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Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., | w (a,b) σ |∞ ≦ C| w (a,b) f|∞, whenever α,β ≧ −1/2 and a, b are within the square around ( α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations. [ABSTRACT FROM AUTHOR]

In 1936 Zygmunt Zalcwasser asked, with respect to the trigonometric system, how "rare" can a sequence of strictly monotone increasing integers (nj) be such that the almost everywhere relation 1N∑j=1NSnjf→f is fulfilled for each integrable function f. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function f and lacunary sequence (nj) of natural numbers. [ABSTRACT FROM AUTHOR]

Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0p. In this paper, the following result is proved:TheoremLet0N(f)˜BN(f)means that there’s a positive constant C, independent of N and f, such thatIn the cased= 2,this result was proved by Belinskii in 1996. [ABSTRACT FROM AUTHOR]

In this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence. [ABSTRACT FROM AUTHOR]

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205-225, ). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C(ℝ) and to evolution operators associated with nonautonomous second-order differential operators in C(ℝ) with time-periodic coefficients. [ABSTRACT FROM AUTHOR]

In 1953 Nash [7] introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro ( C, α)-means ( α∈(−1,0)) of trigonometric Fourier series of the classes H∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown. [ABSTRACT FROM AUTHOR]

Simon [12] proved that the maximal operator of ( C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H p to the space L p for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ is bounded from the martingale Hardy space H1/(1+ α) to the space weak- L1/(1+ α). [ABSTRACT FROM AUTHOR]

Let $\sigma_k^\delta$ denote the Cesaro means of order $\delta > -1$ of the spherical harmonic expansions on the unit sphere $S^{d-1}$, and let $E_j(f, H^1)$ denote the best approximation of $f$ in the Hardy space $H^1(S^{d-1})$ by spherical polynomials of degree at most $j$. It is known that $\lambda:= (d-2)/2$ is the critical index for the summability of the Cesaro means on $H^1(S^{d-1})$. The main result of this paper states that, for $ f\in H^1(S^{d-1})$, $$\sum_{j=0}^N \f 1{j+1} \|\sigma_j^\lambda (f) -f\|_{H^1}\approx \sum_{j=0}^N \f 1{j+1} E_j(f, H^1),$$ where “$\approx$” means that the ratio of both sides lies between two positive constants independent of $f$ and $N$. [ABSTRACT FROM AUTHOR]

In the present paper, the question of Kolmogorov weak type (1.1) for multidimensional operators occurring in Fourier analysis is studied. [ABSTRACT FROM AUTHOR]

We investigate the subsequence { t 2 n f } of Nörlund means with respect to the Walsh system generated by nonincreasing and convex sequences. In particular, we prove that a large class of such summability methods are not bounded from the martingale Hardy spaces H p to the space w e a k − L p for 0 < p < 1 / (1 + α) , where 0 < α < 1 . Moreover, some new related inequalities are derived. As applications, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means. [ABSTRACT FROM AUTHOR]

Let f ∈ L 1 (L ∞ (T) ⊗ ¯ M) , where T is the classical torus, M is a semi-finite von Neumann algebra. We prove the noncommutative weak type maximal inequality ‖ (γ n (f)) n ‖ Λ 1 , ∞ (N , ℓ ∞) ≤ C ‖ f ‖ L 1 (N) , where γ n (f) is the Cesàro means of the subsequence of the partial sums sequence (S n (f)) n ≥ 0 . As a consequence, the Cesàro means γ n (f) converges bilaterally almost uniformly to f whenever n → ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we state and prove some new inequalities related to the rate of Lp approximation by Cesàro means of the quadratic partial sums of double Vilenkin-Fourier series of functions from Lp. [ABSTRACT FROM AUTHOR]

It is proved that the maximal operators of subsequences of Cesàro means with varying parameters of two-dimensional Walsh-Fourier series is bounded from the dyadic Hardy spaces H p I to L p I . This implies an almost everywhere convergence for the subsequences of the summability means. [ABSTRACT FROM AUTHOR]

Let (C (t)) t ∈ R be a strongly continuous cosine operators on a Banach space X into itself and let A be their infinitesimal generator. In this work, we study the uniform convergence of the Abel averages λ ∫ 0 ∞ e - λ t C (t) d t of C(t) when λ tends to 0 + . More precisely, we show that C(t) is uniformly Abel ergodic if and only if X = R (A) ⊕ N (A) , with R (A) and N (A) the range and the kernel of A, respectively. Next, we examine this theory with the uniform power convergence of the Abel average λ 2 R (λ 2 , A) , for some λ > 0 , where R (λ 2 , A) be the resolvent function of A. [ABSTRACT FROM AUTHOR]

Let $\sigma_k^\delta$ denote the Cesaro means of order $\delta > -1$ of the spherical harmonic expansions on the unit sphere $S^{d-1}$, and let $E_j(f, H^1)$ denote the best approximation of $f$ in the Hardy space $H^1(S^{d-1})$ by spherical polynomials of degree at most $j$. It is known that $\lambda:= (d-2)/2$ is the critical index for the summability of the Cesaro means on $H^1(S^{d-1})$. The main result of this paper states that, for $ f\in H^1(S^{d-1})$, $$\sum_{j=0}^N \f 1{j+1} \|\sigma_j^\lambda (f) -f\|_{H^1}\approx \sum_{j=0}^N \f 1{j+1} E_j(f, H^1),$$ where “$\approx$” means that the ratio of both sides lies between two positive constants independent of $f$ and $N$. [ABSTRACT FROM AUTHOR]

We study the strong approximation properties of the Cesáro means of order δ of the Fourier-Laplace expansion of functions integrable on the unit sphere Sn-1, where δ &gT; λ := (n - 2)/2, the latter being the critical index for Cesáo summability of Fourier-Laplace series on Sn-1. The main purpose of this paper is to extend known results from the unit circle S1 to the general sphere Sn-1 with n &gT; 3. We prove six theorems. To prove Theorems 1–3, our machinery is based on the equieonvergent operator ENδ(f) of the Cesáro means σNδ (f) on Sn-1 introduced by Wang Kunyang for δ > -1. We prove in Theorem 6 that ENδ(f) is also equiconvergent with σNδ(f) for δ > 0 in the case of strong approximation. To prove Theorems 4 and 5, we rely on known equivalence relations between K-functionals and moduli of continuity. [ABSTRACT FROM AUTHOR]

We study the strong approximation properties of the Cesáro means of order δ of the Fourier-Laplace expansion of functions integrable on the unit sphere Sn-1, where δ &gT; λ := (n - 2)/2, the latter being the critical index for Cesáo summability of Fourier-Laplace series on Sn-1. The main purpose of this paper is to extend known results from the unit circle S1 to the general sphere Sn-1 with n &gT; 3. We prove six theorems. To prove Theorems 1–3, our machinery is based on the equieonvergent operator ENδ(f) of the Cesáro means σNδ (f) on Sn-1 introduced by Wang Kunyang for δ > -1. We prove in Theorem 6 that ENδ(f) is also equiconvergent with σNδ(f) for δ > 0 in the case of strong approximation. To prove Theorems 4 and 5, we rely on known equivalence relations between K-functionals and moduli of continuity. [ABSTRACT FROM AUTHOR]

In this study, we approach the strain gradient thermoelasticity of bodies with microtemperatures. We define the internal energy corresponding to an arbitrary solution of the mixed problem with boundary and initial values, considered in the context of strain gradient thermoelasticity of bodies with microtemperatures. The Cesaro means of different parts of the internal energy are considered. In our main result, we prove the asymptotic equipartition of the strain and kinetic energies, in the case t → ∞ . [ABSTRACT FROM AUTHOR]

For sine and cosine Fourier serieswithmonotonic coefficients we study L-convergence (1 < p < ∞) of their Cesàro means of a negative order. [ABSTRACT FROM AUTHOR]

The convergence of partial sums and Cesáro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated. [ABSTRACT FROM AUTHOR]

Under study is the convergence of the negative order Cesàro means of the double trigonometric Fourier series of functions of bounded generalized Λ-variation. [ABSTRACT FROM AUTHOR]

In this work, using subordination results for the Cesáro sum of certain analytic functions, a concept called Cesáro stable is given and two conjectures in this direction are also proposed. Using an extension of Vietoris' theorem on positivity of cosine sums, some results on positivity of cosine and sine sums are obtained. These results are useful in discussing some particular cases of the proposed conjectures. [ABSTRACT FROM AUTHOR]

We consider classes of periodic functions of bounded Λ-variation, where Λ has a power growth rate. We show that this class contains a continuous function whose Cesaro means of the Fourier series (whose order depends on the growth rate of Λ) have no localization property. [ABSTRACT FROM AUTHOR]

Let { φ( z)} be a system of Jacobi polynomials orthonormal on the circle | z| = 1 with respect to the weight (1 − cos τ)(1 + cos τ) ( α, β > −1), and let $\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}$ ). We establish relations between the polynomial φ ( z) and the nth ( C, α − 1/2)-mean of the Maclaurin series for the function (1 − z) and also between the polynomial φ ( z) and the nth ( C, α + 1/2)-mean of the Maclaurin series for the function (1 − z). We use these relations to derive an asymptotic formula for φ( z); the formula is uniform inside the disk | z| < 1. It follows that φ ( z) ≠ 0 in the disk | z| ≤ ρ for fixed φ ∈ (0, 1) and α > −1 if n is sufficiently large. [ABSTRACT FROM AUTHOR]

Several problems on Fourier series and trigonometric approximation on regular hexagonal and triangular domains are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation, and best approximation by trigonometric functions with both direct and inverse theorems. One of the objectives of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that permits an extensive theory for Fourier series and approximation. [ABSTRACT FROM AUTHOR]

Cesàro ( C, δ) means are studied for orthogonal expansions with respect to the weight function $\prod_{i=1}^{d}|x_{i}|^{2\kappa_{i}}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the ( C, δ) kernel with δ>−1 and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Cesàro means and the projection operator on these domains. [ABSTRACT FROM AUTHOR]

The behavior of generalized Cesàro ( C, α n )-means ( α n ∈ (0, d), d > 0) of trigonometric Fourier series of H ω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown. [ABSTRACT FROM AUTHOR]

We investigate the convergence of the linear means of the Fourier-Jacobi series of functions ƒ( x) from the weight space L α,β for x = 1 for the case in which this point is a Lebesgue point for ƒ. We establish su.cient summability conditions depending on the behavior of the function on the closed interval [−1, 0] and on the properties of the matrix involved in the summation method. [ABSTRACT FROM AUTHOR]

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also *-outer, is examined in greater detail. The connection to Agler’s model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over $$ \mathbb{R}^m, m \geq 1 $$ , in terms of rational functions with fixed denominators. [ABSTRACT FROM AUTHOR]

The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Several illustrative examples, including those related to the Cesàro mean, are discussed, and connections are made with earlier works. [ABSTRACT FROM AUTHOR]