We investigate a new class of function spaces called variable bounded variation spaces of higher order by generalizing the classical notion of bounded variation. Our approach incorporates concepts from higher-order variations, bounded variation in the sense of generalized norms, and variations defined through modular function spaces. We focus on analysing the approximation properties of convolution-type nonlinear integral operators within these newly defined function spaces. In particular, we study these operators on variable bounded variation spaces of a fixed order in the sense of higher variations. Several approximation results are obtained using regular summability methods, highlighting the effectiveness of the proposed approach in improving convergence behaviour.
Using the J_P^2 -power series method, we redefine the sets of statistical limit points and statistical cluster points for double sequences of real numbers. We compare these sets with the set of usual limit points of double sequences and provide examples to demonstrate that the inclusions between these sets can be strict.
Tonks’ projection from the permutohedron to the associahedron and the Loday-Ronco map both send permutations to planar binary trees. We give a syntactic account of these maps in the equational calculus of the free non-symmetric, non-unital operad on one binary generator. The vertex restriction of Tonks’ projection is obtained by evaluating the head-insertion encoding on the reversed permutation, while the Loday-Ronco map is obtained by evaluating the decreasing encoding. We also give a local operadic proof that Tonks’ vertex map is order-preserving from the weak Bruhat order to the Tamari order.
We prove a Steinhaus-type theorem for four-dimensional matrix transformations of double sequences, establishing that (L_1, L_1, P ) \cap (L_s, L_1) = \emptyset , s > 1 . This extends Fridy?s classical result for single sequences. Our results hold for sequences of bounded variation, bounded, Pringsheim convergent, bounded Pringsheim convergent, and regularly convergent sequence spaces. We show this theorem fails in non-archimedean fields through a counterexample.
If p(z) is a polynomial of degree n having all its zeros in |z| <= k, k >= 1, then for r >= 1, Aziz [J. Approx. Theory, 55 (1988), 232-239] proved { integral(2 pi)(0) |1 + k(n) e(i theta)|(R) d theta}(1/r) max(|z|=1) |p ' (z)| >= n { integral(2 pi)(0) |p(e(i theta))|(R) d theta}(1/r) , whereas Devi et al. [Note Mat., 41 (2021), 19-29] proved that if p(z) is a polynomial of degree n having no zero in |z| < k, k <= 1, then for r > 0, k(n)n{(2 pi) (0) |p(e(i theta))|(R) d theta}(1/r) <= {(2 pi) (0) |e(i theta) + k(n)|(R) d theta} (1/r) {n max (|z|=1) |p(z)| - max (|z|=1) |p ' (z)|} , provided |p ' (z)| and |q ' (z)| attain their maxima at the same point on |z| = 1, where q(z) = z (n)p (1/z) (SIC). We do not only obtain improved extensions of the above inequalities into polar derivative by involving the leading coefficient and the constant term of the polynomial, but also give integral analogues of inequalities on polar derivative recently proved by Mir and Dar [Filomat, 36(16) (2022), 5631- 5640].
