For a positive integer $m$ and a finite non-negative Borel measure $μ$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces $\mathcal H_{μ, m}$. We show that if $α>\frac{1}{2},$ then for any $f$ in $\mathcal H_{μ, m},$ the sequence of generalized Ces{à}ro sums $\{σ_n^α[f]\}$ converges to $f$. We further show that if $α=\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.
We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.
In this paper, we study the (strongly) deferred Cesàro conull FK-spaces and we give some characterizations. We also apply these results to summability domains.
In this paper, we extend the notions of statistically convergence of order $β$ and strong Cesàro summability of order $β,$ and introduce the notions $f-$statistically convergence of order $β$ and strong Cesàro summability of order $β$ for $β\in \left( 0,1\right] $ with respect to an unbounded modulus function $f$ for sequences of fuzzy numbers and give some inclusion theorems.
The nonlinear signal processing has achieved a rapid process in the recent years. A family of nonlinear Fourier bases, as a typical family of mono-component signals, has been constructed and applied to signal processing. In this paper, the approximation properties of the partial sums and Ces?aro summability of series by the nonlinear Fourier basis are investigated in the Lp(T). Furthermore, these results are applied to the prove of Bernstein's inequalities for nonlinear trigonometric polynomials.
In this note we show that the Taylor series of a function in a weighted Dirichlet space is (generalized) Nörlund summable, provided that the sequence determining the Nörlund operator is non-decreasing and has finite upper growth rate. In particular the Taylor series is Nörlund summable for all $α>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The inequality $α>1/2$ is sharp. On the other hand if the Taylor series is Nörlund summable and the partial sums of the determining sequence enjoy a certain growth condition then the determining sequence has finite lower growth rate. An …
In this paper, we construct new important the subspaces $D_{p}^{q}S$, $D_{p}^{q}W$, $D_{p}^{q}F$ and $D_{p}^{q}B$ for a locally convex FK-space $X$ containing $ϕ$, the space of finite sequences. Then we show that there is relation among these subspaces. Also, we study deferred Cesàro conullity of one FK-space with respect to another, and we give some important results. Finally, we examine the deferred Cesàro conullity of the absolute summability domain $l_A$, and show that if $l_A$ is deferred Cesàro conull, then $A$ cannot be $l$-replaceable.
We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.
We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial sums of $f$ nor their Cesàro, Abel, Borel or logarithmic means converge to $f$ in $\mathcal{H}(b)$. A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.
This article is a study on the summability of random Fourier--Jacobi series of some functions in different spaces. We consider the random series $ \sum_{n=0}^\infty a_nA_n(ω)p_n^{(γ,δ)}(y), $ where $p_n^{(γ,δ)}(y),γ,δ>-1$ are orthonormal Jacobi polynomials, the scalars $a_n$ are Fourier--Jacobi coefficients of a function $f$ and the random variables $A_n(ω)$ are Fourier--Jacobi coefficients of the symmetric stable process $X(t,ω)$ of index $α\in [1,2].$ It is established that the random Fourier--Jacobi series is $Θ$--summable in probability, if $a_n$ are the Fourier--Jacobi coefficients of fun…
We show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function $f$ is of exponential type if and only if the formal power series $\sum_{n=0}^{\infty}f^{(n)}(0)z^{n}$ is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor…
Thesis includes review on the large order behaviour of perturbation theory in quantum mechanical and field theory models; generalization of the Borel summability and strong asymptotic conditions to various (including horn-shaped) regions; discussion of analytic aspects of perturbation theory; examples which demonstrate differences between the Borel summability and generalized one; application to the Rayleigh-Schrödinger perturbation theory and to the definition of the operator valued functions. The new summability methods converges in the whole Mittag-Leffeler star of an analytical function an…
It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz summability which is a generalization of the Cesàro summability. Therefore, we give the completeness criteria of these spaces with $c_0(X)$-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through $M^\infty_{R}\big(\sum_k T_k\big)$ which will be assumed that is complete for every $c_0(X)$-multi…
A particular consequence of the famous Carleson-Hunt theorem is that the Taylor series expansions of bounded holomorphic functions on the open unit disk converge almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, Bayart, Konyagin, and Queffélec constructed an example of an ordinary Dirichlet series $\sum a_n n^{-s}$, which on the open right half-plane $[Re >0]$ converges pointwise to a bounded, holomorphic function -- but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane.…
The classical Szegő-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal polynomials determined by the measure. The present paper constructs orthogonal polynomials on the torus of arbitrary finite dimension in order to prove theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings. The results are applied to the one-dimensional Schrödinger equation in impedance form to yield a new trace formula valid for piec…
