The completeness property of the Nyman-Beurling space is closely related to the Riemann hypothesis. Within this context, we consider the Friedrichs angle between two subspaces of the Nyman-Beurling space. We prove that if the Riemann hypothesis is true, then the Friedrichs angle between the subspaces is zero. Moreover, we present an unexpected result that holds regardless of the truth of the Riemann hypothesis. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In the study of the Riemann hypothesis, Báez-Duarte provided a strong version of the Nyman–Beurling criterion by considering the approximation of the characteristic function using natural Beurling functions in L^2 . Later, Balazard and Saias posed a question regarding the coefficients of these natural Beurling functions, which Weingartner answered using the sequence space l^2 . In this paper, we extend Weingartner’s theorem to the Banach space L^p , a setting that is more difficult to handle than the Hilbert space l^2 . We also show that the Beurling functions form the unique best minimal system for approximating the characteristic function in L^p .

It is known that the Riemann hypothesis is true if and only if chi belongs to the Nyman-Beurling space, where chi denotes the characteristic function on the unit interval. Moreover, Baez-Duarte, Balazard, Landreau, and Saias provided a lower bound for the distance between chi and the subspaces of the Nyman-Beurling space. In this paper, we extend the result of Baez-Duarte et al. by considering more general characteristic functions. In addition, we give some applications and raise several questions related to the Riemann hypothesis. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Nyman and Beurling showed that the Riemann hypothesis is equivalent to the density of a certain function space in $L^2(0,1)$, called the Nyman–Beurling space. In this paper, we consider subspaces spanned by generators of the Nyman–Beurling space and give

In the setting of the unit disk we have recently obtained characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators acting from a standard weighted Bergman space into the corresponding weighted Lebesgue space. In this paper we extend those results to the case when the exponents of the domain space and the target space are different.

We define new integral\r\noperators on the Haydy space similar to Szego projection. We show that these operators map\r\nfrom Hp to H2 for some 1 ≤ p 2, where the range of p is depending\r\non a growth condition. To prove that, we generalize the Hausdorff-Young Theorem\r\nto multi-dimensional case.

We define new integral operators on the Haydy space similar to Szegö projection. We show that these operators map from Hp to H2 for some 1 ≤ p 2, where the range of p is depending on a growth condition. To prove that, we generalize the Hausdorff-Young Theorem to multi-dimensional case.

Choe et al. have recently characterized compact double differences formed by four composition operators acting on the standard weighted Bergman spaces over the disk of the complex plane. In this paper, we extend such a result to the ball setting. Our characterization is obtained under a suitable restriction on inducing maps, which is automatically satisfied in the case of the disk. We exhibit concrete examples, for the first time even for single composition operators, which shows that such a restriction is essential in the case of the ball.

Let N be the linear space of functions Sigma(n)(k=1) a(k)rho(theta(k)/x) with a condition Sigma(n)(k=1) a(k)theta(k) = 0 for 0 < theta(k) <= 1. Here rho(x) denotes the fractional part of x. Beurling pointed out that the problem of how well a constant function can be approximated by functions in N is closely related to the zero-free region of the Riemann zeta function. More precisely, Baez-Duarte gave a zero-free region related to a L-p-norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the L-infinity-norm estimation of a constant function and give a wider zero-free region than that of the Baez-Duarte result. (c) 2018 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

It is known that the Riemann hypothesis holds if and only if the function chi((0,1)) can be approximated by linear combinations of u(alpha) in L-2(0,1). Here u(alpha)(x) is defined by [alpha/x] - alpha[1/x] for 0 < alpha < 1. In this note we generalize the Beurling's equivalent condition by replacing the function chi((0,1)) with chi((a,b)) for any 0 <= a < b <= 1. (C) 2016 Elsevier Inc. All rights reserved.

The Nyman–Beurling’s theorem states that the Riemann hypothesis is true if and only if the set of all linear combination of the functions \({u_\alpha(x) := [\alpha/x]-\alpha[1/x]}\) is dense in \({L^2(0,1)}\). In this note we show that if a function has a nonzero limit at the origin, the function is not orthogonal to some \({u_\alpha}\).

It has been known that the difference of two composition operators induced by linear fractional self-maps of a ball cannot be nontrivially compact on either the Hardy space or any standard weighted Bergman space. In this paper we extend this result in two significant directions: the difference is extended to general linear combinations and inducing maps are extended to linear fractional maps taking a ball into another possibly of different dimension.

It is known that the partial sum of the Taylor series of an holomorphic function of one complex variable converges in norm on H-p (D) for 1 < p < infinity. In this paper, we consider various type of partial sums of a holornorphic function of several variables which also converge in norm on H-p (B-n) for 1 < p < infinity. For the partial sums in several variable cases, some variables could be chosen slowly (fastly) relative to other variables. We prove that in any cases the partial sum converges to the original function, regardlessly how slowly (lastly) some variables are taken.

In the setting of the Fock space over the complex plane, Bauer and Lee have recently characterized commutants of Toeplitz operators with radial symbols, under the assumption that symbols have at most polynomial growth at infinity. Their characterization states: If one of the symbols of two commuting Toeplitz operators is nonconstant and radial, then the other must be also radial. We extend this result to the Fock–Sobolev spaces.