Search

In this article, the complete convergence for the partial sum of the moving average process { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } is established under some mild conditions, where { Y i , − ∞ < i < ∞ } is a sequence of extended negatively dependent and identically distributed random variables that is stochastically dominated by a random variable Y under a sub-linear expectation (Ω , H , E ̂) , and { a i , − ∞ < i < ∞ } is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results of moving the average process under extended negatively dependent random variables from the traditional probability space to the sub-linear expectation space. [ABSTRACT FROM AUTHOR]

Inequalities are obtained which connect the probability tails and moments of functions of the nth partial sums of independent random variables taking values in a separable Banach space and those for the accompanying infinitely divisible laws. Some applications to empirical processes are studied. [ABSTRACT FROM AUTHOR]

In this present investigation, for functions of the form f(z) = = 1/zp + ∑∞ k=1−p akz k unit disk U∗ = {z ∈ C : 0 < |z| < 1} we introduce a new subclass of meromorphically p-valent functions and investigate convolution properties, Coefficient estimates and contianment for this subclass. [ABSTRACT FROM AUTHOR]

Students’ development of reasoning through algebraic symbols is a crucial component of most mathematics courses. This paper reports the journey of one second-semester calculus student, Cedric, as he attempted to reason about and create algebraic representations for arbitrary partial sums and infinite series through two exploratory teaching interviews. We report Cedric’s symbolizing activity in terms of Eckman's (2023) expression framework, focusing on Cedric’s development and attribution of meaning to a personal expression template for denoting partial sums and series. Specifically, we describe how Cedric leveraged his initial meanings for partial sums to create personal expressions to reason about infinite series. We then describe Cedric's construction of a general personal expression template for partial sums and series, including cognitive and physical modifications he made to his personal expression template as he symbolized various series. Our results show potentially profitable task sequences and lines of questioning that instructors might utilize to help students be more successful in constructing productive meanings for algebraic symbols. [ABSTRACT FROM AUTHOR]

Let λ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sárközy asks to show that if P(x) ∈ Z[x], then the sequence λ(P(n)) changes sign infinitely often, assuming only that P(x) is not the square of another polynomial. We show that the sequence λ(P(n)) indeed changes sign infinitely often, provided that either (i) P factorizes into linear factors over the rationals; or (ii) P is a reducible cubic polynomial; or (iii) P factorizes into a product of any number of quadratics of a certain type; or (iv) P is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of g(P(n)) for g a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on g. This establishes a "99% version" of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of g(P(n)) and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value. [ABSTRACT FROM AUTHOR]

In this paper, a partial Bregman alternating direction method of multipliers (ADMM) with a general relaxation factor α ∈ (0 , 1 + 5 2) is proposed for structured nonconvex and nonsmooth optimization, where the objective function is the sum of a nonsmooth convex function and a smooth nonconvex function without coupled variables. We add a Bregman distance to alleviate the difficulty of solving the nonsmooth subproblem. For the smooth subproblem, we directly perform a gradient descent step of the augmented Lagrangian function, which makes the computational cost of each iteration of our method very cheap. To our knowledge, the nonconvex ADMM with a relaxation factor α ≠ 1 in the literature has never been studied for the problem under consideration. Under some mild conditions, the boundedness of the generated sequence, the global convergence and the iteration complexity are established. The numerical results verify the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Considering the approximation of , with , by the partial sums of the Fourier series in a system of polynomials orthogonal in the sense of Sobolev and generated by a system of classical Laguerre polynomials, we obtain some estimates for the convergence rate of to . [ABSTRACT FROM AUTHOR]

A random walk with counterbalanced steps is a process of partial sums Š(n) = X1 + • • • + Xn whose steps Xn are given recursively as follows. For each n ≥ 2, with a fixed probability p, Xn is a new independent sample from some fixed law μ, and with complementary probability 1 - p, Xn = -Xυ(n) counterbalances a previous step, with υ(n) uniform random pick from {1,...,n -1} We determine the asymptotic behavior of Š(n) in terms of p and the first two moments of μ. Our approach relies on a coupling with a reinforcement algorithm due to H. A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution μ belongs to the domain of attraction of a stable law. [ABSTRACT FROM AUTHOR]

This article, titled "Functions Almost Universal in the Sense of Signs with Respect to the Trigonometric System and the Walsh System," discusses the existence and structure of functions whose Fourier series are almost universal in the sense of signs with respect to the trigonometric system and the Walsh system. The paper explores various definitions and theorems related to universal functions and poses unanswered questions about the existence of certain types of universal functions. The authors provide examples and results from previous studies in the field of complex and real analysis. The article concludes with acknowledgments and funding information. [Extracted from the article]

This article discusses new representations of the values of the Dirichlet beta function at even points. The Dirichlet beta function is a well-known mathematical function, and it is known that the values of this function are transcendental numbers. However, there is limited knowledge about the arithmetic nature of these numbers. The article presents new representations of certain linear combinations of these numbers in the form of series containing logarithms and limits of sequences containing trigonometric functions. The results of the article have applications in various representations of mathematical constants, such as Catalan's constant. The article also acknowledges funding from the Russian Science Foundation and declares no conflicts of interest. [Extracted from the article]

It is known that there exist functions in certain de Branges–Rovnyak spaces whose Taylor series diverge in norm, even though polynomials are dense in the space. This is often proved by showing that the sequence of Taylor partial sums is unbounded in norm. In this note we show that it can even happen that the Taylor partial sums tend to infinity in norm. We also establish similar results for lower-triangular summability methods such as the Cesàro means. [ABSTRACT FROM AUTHOR]

The partial sums of the series expansion of the Bessel function Jα of the first kind provide lower and upper bounds on the positive real axis for order α ≥-1=2. [ABSTRACT FROM AUTHOR]

We study the convergence of Fourier series in the polynomial system orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of in this system converges to pointwise on the grid as . In addition, we study the approximation properties of partial sums of Fourier series in the system . [ABSTRACT FROM AUTHOR]

This study focuses on the rate of uniform approximation for continuous functions g(x,y),2π$$ g\left(x,y\right),2\pi $$‐periodic in each variable, using double matrix means of the rectangular partial sums of double Fourier series. The importance of the weighted integral modulus of symmetric smoothness lies in its ability to capture the local behavior of a function. So we obtain the results in terms of the weighted integral modulus of symmetric smoothness. In addition, we introduce the concept of the weighted Lipschitz and Zygmund classes as an extension of the existing Lipschitz classes Lip(α,β)$$ Lip\left(\alpha, \beta \right) $$ and Lip(α,β;p)$$ Lip\left(\alpha, \beta; p\right) $$ and Zygmund classes Z(α,β)$$ Z\left(\alpha, \beta \right) $$ and Z(α,β;p)$$ Z\left(\alpha, \beta; p\right) $$, respectively. These weighted classes allow us a more detailed analysis of the approximation rates of functions by assigning the weights to different regions of the domain of function. Further, we prove two theorems about the degree (error) of approximation of functions belonging to these classes using double matrix means. We also discuss some corollaries from our results. [ABSTRACT FROM AUTHOR]

In the paper, with the aid of the Faà di Bruno formula, in terms of the central factorial numbers and the Stirling numbers of the second kinds, the authors derive several series expansions for any positive integer powers of the sinc and sinhc functions, discover several closed-form expressions for partial Bell polynomials of all derivatives of the sinc function, establish several series expansions for any real powers of the sinc and sinhc functions, and present several identities for central factorial numbers of the second kind and for the Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions f , utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained initial coefficients of Taylor series expansion of f , coefficient inequalities for f − 1 and the Fekete–Szegö problem. We also covered some key geometric properties for functions f in this newly formed class, such as the necessary and sufficient condition, convex combination, sequential subordination and partial sum findings. [ABSTRACT FROM AUTHOR]

The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines formed by the fundamental system of solutions to the Cauchy problems for the homogeneous equation of the original problem. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes formed by partial sums of series in even powers of the grid step for the solutions to Cauchy problems. [ABSTRACT FROM AUTHOR]

Let \mathcal {N}=L_{\infty }(G_{\mathbf {m}})\bar {\otimes }\mathcal {M} where \mathcal {M} is a semifinite von Neumann algebra and G_{\mathbf {m}} is a bounded Vilenkin group. Considering the partial sums S_n(f) of the Vilenkin-Fourier series of f\in L_1(\mathcal {N}), we prove that there is a universal constant c independent of f, n and \mathcal {M} such that \begin{equation*} \|S_n(f)\|_{L_{1,\infty }(\mathcal {N})}\leq c\|f\|_{L_1(\mathcal {N})}. \end{equation*} Consequently, by transference argument, we obtain Scheckter and Sukochev's result in the noncommutative bounded Vilenkin system setting. We also show the H_1-L_1 boundedness for modified partial sums. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of one kind of random differential equations with a Caputo fractional time derivative driven by colored noise are investigated by using approximating sequences. To this end, a generalized Gronwall inequality containing singular kernel which is first proved as an auxiliary tool to handle random time fractional differential equations, which is based on the ideas mentioned in [5]. [ABSTRACT FROM AUTHOR]

By means of the weight coefficients, Hermite-Hadamard's inequality, Euler-Maclaurin summation formula and Abel's summation by parts formula, a more accurate reverse Hardy- Hilbert's inequality with two partial sums is given. The equivalent statements of the best possible constant factor related to several parameters are provided, and some particular inequalities are deduced. [ABSTRACT FROM AUTHOR]

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n → ∞ cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order. That opens a new possibility to model the beta distributions using divisors of permutations. [ABSTRACT FROM AUTHOR]

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code. [ABSTRACT FROM AUTHOR]

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the Landau–Selberg–Delange method. In this paper, we are interested in the opposite direction. In particular, we prove that when |$f$| is a suitable divisor-bounded multiplicative function with small partial sums, then |$f(p)\approx -p^{i\gamma _1}-\ldots -p^{i\gamma _m}$| on average, where the |$\gamma _j$| s are the imaginary parts of the zeros of the Dirichet series of |$f$| on the line |$\textrm {Re}(s)=1$|⁠. This extends a result of Koukoulopoulos and Soundararajan and it builds upon ideas coming from previous work of Koukoulopoulos for the case where |$|f|\leqslant 1$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we have done some research studies on the fractal dimension of the sum of two continuous functions with different K -dimensions and approximation of s -dimensional fractal functions. We first investigate the K -dimension of the linear combination of fractal function whose K -dimension is s and the function satisfying Lipschitz condition is still s -dimensional. Then, based on the research of fractal term and the Weierstrass approximation theorem, an approximation of the s -dimensional continuous function is given, which is composed of finite triangular series and partial Weierstrass function. In addition, some preliminary results on the approximation of one-dimensional and two-dimensional fractal continuous functions have been given. [ABSTRACT FROM AUTHOR]

By means of the techniques of real analysis, applying some basic inequalities and formulas, a new reverse Mulholland's inequality with one partial sum in the kernel is given. We obtain the equivalent conditions of the parameters related to the best value in the new inequality. As applications, we reduce to the equivalent forms and a few inequalities for particular parameters. [ABSTRACT FROM AUTHOR]

It is well known that Luzin's conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin's conjecture has been achieved by considering simpler problems. In this paper, we consider three of these problems for spherical partial sums: the principle of generalized localization, summability almost everywhere, and convergence almost everywhere of multiple Fourier series of smooth functions. A brief overview of the work in these areas is given and unsolved problems are mentioned and new problems are formulated. Moreover, at the end of the work, a new result on the convergence of spherical sums for functions from Sobolev classes is proved. [ABSTRACT FROM AUTHOR]

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion $ a_0({\boldsymbol {x}})+\sum _{j\in {\mathbb N}}y_ja_j({\boldsymbol {x}}) $ a 0 (x) + ∑ j ∈ N y j a j (x) , where elements of the parameter vector $ {\boldsymbol {y}}=(y_j)_{j\in {\mathbb N}}\in U^\infty $ y = (y j) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on $ U:=[-\frac {1}{2},\frac {1}{2}] $ U := [ − 1 2 , 1 2 ]. Under the assumption $ \|\sum _{j\in {\mathbb N}}\rho _j|a_j|\|_{L_\infty (D)} ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ (D) < ∞ with some positive sequence $ (\rho _j)_{j\in {\mathbb N}}\in \ell _p({\mathbb N}) $ (ρ j) j ∈ N ∈ ℓ p (N) for $ p\in (0,1] $ p ∈ (0 , 1 ] we show that for any $ {\boldsymbol {y}}\in U^\infty $ y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues $ (\lambda _j({\boldsymbol {y}}))_{j\in {\mathbb N}} $ (λ j (y)) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap $ \lambda _2({\boldsymbol {y}})-\lambda _1({\boldsymbol {y}}) $ λ 2 (y) − λ 1 (y) is uniformly positive in $ U^\infty $ U ∞ . From this, we prove the holomorphic extension property of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y) to a complex domain in $ {\mathbb C}^\infty $ C ∞ and estimate partial derivatives of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y) with respect to the parameter $ {\boldsymbol {y}} $ y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y). [ABSTRACT FROM AUTHOR]

Applying r-repeated de la Vallée Poussin sums, we have proved four theorems which show the upper bound of the r-repeated de la Vall'ee Poussin kernel, their convergence at a point, the deviation between a continuous function and the r-repeated de la Vallée Poussin sums of partial sums of its Fourier series, and finally we determine the degree of approximation of functions belonging to ordinary Lipschitz class. [ABSTRACT FROM AUTHOR]

The existing rate of convergence of series requires a known limit of that series. Direct computation of Brun's constant as the sum of twin prime reciprocals is challenging as the convergence is agonizingly slow. Here, I introduce the relative increase of the partial sum of a series as a new rate of convergence regardless of the limit of the series. With this concept, I extract a slowly convergent part from the original sum for Brun's constant so that the remaining one converges faster. Computed data demonstrate the usefulness of the concept. As a result, I obtain an improved lower bound for Brun's constant using less computing resource and establish an approximate mapping that gives a new estimate for the constant at 1.89558413. [ABSTRACT FROM AUTHOR]

Considering the approximation of a function from a Sobolev space by the partial sums of Fourier series in a system of Sobolev orthogonal polynomials generated by classical Laguerre polynomials, we obtain an estimate for the convergence rate of the partial sums to . [ABSTRACT FROM AUTHOR]

We argue that a classical inequality due to Fan, Taussky and Todd (1955) is equivalent to the dissipativity of a Jordan block. As the latter can be characterised via the zeros of Chebyshev polynomials, we obtain a short new proof of the inequality. Three other inequalities of Fan-Taussky-Todd are reproven similarly. By the Lumer-Phillips theorem, the matrix semigroup generated by the Jordan block is contractive. This yields new extensions of the classical Fan-Taussky-Todd inequalities. As applications, we give an estimate for the partial sums of a Bessel function, and a contribution to the classification of self-similar Gaussian Markov processes. [ABSTRACT FROM AUTHOR]

The main goal of this paper is a study of a class of more accurate Hilbert-type inequalities involving partial sums. We first derive a discrete Hilbert-type inequality involving two partial sums and the kernel 1/ max(mλ, nλ). Then, by virtue of the Hardy inequality, we establish a weaker version of the latter relation involving only one partial sum, as well as the corresponding equivalent form. In addition, we prove that the constants appearing on the right-hand sides of the established inequalities are the best possible. Finally, we also give integral analogues of the corresponding discrete results. [ABSTRACT FROM AUTHOR]

Let χ$\chi$ be a primitive character modulo a prime q$q$, and let δ>0$\delta > 0$. It has previously been observed that if χ$\chi$ has large order d⩾d0(δ)$d \geqslant d_0(\delta)$ then χ(n)≠1$\chi (n) \ne 1$ for some n⩽qδ$n \leqslant q^{\delta}$, in analogy with Vinogradov's conjecture on quadratic non‐residues. We give a new and simple proof of this fact. We show, furthermore, that if d$d$ is squarefree then for anyd$d$th root of unity α$\alpha$ the number of n⩽x$n \leqslant x$ such that χ(n)=α$\chi (n) = \alpha$ is od→∞(x)$o_{d \rightarrow \infty}(x)$ whenever x>qδ$x > q^\delta$. Consequently, when χ$\chi$ has sufficiently large order the sequence (χ(n))n⩽qδ$(\chi (n))_{n \leqslant q^\delta}$ cannot cluster near 1$\hskip.001pt 1$ for any δ>0$\delta > 0$. Our proof relies on a second moment estimate for short sums of the characters χℓ$\chi ^\ell$, averaged over 1⩽ℓ⩽d−1$1 \leqslant \ell \leqslant d-1$, that is non‐trivial whenever d$d$ has no small prime factors. In particular, given any δ>0$\delta > 0$ we show that for all but o(d)$o(d)$ powers 1⩽ℓ⩽d−1$1 \leqslant \ell \leqslant d-1$, the partial sums of χℓ$\chi ^\ell$ exhibit cancellation in intervals n⩽qδ$n \leqslant q^\delta$ as long as d⩾d0(δ)$d \geqslant d_0(\delta)$ is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime 3⩽d⩽q−1$3 \leqslant d \leqslant q-1$, the Pólya–Vinogradov inequality may be improved for χℓ$\chi ^\ell$ on average over 1⩽ℓ⩽d−1$1 \leqslant \ell \leqslant d-1$, extending work of Granville and Soundararajan. [ABSTRACT FROM AUTHOR]

In this paper, the harmonic function related to the q-Srivastava–Attiya operator is described to introduce a new class of complex harmonic functions that are orientation-preserving and univalent in the open-unit disk. We also cover some important aspects such as coefficient bounds, convolution conservation, and convexity constraints. Next, using sufficiency criteria, we calculate the sharp bounds of the real parts of the ratios of harmonic functions to their sequences of partial sums. In addition, for the first time some of the interesting implications of the q-Srivastava–Attiya operator in harmonic functions are also included. [ABSTRACT FROM AUTHOR]

We consider the problem of Ω bounds for the partial sums of a modified character, i.e. , a completely multiplicative function f such that |$f(p)=\chi(p)$| for all but a finite number of primes p , where χ is a primitive Dirichlet character. We prove that in some special circumstances, |$\sum_{n\leq x}f(n)=\Omega((\log x)^{|S|})$|⁠ , where S is the set of primes p , where |$f(p)\neq \chi(p)$|⁠. This gives credence to a corrected version of a conjecture of Klurman et al. Trans. Amer. Math. Soc. 374 (11), 2021, 7967–7990. We also compute the Riesz mean of order k for large k of a modified character and show that the Diophantine properties of the irrational numbers of the form |$\log p / \log q$|⁠ , for primes p and q , give information on these averages. [ABSTRACT FROM AUTHOR]

This document is a summary of a special issue of the scientific journal Axioms titled "Recent Advances in Fractional Calculus." The issue contains 10 articles that explore various aspects of fractional calculus, a field that expands upon classical calculus and has applications in physics, engineering, biology, economics, and other disciplines. The articles cover topics such as boundary value problems, meromorphic functions, non-linear equations, evolution equations, COVID-19 modeling, reaction-diffusion problems, convex functions, and inequalities. The authors present new mathematical models, methods, and results, and provide examples and applications to illustrate the practical significance of their findings. The document concludes with acknowledgments to the authors, reviewers, and editors involved in the publication. [Extracted from the article]

It is well-known that there exist holomorphic functions f on the open unit disc D for which the sequence (Sn(f)) of partial sums of its Taylor series with center 0 approximate all the polynomials on all compact subsets K ⊂ C\D with connected complement. This result fails if we consider the sequence of the weighted partial sums (2nSn(f)) instead of (Sn(f)). In the present paper we show that there exist holomorphic functions f on D for which the sequence (2nSn(f)) approximate all the polynomials on only some specific compact sets K. The geometry of K plays a crucial role. We also generalize these results for weights other than 2n and on arbitrary simply connected domains. [ABSTRACT FROM AUTHOR]

We find two series expansions for Legendre's second incomplete elliptic integral E(\lambda, k) in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the (\lambda, k) plane. Partial sums of the proposed expansions form a sequence of approximations to E(\lambda,k) which are asymptotic when \lambda and/or k tend to unity, including when both approach the logarithmic singularity \lambda =k=1 from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for E(\lambda, k). For the reader's convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses the absolute convergence of Fourier series of functions of two variables in the space C1, ¿. The authors explore the conditions that ensure the absolute convergence of the series and define the space A, which consists of functions for which the series converges. They also discuss the classical results of Bochner and Weinger, who extended the Bernstein theorems to functions of two variables. The authors refine Weinger's result using probabilistic considerations and provide a theorem that demonstrates the existence of a function in the space C1, ¿ that is not included in A. The article also mentions the sufficient conditions on the modulus of continuity ¿(¿) that ensure the embedding C1, ¿ ¿ A. The work was financially supported by the Russian Science Foundation. [Extracted from the article]

This paper mainly considers the long-term behavior of p-Laplace equations with infinite delays driven by nonlinear colored noise. We firstly prove the existence of weak solutions to the equation, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity conditions, and thus generate a multi-valued dynamical system. Moreover, the regularity of solutions is also proved. Then we prove the existence of a pullback attractor. Subsequently, the measurability of the pullback attractor and the multi-valued dynamical system are also proved. [ABSTRACT FROM AUTHOR]

Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and closely related research on properties of matrix-based signal filters are challenging and are attracting increased attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh--Ritz method in [A. V. Knyazev and M. E. Argentati, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521-1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4-13] to derive sharp convergence rate bounds of abstract block iterations, wherein tuples of tangents of principal angles or relative errors of Ritz values are bounded using majorization in terms of arranged partial sums and tuples of convergence factors. Our novel bounds are robust in the presence of clusters of eigenvalues, improve previous results, and are applicable to most known block iterative solvers and matrix-based filters, e.g., to block power, Chebyshev, and Lanczos methods combined with polynomial filtering. The sharpness of our bounds is fundamental, implying that the bounds cannot be improved without further assumptions. [ABSTRACT FROM AUTHOR]

Let f(z) be a power series with a non-zero radius of convergence x: 0 < - ≤ +∞. Denote by fn(z) the n-th partial sum of f, and 0, 1, 2, ... . By the general result of Hendriksen and Van Rossum there exists a unique linear functional L on Laurent polynomials, such that L(RnRm) = 0, when n 6= m, while L(R2 n) 6= 0, and L(1) = 1. We present an explicit integral representation for L in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well. [ABSTRACT FROM AUTHOR]

We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale on a computational tree, which leads to explicit and interpretable bounds with nonlinear terms controlled explicitly rather than by big-O terms. Two probability parameters strengthen the precision-awareness of our bounds: one parameter controls the first order terms in the summation error, while the second one is designed for controlling higher order terms in low precision or extreme-scale problem dimensions. Our systematic approach yields new deterministic and probabilistic error bounds for three classes of mono-precision algorithms: general summation, shifted general summation, and compensated (sequential) summation. Extension of our systematic error analysis to mixed-precision summation algorithms that allow any number of precisions yields the first probabilistic bounds for the mixed-precision FABsum algorithm. Numerical experiments illustrate that the probabilistic bounds are accurate, and that among the three classes of mono-precision algorithms, compensated summation is generally the most accurate. As for mixed precision algorithms, our recommendation is to minimize the magnitude of intermediate partial sums relative to the precision in which they are computed. [ABSTRACT FROM AUTHOR]

This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we suppose that the boundary of a domain G in the complex plane C belongs to a special subclass of smooth curves and that the canonical domain GR, R > 1 is the largest domain where a function f is analytic. We investigate the rate of convergence to the function f by the partial sums of Faber series of the function f on the domain Ġ. Under the boundary conditions of the domain G, we obtain some results which characterize the maximal convergence of the Faber expansion of the function f which belongs to the weighted Smirnov class with variable exponent Ewp(.) (GR). [ABSTRACT FROM AUTHOR]

By the use of the techniques of analysis and some useful formulas, we give a new extension of Hilbert-type inequality with two internal variables involving one partial sums, which is a refinement of a published inequality. We provide a few equivalent conditions of the best possible constant related to multi parameters. We obtain the equivalent inequalities, the operator expressions as well as a few inequalities with the particular parameters as applications. [ABSTRACT FROM AUTHOR]