We study product-decay conditions for backward trajectories in nonstationary fixed-point systems and revisit two statements from earlier work. Backward images of the terminal spaces are nested, so uniform convergence with respect to the terminal point holds if and only if their diameters tend to zero. Consequently, on bounded complete spaces, decay of the relevant products of Lipschitz constants is enough; summability of these products is not needed. We distinguish convergence in the first space from the existence of a compatible thread at every level and apply the same argument to staircase systems and prefix-dependent trees of maps. We also give error estimates, prove that the product condition is optimal for maps of the interval, and construct a staircase system with infinitely many expansive bonding maps. The nested-set and point-fibred arguments used here are classical; the contribution is their application to the precise summability assumptions in the cited nonstationary results, together with the necessary qualifications and sharp estimates.
In the theory of orthonormal systems, the study of summability methods plays a central role. For the bounded Vilenkin system, a fundamental theorem asserts that the $L_{p}$-norm convergence of certain operator sequences follows from a mild $\ell_{\gamma}$-type condition on the coefficients, without requiring any positivity or monotonicity assumptions. In this paper, we extend this classical result (Theorem~V) to the setting of almost everywhere convergence. In particular, we show that if the coefficients $\{t_{k,n}\}$ satisfy \[ \left\Vert (t_{k,n})_{k=0}^{n}\right\Vert_{\ell_{\gamma}} = O\!\left(n^{-1/\gamma'}\right), \qquad 1 < \gamma \le 2, \] then the corresponding sequence converges almost everywhere with respect to the bounded Vilenkin system. This work continues the line of research initiated by Blahota and Gát (J. Geom. Anal., 2025). Moreover, we address an open problem posed by Blahota and Gát, providing both affirmative and negative results. Finally, we present several applications of the main theorem to classical and generalized summability methods, thereby illustrating its broad applicability within the context of Fourier analysis.
The paper investigates the numerical stability of branched continued fraction expansions of Appell hypergeometric functions $F_2$ and their ratios in special cases. This is important when computing the values of these special functions, as they are widely used in science and engineering. Using the proposed new approach for estimating the so-called tails of approximants of branched continued fraction, we established the sets of numerical stability of the above-mentioned expansions in the cases of real and complex elements. Numerical experiments presented at the end confirm the obtained theoretical results.
This article investigates the extended best polynomial approximation operator, defined on the space of measurable functions, within the setting of Lorentz Gamma spaces \(\Gamma_{ w,q}\) for \(0 < q \le 1\). We derive quantitative estimates for extended best approximations and the outer operator quasi-norm. As a consequence, we show that extended best approximations in \(\Gamma_{ w,1}\) are near-best approximations in \(\Gamma_{ w,q}\).
This paper is devoted to the investigation of several notions associated with the shadowing behavior of ${\mathscr{G}}$-processes. We introduce and explore various types of shadowing, including standard shadowing, $h$-shadowing, limit shadowing, $s$-limit shadowing, and exponential limit shadowing for ${\mathscr{G}}$-processes. It is shown that if two ${\mathscr{G}}$-processes are uniformly conjugate, then the presence of any of these shadowing properties in one implies the same property holds for the other. Moreover, we establish that a finite direct product of ${\mathscr{G}}$-processes possesses the $h$-shadowing property if and only if each constituent ${\mathscr{G}}$-process in the product exhibits this property. The same equivalence also holds for the other types of shadowing considered.
