CESRO MEANS CONNECTED WITH THE ALLIED SERIES OF A FOURIER SERIES

<正> 1.Introduction.It is classical that,if the Fourier series of a function ψ(t) ofbounded variation is (?) sin nt,the sequence (?) is bounded.This resultcannot be improved.“For,the function (?) whoseFourier series is (?) sin nt is of bounded variation;but we have(?).We know,however,that,if ψ(t) is of bounded varia-tion in (0, 2π),the sequence (?) converges(c,β) to 2π~(-1)ψ(+0) for everyα>0.In this paper Ⅰ shall give some results concerning the convergence(C) of the sequence (?),by studying the Ces(?)ro means ψ(t) of ψ(t).     

ON A THEOREM OF CHUI

Let f∈C_(2π) and let α_0/2+sum from k=1 to ∞ (a_k coskx+b_k sinkx) (1)be its Fourier series. Denote by S_n(f, x) the n-th partial sumof series (1) and by ω(f,δ) the modulus of continuity of functionf(x). For each q0, the Euler (E, q)-mean of series (1) is     

On the divergence of Nörlund logarithmic means of Walsh-Fourier series

It is well known in the literature that the logarithmic means 1/logn ^n-1∑k=1 Sk（f）/k of Walsh or trigonometric Fourier series converge a.e. to the function for each integrable function on the unit interval. This is not the case if we take the partial sums. In this paper we prove that the behavior of the so-called NSrlund logarithmic means 1/logn ^n-1∑k=1 Sk（f）/n-k is closer to the properties of partial sums in this point of view.     

On the Best Degree of Approximation by Euler (E,q)-Means

Let f(x)∈L_(2π) and its Fourier series by f(x)～α_0/2+sum from n=1 to ∞(α_ncosnx+b_nsinx)≡sum from n=0 to ∞(A_n(x)). Denote by S_n (f,x) its partial sums and by E_n~q(f,x) its Euler (E, q)-means, i. e. E_n~q(f,x)=1/(1+q)~π sum from m=0 to n((?)q~(n-m)S_m(f,x)), with q≥0 (E_n~0≡S_n). In [1] Holland and Sahney proved the following theorem. THEOREM A Ifω(f,t) is the modulus of continuity of f∈C_(2π), then the degree of approximation of f by the (E,q)-means of f is givens by##特殊公式未编改     

Convergence in L p -norm of Fourier series on the complete product of quaternion groups with bounded orders

A well-known result for Vilenkin systems is the fact that for all 1 p ∞ the n-th partial sums of Fourier series of all functions in the space Lpconverge to the function in Lp-norm.This statement can not be generalized to any representative product system on the complete product of finite non-abelian groups,but even then it is true for the complete product of quaternion groups with bounded orders and monomial representative product system ordered in a specific way.     

Vilenkin-Like系统的不等式

For bounded Vilenkin-Like system, the inequality is also true：
（∑ k=1 ^∞ kp-2｜f^^（k）｜^p）^1/p ≤ C｜｜f｜｜Hp, 0 〈 p ≤ 2,
where f^^（·） denotes the Vilenkin-Like Fourier coefficient of f and the Hardy space Hp（Gm） is defined by means of maximal functions. As a consequence, we prove the strong convergence theorem for bounded Vilenkin-Like Fourier series, i.e.,
（∑ k=1 ^∞ k^p-2｜｜Skf｜｜p^p）^1/p≤C｜｜f｜｜Hp,0〈p〈1.     

On the fourier-vilenkin coefficients

In this article, we prove the following statement that is true for both unbounded and bounded Vilenkin systems: for any ε∈(0, 1), there exists a measurable set E [0, 1)of measure bigger than 1-ε such that for any function f ∈ L~1[0, 1), it is possible to find a function g ∈ L~1[0, 1) coinciding with f on E and the absolute values of non zero Fourier coefficients of g with respect to the Vilenkin system are monotonically decreasing.     

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f ∈ L~2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator S_(N(x,y))f(x, y).Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio ■≌2~(n_0) for all x and y, where n_0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n_0, are then extended to L~r, 1 r 2.We give an application to the family N(x, y) = λxy on [0, 1) × [0, 1), any λ 10.     

DISTRIBUTION OF THE（0，∞）ACCUMULATIVE LINES OF MEROMORPHIC FUNCTIONS

Suppose that f(z)is a meromorphic function of order λ（0&lt;λ&lt;+∞）and of lower order μ in the plane.Let ρ be a positive number such that μ≤ρ≤λ.(1)If f^(l)(z)(0≤l&lt;+∞）has p(1≤p&lt;+∞）finite nonzero deficient valnes αi(i=1,…，p)with deficiencies δ（αi,f^(l)),then f(z)has a (0,∞）accumulative line of order ≥ρin any angular domain whose vertex is at the origin and whose magnitude is larger than max(π/ρ，2π-4/ρ ∑i=1^p arcsin √δ（αi,f^(l))/2).(2)If f(z) has only p(0&lt;p&lt;+∞）（0,∞）,accumulative lines of order≥ρ：arg z=θk(0≤θ1&lt;θ2&lt;…&lt;θp&lt;2π，θp+1=θ1+2π）,then λ≤π/ω，where ω=min I≤k≤p(θk+1-θk),provided that f^(l)(z)(0≤l&lt;+∞）has a finite nonzero deficient value.     

Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution

Here we consider the following strongly singular integral T_(?,γ,α,β)f (x, t) =∫ _(R~n)e~[i|y|~(-β)]?(y/|y|)/|y|~(n+α)f (x - y, t - γ(|y|))dy,where ? ∈ L~p(S~(n-1)), p 1, n 1, α 0 and γ is convex on(0,∞).We prove that there exists A( p, n) 0 such that if β A( p, n)(1 + α), then T_(?,γ,α,β)is bounded from L~2(R~(n+1)) to itself and the constant is independent of γ. Furthermore,when ? ∈ C~∞(S~(n-1)), we will show that T?,γ,α,βis bounded from L~2(R~(n+1)) to itself only if β 2α and the constant is independent of γ.     

A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Well-posedness of equations with fractional derivative via the method of sum

We study the well-posedness of the equations with fractional derivative Dαu(t)=Au(t)+f(t)(0 ≤t≤2π),where A is a closed operator in a Banach space X,0α1 and Dα is the fractional derivative in the sense of Weyl.Although this problem is not always well-posed in Lp(0,2π;X) or periodic continuous function spaces Cper([0,2π];X),we show by using the method of sum that it is well-posed in some subspaces of L p(0,2π;X) or C per([0,2π];X).     

On operator-valued Fourier multipliers

We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0,2π]d;X) using an induction argument based on a known result when d=1.     

The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients

Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ ( $ \tfrac{1} {2} Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ (, 1). Namely, the following assertion is true. Let α ∈ (, 1), < p < 2, a sequence a ∈ M _α, and . Then the series cos nx converges on (0,2π) to a finite function f(x) and f(x) ∈ L _p (0,2π). Original Russian Text ? M.I. D’yachenko, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 2008, No. 5, pp. 38–47.     

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved 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^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .     

Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L ¹(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L ^p (0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L ¹(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators if one of them generates a C ₀-semigroup. Günter Lumer in memoriam     

Generalized absolute Cesaro summability of a Fourier series

In an attempt to study the scope of a theorem due to Pati, the authors have established that φ(t) logK|t∈B _u V in (0,π)⟹ΣA _n (x) is |C, 0,β| forβ>1, at the pointt = x.     

On double sine and cosine transforms,Lipschitz and Zygmund classes

We consider complex-valued functions f ∈ L 1 (R+2),where R +:= [0,∞),and prove sufficient conditions under which the double sine Fourier transform f ss and the double cosine Fourier transform f cc belong to one of the two-dimensional Lipschitz classes Lip(α,β) for some 0 α,β≤ 1;or to one of the Zygmund classes Zyg(α,β) for some 0 α,β≤ 2.These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L 1 (R+2).     

Embedding Theorems in B-Spaces and Applications

This study focuses on the anisotropic Besov-Lions type spaces B^lp,θ（Ω;E0,E） associated with Banach spaces E0 and E. Under certain conditions, depending on l =（l1,l2,…,ln）and α=（α1,α2,…,αn）,the most regular class of interpolation space Eα between E0 and E are found so that the mixed differential operators D^α are bounded and compact, from B^l＋s p,θ（Ω;E0,E） to B^s p,θ（Ω;Eα）.These results are applied to concrete vector-valued function spaces and to anisotropic differential-operator equations with parameters to obtain conditions that guarantee the uniform B separability with respect to these parameters. By these results the maximal B-regularity for parabolic Cauchy problem is obtained. These results are also applied to infinite systems of the quasi-elliptic partial differential equations and parabolic Cauchy problems with parameters to obtain sufficient conditions that ensure the same properties.     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.