The main object of this article is to provide closed-form expressions for each of the following sums: K-q := Sigma(alpha is an element of J) (2N)(-q alpha) and (K) over bar (q) := Sigma(alpha is an element of J) (2N + 1)(-q alpha), which appear in the analysis of stochastic partial differential equations. Our main results express these sums as finite products of Gamma functions: K-q = Pi(q-1)(j=0) Gamma(1 - omega(j)/2) and (K) over bar (q) = 2(q)/pi(q/2) Pi(q-1)(j=1) Gamma(3/2 - omega(j)/2), where omega = e(2 pi i/q) is a primitive qth root of unity and q >= 2 is a natural number. The relevance and potential uses of explicit evaluations of such product series in, for example, white noise theory and L & eacute;vy noise analysis are also indicated.
Translation generated spaces of square-integrable functions on the Vilenkin group are studied using a C-bracket. Principal shift-invariant spaces are characterized via the periodization function and the C-bracket. Conditions for semi-orthogonal Parseval wavelets are established, leading to a generalized multiresolution analysis (GMRA) of the semi-orthogonal Parseval frame. Examples are provided to support and illustrate the theoretical results.
We consider H 1 {H<^>{1}} -Sobolev spaces associated with the Calder & oacute;n-Torchinsky Hardy spaces H 1 {H<^>{1}} and characterize these spaces by means of certain square functions. This characterization generalizes and, in fact, improves upon previously known results in the Euclidean setting.
We study a periodic problem for the non-autonomous Duffing equation u'' = p(t)u - h(t)|u|(lambda )sgn u (lambda > 1) and unforced undamped periodic motions of the mechanical part of the energy harvester described by the equation my'' + by' + ky + F-mag(t, y) = f(t).
Abstract Sequences, or more generally, nets, are important in numerous branches of mathematics, such as analysis, number theory and topology. For the reasons explained in introduction, in some works we move from sequences to nets. Considering limits of sequences in a Hausdorff space X assigns a function lim {\lim} from the set c ( X ) {c(X)} of convergent sequences to X . Inspired by this, a G - method , as a generalization of the function lim {\lim} , is defined by replacing the set c ( X ) {c(X)} with a certain class of sequences. Originated by such G -methods, the sequential versions of some fundamental properties in topology, such as continuity, compactness, and connectedness, have been recently generalized by G -methods defined for sequences in sets. This paper presents a study of convergent G 𝔫 {G_{\mathfrak{n}}} -methods involving nets rather than sequences as generalizations of G -methods; and then gives some properties and characterizations of G 𝔫 {G_{\mathfrak{n}}} -compactness and G 𝔫 {G_{\mathfrak{n}}} -countably compactness.
This paper develops a comprehensive group-theoretic framework for analyzing transformations and connections between polynomial sets. Building upon the fundamental characterization [Y. Ben Cheikh, Some structures on the polynomial sets, Dolomites Res. Notes Approx., to appear] of the space & Popf; {\mathbb{P}} of all polynomial sets as the Cartesian product & Popf; congruent to Lambda ( - 1 ) / & Rscr; & times; & Sscr; 1 & times; & Sscr; 0 {\mathbb{P}\cong\Lambda<^>{(-1)}/\mathscr{R} imes\mathcal{S}_{1} imes\mathcal{% S}_{0}} , we introduce and systematically study T-operators acting on this space. These operators provide a unified mechanism for connecting arbitrary polynomial sequences, while stabilizer analysis reveals the inherent symmetry properties of classical polynomial families. Our approach yields powerful new tools for addressing fundamental problems in polynomial theory, including connection coefficients, transformation classification, and the systematic study of relationships between diverse polynomial families. The framework provides both theoretical insights and practical computational methods for working with polynomial sets and their interconnections.
In this paper, the authors establish variable CBMO estimates for the commutators of singular integrals with rough kernels on variable Herz-type spaces, which extend some known results.
Let X(IR+) be a Lorentz space L-p,L-q(IR+) with 1 < p, q < oo or a reflexive Orlicz space L-Phi(IR+). We establish a Fredholm criterion for operators from the Banach algebra generated by all Wiener-Hopf operators with continuous symbols on the space X(IR+).
This paper presents the concept of the 2-Toeplitz operator on K & ouml;the spaces, which is a subclass of Fr & eacute;chet spaces that has recently gained significant prominence in the study of functional analysis and operator theory. We establish detailed characterizations of the continuity and compactness of this operator, providing both necessary and sufficient conditions. Additionally, several concrete examples are provided to illustrate the theoretical results.
This paper investigates the existence and multiplicity of normalized solutions of the double critical Schr & ouml;dinger-Poisson system on the first Heisenberg group. Together with the truncation techniques, concentration-compactness principle, and genus theory, we establish the existence and multiplicity of normalized solutions for this system. To some extent, the results presented in this paper can be regarded as a supplement and enrichment of previous studies.
In this paper, we characterize the uniform polynomial stability of evolution families by means of Banach spaces of sequences or functions. This approach provides a framework that not only unifies existing versions of Barbashin-type theorems, but also provides new characterizations.
This paper provides a complete characterization of the closure of analytic tent spaces within the logarithmic Bloch-type space.
The paper addresses the periodic problem u '' = p(t)u - q(t, u)u, u(0) = u(omega), u '(0) = u '(omega), where p : [0, omega] -> R is a Lebesgue integrable function and q : [0, omega] x R -> R is a function satisfying Carath & eacute;odory conditions. By combining the maximum and antimaximum principles with the method of non-ordered lower and upper functions, sufficient and necessary conditions are found for the existence of a positive solution to the given problem. The main results are applied and further refined for the equation u '' = p(t)u - h(t)phi(u)u, whose non-linearity has separated arguments.
We establish a characterization of the rate of approximation of the generalized sampling operator for Lipschitz functions. Our primary goal is to weaken assumptions on the kernel of the sampling operator in order to be able to cover cases to which the already known general theorems are not applicable. In particular, we consider sampling operators generated by the Fej & eacute;r kernel and low-order Bochner-Riesz kernels.
In this paper we prove a Poincar & eacute;-type inequality within the framework of Luxemburg norms constructed by using some continuous and bounded functions which depend on the test functions.
In this paper, we present new abstract versions of Korovkin-type approximation theorems in the framework of modular spaces by employing sequences of monotone and sublinear operators with respect to power series methods. We further calculate the rate of convergence in terms of modulus of continuity and demonstrate the applicability of our abstract results through Bernstein-Kantorovich and Bernstein-Kantorovich-Choquet operators in Orlicz spaces under non-additive measures. Finally, we relax the monotonicity of the sublinear operators in the modular Korovkin theorem.
The works of the Swiss mathematician Karl Rudolf Fueter and the Romanian mathematicians Grigore Constantin Moisil and Nicolae Victor Teodorescu marked the starting point of a hypercomplex analysis over the skew field of real quaternions. Nowadays, this function theory, so-called quaternionic analysis, is the most attractive and close generalization of complex analysis since it preserves many of its key features. After the discovery of the quaternions by William Rowan Hamilton occurred in 1843, the concept of reduced quaternions, referred to quaternions with vanishing third imaginary unit-coefficient, emerged. A modification of quaternionic analysis, based on functions that map from domains of three-dimensional real space & Ropf; 3 {{\mathbb{R}}<^>{3}} to the set of reduced quaternions, denoted by & Ascr; {{\mathcal{A}}} , has recently been developed. The primary aim of the present paper is to derive an analogue of the Sokhotski-Plemelj formulas for the reduced quaternionic Cauchy-like integral in the context of & Ascr; {{\mathcal{A}}} -valued functions on domains of & Ropf; 3 {{\mathbb{R}}<^>{3}} with Ahlfors-regular boundary, which is considered distinct from that of quaternionic analysis, despite their natural similarities. Our second task is to establish a Poincar & eacute;-Bertrand formula for interchanging the order of integration of two repeated singular reduced quaternionic Cauchy-like integrals, drawing on arguments involving quaternionic analysis. Finally, a Geza Freud-type theorem is proved, as well as some local properties of the singular reduced quaternionic Cauchy-like integral.
Let A be a nonzero positive operator on a complex Hilbert space (H, ), inducing the semi-inner product < x, y >(A) := < Ax, y > for x, y is an element of H. The space (H, parallel to center dot parallel to(A)), where parallel to x parallel to(A) = root < x, x >(A), is a semi-Hilbert space equipped with a seminorm induced by A. In this paper, we establish the inequality vertical bar Sigma(n)(i=1) ci < x, yi >(A)vertical bar(2) <= parallel to x parallel to(2)(A) (Sigma(n)(i=1) vertical bar c(i)vertical bar(p) (Sigma(n)(j=1) vertical bar < y(i), y(j)>(A)vertical bar))(1/p) (Sigma(n)(i=1) vertical bar c(i)vertical bar(q)(Sigma(n)(j=1) vertical bar < y(i), y(j)>(A)vertical bar))(1/q), where x, y(1),..., y(n) is an element of H, c(1),..., c(n) are complex numbers, and p, q > 1 satisfy 1/p + 1/q = 1. We derive analogues of the Bombieri, Selberg, and Heilbronn inequalities in the context of semi-Hilbert spaces. Applications to A-seminorms associated with n-tuples of operators in semi-Hilbert spaces, including the joint A-numerical radius and the Euclidean A-seminorm, are also presented.
Inspired by a combinatorial identity given by Akyuz and Halici, we present three closely related summation formulas. One of our results states that Sigma([n/2]-j)(nu=0) (nu+j m)(nu+j-m nu)(n 2(nu+j)) = 2(n-2j-1) n/n-j(n-j j)(j m), where m >= 0, j >= 0 and n >= 1 are integers with m <= j <= [n/2]. We use properties of hypergeometric functions and the method "comparison of coefficients" to prove our theorems.
Let R be a ring, (S , <=) a strictly totally ordered monoid and suppose also omega : S -> End (R) monoid homomorphism. A skew generalized power series ring R [[S, omega, <=]] consists of all functions from a monoid S to a coefficient ring R whose support contains neither infinite descending chains nor infinite anti-chains, equipped with point-wise addition and with multiplication given by convolution twisted by an action omega of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are the skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Malcev-Neumann series rings and generalized power series rings as well as the untwisted versions of all of these objects. In the present article, we study the so-called (S , omega)-McCoy condition on R that is a generalization of the standard McCoy condition from polynomials to skew generalized power series, thereby generalizing some of the existing results in the literature relevant to the subject.
