Partial Sums Sn Research Articles

Weak type estimate for the partial sums of Noncommutative Vilenkin-Fourier series

Let N = L ∞ ( G m ) ⊗ ¯ M \mathcal {N}=L_{\infty }(G_{\mathbf {m}})\bar {\otimes }\mathcal {M} where M \mathcal {M} is a semifinite von Neumann algebra and G m G_{\mathbf {m}} is a bounded Vilenkin group. Considering the partial sums S n ( f ) S_n(f) of the Vilenkin-Fourier series of f ∈ L 1 ( N ) f\in L_1(\mathcal {N}) , we prove that there is a universal constant c c independent of f f , n n and M \mathcal {M} such that ‖ S n ( f ) ‖ L 1 , ∞ ( N ) ≤ c ‖ f ‖ L 1 ( N ) . \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 H_1 - L 1 L_1 boundedness for modified partial sums.

Bernstein-type constants for approximation of [formula omitted] by partial Fourier–Legendre and Fourier–Chebyshev sums

In this paper, we study the approximation of fα(x)=|x|α,α>0 in L∞[−1,1] by its Fourier–Legendre partial sum Sn(α)(x). We derive the upper and lower bounds of the approximation error in the L∞-norm that are valid uniformly for all n≥n0 for some n0≥1. Such an optimal L∞-estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can obtain the explicit Bernstein-type constant B∞(α)≔limn→∞nα‖fα−Sn(α)‖L∞=2Γ(α)π|sinαπ2|. Interestingly, using a similar argument, we can show that the Fourier–Chebyshev sum has the same Bernstein-type constant B∞(α) as the Legendre case.

Convergence rates in the functional CLT for [formula omitted]-mixing triangular arrays

We obtain convergence rates (in the Levi–Prokhorove metric) in the functional central limit theorem (CLT) for partial sums Sn=∑j=1nξj,n of triangular arrays {ξ1,n,ξ2,n,…,ξn,n} satisfying some mixing and moment conditions (which are not necessarily uniform in n). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of Sn. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of Sn is “not negligible” in comparison with the (new) number of summands.

An almost sure invariance principle for some classes of non-stationary mixing sequences

In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast ϕ-mixing. The obtained rate is of order o(Vn14+δ) for an arbitrary δ>0, where Vn is the variance of the underlying partial sums Sn. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.

Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs

For a function f continuous on a closed interval, its modulus of fractality ν(f, e) is defined as the function that maps any e > 0 to the smallest number of squares of size e that cover the graph of f. The following condition for the uniform convergence of the Fourier series of f is obtained in terms of the modulus of fractality and the modulus of continuity ω(f, δ): if $$\begin{array}{*{20}{c}} {\omega (f,\pi /n)\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right) \to 0}&{\text{as}}&{n \to + \infty ,} \end{array}$$ then the Fourier series of f converges uniformly. This condition refines the known Dini–Lipschitz test. In addition, for the growth order of the partial sums Sn(f, x) of a continuous function f, we derive an estimate that is uniform in x ∈ [0, 2π]: $${S_n}(f,x) = o\left( {\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right)} \right).$$. The optimality of this estimate is shown.

A continuous function with universal Fourier series on a given closed set of Lebesgue measure zero

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2].

Приближение функций частными суммами ряда Фурье по многочленам, ортогональным на произвольных сетках

for arbitrary continuous function f(t) on the segment [−1,1] we construct discrete Fourier sums Sn,N(f,t) on a system of polynomials forming an orthonormal system on non-uniform grids $${T_N} = \left\{{{t_j}} \right\}\matrix{{N - 1} \cr {j = 0} \cr}$$ of N points from segment [−1,1] with weight Δtj=tj+1−tj. Approximation properties of the constructed partial sums Sn,N(f,t) of order n ≤ N−1 are investigated. A two-sided pointwise estimate is obtained for the Lebesgue function Ln,N(t) of considered discrete Fourier sums for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right),{\delta _N} = {\max _0} \le j \le N - 1\Delta {t_j}$$. The question of the convergence of Sn,N(f,t) to f(t) is also investigated. In particular, we obtain the deflection estimation of a partial sum Sn,N(f,t) from f(t) for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right)$$, which depends on n and the position of a point t ∊ [−1,1].

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {pk,N(x)} =0 –1 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1–tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ −2/7 ), δN = max0≤ j≤N−1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].

The limit law of the iterated logarithm for linear processes

In this short note, we established the limit law of the iterated logarithm for linear process. Let {ξi,−∞<i<∞} be a sequence of independent identically distributed random variables with Eξ1=0 and Eξ12=1. Define the linear process by Xt=∑j=−∞∞ajξt−j,t≥1 and the partial sum Sn=∑t=1nXt, where {aj,−∞<j<∞} is a sequence of real numbers with ∑j=−∞∞aj≠0 and ∑j=−∞∞|aj|<∞. Then, we have limn→∞12loglognmax1≤k≤n|Sk|k=|∑j=−∞∞ai|a.s.

Existence of moments of a counting process and convergence in multidimensional time

AbstractStarting with independent, identically distributed random variables X1,X2... and their partial sums (Sn), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑n1(Sn&gt;b(n)) and aim for necessary and sufficient conditions on X1 in order to obtain the existence of certain moments for N, as well as for generalized counting variables with weights, and multi-index random variables. The existence of the first moment of N when b(n)=εn, i.e. ∑n=1∞ℙ(|Sn|&gt;εn)&lt;∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.

A strong law of large numbers for nonnegative random variables and applications

For a sequence of nonnegative random variables {Xn,n≥1} with finite means and partial sums Sn=∑i=1nXi,n≥1, and a sequence of positive numbers {bn,n≥1} with bn↑∞, sufficient conditions are given under which (Sn−ESn)/bn→0 almost surely. Our result generalizes the strong law of large numbers obtained by Korchevsky (2015). Some applications for dependent random variables are also provided.

Convergence and localization in Orlicz classes for multiple Walsh–Fourier series with a lacunary sequence of rectangular partial sums

For functions f in Orlicz classes, we consider multiple Walsh–Fourier series for which the rectangular partial sums Sn(x;f) have indices n=(n1,…,nN)∈ZN (N≥3), where either N or N−1 components are elements of (single) lacunary sequences. For this series, we prove the validity of weak generalized localization almost everywhere on an arbitrary measurable set A⊂IN={x∈RN:0≤xj<1,j=1,2,…,N}, in the case when the structure and geometry of A are defined by the properties Bk, 2≤k≤N. We define the relation between the parameter k and the “smoothness” of functions in terms of the Orlicz classes. As a consequence, we obtain some results on the “local smoothness conditions.” In particular, the theorem is proved for the convergence of Walsh–Fourier series on an arbitrary open set Ω⊂IN under the minimal conditions imposed on the smoothness of the function on this set.

Limit ultraspherical series and their approximation properties

We study new series of the form \(\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} \) in which the general term \(f_k^{ - 1} \hat P_k^{ - 1} (x)\), k = 0, 1, …, is obtained by passing to the limit as α→−1 from the general term \(\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)\) of the Fourier series \(\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} \) in Jacobi ultraspherical polynomials \(\hat P_k^{\alpha ,\alpha } (x)\) generating, for α> −1, an orthonormal system with weight (1 − x2)α on [−1, 1]. We study the properties of the partial sums \(S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} \) of the limit ultraspherical series \(\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} \). In particular, it is shown that the operator Sn−1(f) = Sn−1 (f, x) is the projection onto the subspace of algebraic polynomials pn = pn(x) of degree at most n, i.e., Sn(pn) = pn; in addition, Sn−1(f, x) coincides with f(x) at the endpoints ±1, i.e., Sn−1 (f,±1) = f(±1). It is proved that the Lebesgue function Λn(x) of the partial sums Sn−1(f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that \(\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1\).

Trigonometric Fourier series and Walsh-Fourier series with lacunary sequence of partial sums

I N = {x ∈ R : 0 ≤ xj 0, (μ = μN isN-dimensional Lebesgue measure), and let f(x) = 0 on A. In the present paper, we study the behavior of the rectangular partial sums Sn(x, f ; Ψ) of multiple Fourier series with respect to the systemΨ as n → ∞, i.e.,min1≤j≤N nj → ∞, on the set A in the case where some of the components n1, . . . , nN of the vector n (the “number” of the partial sums Sn(x, f ; Ψ)) are elements of (single) lacunary sequences (a sequence {n(s)}, n(s) ∈ Z1, is said to be lacunary if n(s+1)/n(s) ≥ q > 1, s = 1, 2, . . . ). As is well known, subsequences of partial sums of Fourier series possess better properties of convergence almost everywhere (a.e.) compared to the sequence of partial sums itself. For trigonometric Fourier series, in the one-dimensional case, Kolmogorov established in [1] as early as 1922 that, for any function f ∈ L2(I), the sequence of partial sums Sn(k)(x, f ; E), where {n(k)} is a lacunary sequence, converges a.e. on I1 to f(x). This result due to Kolmogorov was extended in 1931 by Littlewood and Paley [2] to the classes Lp(I), p > 1. Later (see, for example, [3]) it was established that this result is not valid in L1(I). For a single Walsh–Fourier series, Shneider [4] found in 1950 that, for any sequence {n(k)} such that the Lebesgue constants Ln(k) are uniformly bounded (in particular, for {n(k)} = {2k}) and for any function f ∈ L1(I), the sequence of partial sums Sn(k)(x, f ;W) converges to f(x) a.e. on I1. Konyagin [5] showed in 1993 that, for the convergence a.e. of Sn(k)(x, f ;W), the boundedness condition for the Lebesgue constants is not essential. In turn, the first result for multiple trigonometric series (for N = 2) involving “lacunary sequence of partial sums”, was obtained by Sjolin in 1971 in [6], where it was proved that if f ∈ Lp(I), p > 1, and {n1 1 } is a lacunary sequence, then a.e. on I2, lim ν1, n2→∞ S n (ν1) 1 , n2 (x, f ; E) = f(x).

Convergence rates in precise asymptotics

Let X1,X2,… be i.i.d. random variables with partial sums Sn, n⩾1. The now classical Baum–Katz theorem provides necessary and sufficient moment conditions for the convergence of ∑n=1∞nr/p−2P(|Sn|⩾εn1/p) for fixedε>0. An equally classical paper by Heyde in 1975 initiated what is now called precise asymptotics, namely asymptotics for the same sum (for the case r=2 and p=1) when, instead, ε↘0. In this paper we extend a result due to Klesov (1994), in which he determined the convergence rate in Heydeʼs theorem.

Large deviations for dependent heavy tailed random variables

Let {Xn,n≥1} be a stationary sequence of random variables with heavy tails. In this paper, we study the logarithmic asymptotic behaviors for the distributions of the partial sums Sn=∑i=1nXi under the assumption that {Xn,n≥1} is a sequence of dependent random variables. Our main interest is in the crude estimates P(|Sn|>nx)≈n−αx+1 for appropriate values x where α is a specific parameter. Some results in this paper extend the works of Hu and Nyrhinen (2004).

Limit theorems for the partial sum in Engel continued fractions

Let x=[d1,···dn,···]E be the Engel continued fraction of x∈[0,1].Call Sn(x)=Σ0≤k<n(x)(dk+1/dk) the normalized partial sum of x. We prove that the partial sum Sn can not satisfy the strong law of large numbers for any reasonably growing norming sequence.

On Gibbs-Wilbraham Phenomenon and the Arclength of Fourier Series

The arclength of the graphs Γ(S N (f)) of the partial sums S N (f) of the Fourier series of a piecewise C 1 function f with jump discontinuities is equal asymptotically to $(\hbox{the sum of all jumps of $f$})\times L_{N}$ , where L N is the Lebesgue constant. This is an improvement of R. Strichartz (J. Fourier Anal. Appl. 6, 533–536, 2000).

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables “Xn; n ≥ 1” with partial sums Sn = Σi=1nXi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form $$ \sum\limits_{n \geqslant n_0 } {n^{r - 2 - \tfrac{1} {{pq}}} a_n E\left( {\mathop {\max }\limits_{1 \leqslant k \leqslant n} \left| {S_k } \right|^{\tfrac{1} {q}} - \varepsilon b_n^{\tfrac{1} {{pq}}} } \right)^ + < \infty } $$ to hold where r > 1, q > 0 and either n0 = 1, 0 < p < 2, an = 1, bn = n or n0 = 3, p = 2, an = (log n)−1/2q, bn = n log n. These results extend results of Chow and of Li and Spataru from the independent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.

Intermediate convergents and a metric theorem of Khinchin

A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function f defined on the positive integers and a real number x, and form the partial sums sn of f evaluated at the partial quotients a1, …, an in the continued fraction expansion for x. Does the sequence {sn/n} have a limit as n → ∞? In 1935 Khinchin proved that the answer is yes for almost every x, provided that the function f does not grow too quickly. In this article we are going to explore a natural reformulation of this problem in which the function f is defined on the rationals and the partial sums in question are over the intermediate convergents to x with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.