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We improve and complement a result by Móricz and Siddiqi [20]. In particular, we prove that their estimate of the Nörlund means with respect to the Vilenkin system holds also without their additional condition. Moreover, we prove a similar approximation result in Lebesgue spaces for any 1 ≤ p < ∞ .

We consider the Lipschitz class functions on [0, 1] and special series of their Fourier coefficients with respect to general orthonormal systems (ONS). The convergence of classical Fourier series (trigonometric, Haar, Walsh systems) of Lip 1 class functions is a trivial problem and is well known. But general Fourier series, as it is known, even for the function f ( x ) = 1 does not converge. On the other hand, we show that such series do not converge with respect to general ONSs. In the paper we find the special conditions on the functions φ n of the system ( φ n ) such that the above-mentioned series are convergent for any Lipschitz class function. The obtained result is the best possible.

We show the uniformly boundedness of the L1 norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means are the well-known Cesàro, Riesz, Bohner-Riesz means. Under some conditions, we verify that the kernels KnT=∑k=1ntk,nDk , (where Dk is the kth Dirichlet kernel) satisfy ‖KnT‖1≤c. As a result of this we prove that for any 1 ≤ p < ∞ and f ∈ Lp the Lp-norm convergence ∑k=1ntk,nSk(f)→f holds. Besides, for each integrable function f we have that these means converge to f almost everywhere.

The accurate calculation of reaction rates from experimental data is crucial for understanding and characterizing chemical processes. However, the presence of noise in experimental data can introduce errors in rate calculations. In this study, we introduced a novel approach that utilizes the Legendre series expansion method to directly derive smooth reaction rates from noisy experimental data, eliminating the need for numerical differentiation methods. This approach proves to be highly effective in handling noisy thermogravimetric analysis (TGA) data obtained from the thermal decomposition of specific polymers. We demonstrated the robustness and reliability of this method and provided Gnu Octave codes as a free alternative to MATLAB, making the implementation more accessible. Furthermore, the smooth reaction rates obtained were used to evaluate the activation energy using the Friedman isoconversional method. The results showed excellent agreement with those obtained using the Vyazovkin integral method. Additionally, the proposed method can be applied to obtain smooth derivative thermogravimetric (DTG) curves using noisy TGA data set.

We study the injectivity of the spherical mean operator associated to the Gelfand pairs ( U ,  N ), where N is a Heisenberg type group and U the subgroup of the group of orthogonal transformations of N that act trivially on its centre. We prove that when the dimension of the centre of N is 3, these spherical mean operator is injective on L p ( N ) for the optimal range 1 ≤ p ≤ 3 .

In this paper, the almost everywhere convergence of Cesàro means of Walsh–Kaczmarz–Fourier series in a varying parameter setting is investigated. In particular, we define subsequence of natural numbers and prove that the maximal operator is of strong type ( ), where is a Hardy space.

We consider the norm convergence for a special matrix-based de la Vallée Poussin-like mean of Fourier series with respect to the Vilenkin system. We estimate the difference between the named mean above and the corresponding function in norm, and the upper estimation is given by the modulus of continuity of the function. We also give theorems with respect to norm and almost everywhere convergences.

We obtain recovery formulas for coefficients of orthonormal spline series by means of its sum, if the partial sums of an orthonormal spline series converge in measure to a function and the majorant of partial sums satisfies some necessary condition, provided that the spline system corresponds to a “regular” sequence. Additionally, it is proved that the regularity of the sequence is essential.

In this paper the Walsh system will be considered in the Kaczmarz rearrangement. We estimate the difference between matrix transform means of Walsh–Kaczmarz–Fourier series and the corresponding function in norm, and the upper estimation is given by the modulus of continuity of the function. We also prove norm convergence with similar conditions.

The present note is an addition to the author’s recent paper [44], concerning general multiplicative systems of random variables. Using some lemmas and the methodology of [13], we obtain a general extremal inequality, with corollaries involving Rademacher chaos sums and those analogues for multiplicative systems. In particular we prove that a system of functions generated by bounded products of a multiplicative system is a convergence system.