. Wang. First, it is shown that, for each f E C[0, 1], the sequence {Mn,q f}nEN converges to M infinity,q f uniformly on [0, 1] with a rate not slower than Cq, fqn, which refines the previously available result by V. Gupta and H. Wang, and implies the possibility of an analytic continuation for M infinity,q f into a neighbourhood of [0, 1]. Further investigation shows that M infinity,q f admits an analytic continuation as an entire function regardless of f E C[0, 1]. Finally, the growth estimates for these functions are received and applied to describe the point spectrum of M infinity,q. The paper also addresses the significant differences between the properties of M infinity,q and the previously (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Here, smoothness analysis of almost periodic functions is studied. Analogously to the case of L2(R), the smoothness of the class of Besicovitch almost periodic functions is measured in a classic form by controlling, in some sense, the increments f (x + h) - f (x) and in a dual form by the decay of its Fourier-Bohr transform or by its approximation properties. The same problem is also treated considering the time-frequency representation given by the Gabor transform. Some results are given as equivalence of norms between appropriate function spaces. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The aim of this note is to introduce two new generic Banach lattice couples based on weighted spaces of continuous functions, and weighted Radon measures on the positive real-line, denoted by ->-C and--> M respectively. This leads to a new approach, based on these couples, of the Sedaev-Semenov result regarding the Calder & oacute;n-Mityagin property for weighted L1 spaces. As a consequence is obtained a formal equivalence between the concept of K divisibility and the relative Calder & oacute;n-Mityagin Property between--> M and general Banach couples.
