We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2π$-periodic continuous functions $f$ from the classes $C^ψ_{β,p}$.These classes are defined by the convolutions of functions from the unit ball in the space
$L_{p}$, $1\leq p<\infty$, with generating fixed kernels
$Ψ_β(t)=\sum_{k=1}^{\infty}ψ(k)\cos\left(kt+\frac{βπ}{2}\right)$,
$Ψ_β\in L_{p'}$, $β\in \mathbb{R}$, $1/p+1/p'=1$. We additionally assume that the product
$ψ(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\infty$ it holds that
$\sum_{k=n}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$, and for $p=1$ the following condition is true
$\sum_{k=n}^{\infty}ψ(k)<\infty$.It is shown that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejer sums \linebreak$σ_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes.
We obtain estimates exact in order for deviations of Zygmund sums in metrics of spaces $L_{q}$, $1<q<\infty$, on classes of $2π$-periodic functions, that admit the representation in the form of convolution of functions that belong to unit ball of the space $L_{1}$ with fixed kernel $Ψ_β$. We show that at certain values of the parameters that define the class $L^ψ_{β,1}$ and method of approximation, Zygmund sums provide the order of best approximation of given classes by trigonometric polynomials in metric $L_{q}$
We obtain order-exact estimates for uniform approximations by using Zygmund sums $Z^{s}_{n}$ of classes $C^ψ_{β,p}$ of $2π$-periodic continuous functions $f$ representable by convolutions of functions from unit balls of the space $L_{p}$, $1< p<\infty$, with a fixed kernels $Ψ_β\in L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$. In addition, we find a set of allowed values of parameters (that define the class $C^ψ_{β,p}$ and the linear method $Z^{s}_{n}$) for which Zygmund sums and Fejer sums realize the order of the best uniform approximations by trigonometric polynomials of those classes.
We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices $α$ in this context. This generalizes a simpler method existing in the literature, but being valid for a restricted range of $α$. As an application, we prove that several fundamental operators in harmonic analysis of the Laguerre expansions, including maximal operators related to the heat and Poisson semigroups, Riesz transforms, Littlewood-Paley-Stein type square functions and multipliers of Laplace and Laplace-Stieltjes tra…
We present an intrinsically defined algebra of operators containing the right and left invariant Calderón-Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<\infty). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón-Zygmund theory.
The approximation properties of the trigonometric sums U_{n,p}^ψof a special type are investigated on the classes C^ψ_{β, \infty} of (ψ,β)-differentiable (in the sense of Stepanets) periodical functions. The solution of Kolmogorov-Nikol'skii problem in a sufficiently general case is found as a result of consistency between the parameters of approximating sums and approximated classes. It is shown that, in some important cases the sums under consideration provide higher order of approximation in the uniform metric on the classes C^ψ_{β, \infty} than Fourier sums, Zygmund sums and de la Valle Po…
We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2π$-periodic functions, which are represented by convolutions of functions $φ(φ\bot 1)$ from unit ball of the space $L_{1}$ with fixed kernels $Ψ_β$ of the form $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k) \cos\left(kt-\frac{βπ}{2}\right)$, $\sum\limits_{k=1}^{\infty}kψ(k)<\infty$, $ψ(k)\geq 0$, $β\in\mathbb{R}$.
We obtain exact for order estimates of best uniform approximations and uniform approximations by Fourier sums of classes of convolutions the periodic functions belong to unit balls of spaces $L_{p}, \ {1\leq p<\infty}$, with generating kernel $Ψ_β$, whose absolute values of Fourier coefficients $ψ(k)$ are such that $\sum\limits_{k=1}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$, and product $ψ(n)n^{\frac{1}{p}}$ can't tend to nought faster than power functions.
On the sets of $2π$-periodic functions $f$, which are defined with a help of $(ψ, β)$-integrals of the functions $φ$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $φ$. We prove that obtained estimates are best possible, in the case when the sequences $ψ(k)$ decrease to zero faster than any power function. In some important cases we establish the asymptotic equalities for the exact upper boundaries of uniform approximations by Fourier sums on …
In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k)\cos(kt-\frac{βπ}{2})$, $β\in\mathbb{R}$, whose coefficients $ψ(k)$ are such that product $ψ(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum\limits_{k=1}^{\infty}ψ^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum\limits_…
In this paper, we study the boundedness theory for maximal Calderón-Zygmund operators acting on noncommutative $L_p$-spaces. Our first result is a criterion for the weak type $(1,1)$ estimate of noncommutative maximal Calderón-Zygmund operators; as an application, we obtain the weak type $(1,1)$ estimates of operator-valued maximal singular integrals of convolution type under proper {regularity} conditions. These are the {\it first} noncommutative maximal inequalities for families of linear operators that can not be reduced to positive ones. For homogeneous singular integrals, the strong type …
We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2π$-periodic functions, representable in the form of convolutions of functions $φ$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.
There were established the exact-order estimations of the best uniform approximations byψ the trigonometrical polynoms on the $C^ψ_{β,p}$ classes of $2π$-periodic continuous functions $f$, which are defined by the convolutions of the functions, which belong to the unit ball in $L_p$, $1\leq p <\infty$ spaces with generating fixed kernels $Ψ_β\subset|L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, whose Fourier coeficients decreasing to zero approximately as power functions. The exact order estimations were also established in $L_p$-metrics, $1 < p \leq\infty$ for $L^ψ_{β,1}$ classes of $2π$-periodic fu…
In this paper asymptotic equalities are found for the least upper bounds of deviations in the uniform metric of de la Vallee Poussin sums on classes of 2π-periodic (ψ,β)-differentiable functions admitting an analytic continuation into the given strip of the complex plane. As a consequence, asymptotic equalities are obtained on classes of convolutions of periodic functions generated by the Neumann kernel and the polyharmonic Poisson kernel.
We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical Lusin area integral $S(f)$, the Littlewood-Paley $g$-function, and their continuous analogs $S_ψ$ and $g_ψ$. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón-Zygmund operator for all $1<p\le 3/2$ and $3\le p<\infty$, and for its maximal truncations for $3\le p<\infty$.
Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagona…
N. Kishore, Proc. Amer. Math. Soc. 14 (1963), 523, considered the Rayleigh functions sigma_n, sums of the negative even powers of the (non-zero) zeros of the Bessel function J_nu(z) and provided a convolution type sum formula for finding sigma_n in terms of sigma_1, ...,sigma_{n-1}. His main tool was the recurrence relation for Bessel functions. Here we extend this result to a larger class of functions by using Riccati ifferential equations. We get new results for the zeros of certain combinations of Bessel functions and their first and second derivatives as well as recovering some results of …
