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We study Jackson's inequality on high-dimensional spheres with respect to the modulus of smoothness defined via the rotation group. We obtain a version of Jackson's inequality with a dimension-free constant, extending Newman and Shapiro's well-known results in 1964 from the case of r = 1 and p = ∞ to more general cases. Our results partially overcome the curse of dimensionality. We also establish similar results on the equivalence of the K-functional and modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new discrete inequalities with weights of Hardy’s type and employ them to establish some properties of discrete weighted Gehring class of sequences satisfying reverse Hölder inequality. We also consider the weighted class of sequences satisfying a reverse inequality for nonincreasing sequences and prove some higher summability theorems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the notion of V a -deformed free convolution, introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of V a -deformed free convolution. Then we provide an approximation of elements of the CSK family generated by V a -deformed free Poisson distribution. [ABSTRACT FROM AUTHOR]

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in L p (G 2) space ( 1 ≤ p < ∞ ) and in C (G 2) . It implies in some special case norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We establish injectivity results for three different spherical means on an H-type group, G. The first one is the standard spherical mean operator, which is defined as the average of a function over the spheres in the complement of the center, the second one is the average over the product of spheres in the center and its complement, and the third one is the average over the spheres defined by a homogeneous norm on G. If m is the dimension of the center of G, injectivity of these spherical means is proved for the range 1 ≤ p ≤ 2 m m - 1 . Examples are provided to show the sharpness of our results in the first two cases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we give direct theorems on approximation by Haar and Walsh polynomials in variable exponent grand Lebesgue space. Also the degree of approximation by Borel, Abel-Poisson, Riesz-Zygmund and Euler linear means of Haar-Fourier series are treated in the above-cited space. [ABSTRACT FROM AUTHOR]

We calculate the rate of convergence of the double rational Fourier series for regular, bounded, measurable, and two-variable functions. The rectangular oscillation of the two-variable function is used to quantify this rate. Additionally, we give an approximation of convergence rate of the double rational Fourier series for continuous functions with generalized bounded variation. [ABSTRACT FROM AUTHOR]

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 < ∞ . [ABSTRACT FROM AUTHOR]

Revisiting the main point of the almost everywhere convergence, it becomes clear that a weak (1,1)-type inequality must be established for the maximal operator corresponding to the sequence of operators. The better route to take in obtaining almost everywhere convergence is by using the uniform boundedness of the sequence of operator, instead of using the mentioned maximal type of inequality. In this paper it is proved that a sequence of operators, defined by matrix transforms of the Walsh–Fourier series, is convergent almost everywhere to the function f ∈ L 1 if they are uniformly bounded from the dyadic Hardy space H 1 I to L 1 I . As a further matter, the characterization of the points are put forth where the sequence of the operators of the matrix transform is convergent. [ABSTRACT FROM AUTHOR]

The notion of supershift generalizes that one of superoscillation and expresses the fact that the sampling of a function in an interval allows to compute the values of the function outside the interval. In a previous paper, we discussed the case in which the sampling of the function is regular and we are considering supershift in a bounded set, while here we investigate how irregularity in the sampling may affect the answer to the question of whether there is any relation between supershift and real analyticity on the whole real line. We show that the restriction to R of any entire function displays supershift, whereas the converse is, in general, not true. We conjecture that the converse is true as long as the sampling is regular, we discuss examples in support and we prove that the conjecture is indeed true for periodic functions. [ABSTRACT FROM AUTHOR]

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{\rm{K}}_{\rm{n}}^{\rm{T}} = \sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{D}}_{\rm{k}}}}, (where Dk is the kth Dirichlet kernel) satisfy ‖KnT‖1≤c.{\left\| {{\rm{K}}_{\rm{n}}^{\rm{T}}} \right\|_1} \le {\rm{c}}{\rm{.}}

In this article, we develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for the fourth-order equation with singular potential and simply supported plate (SSP) boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.

We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details. [ABSTRACT FROM AUTHOR]

Let u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in H r (u) , the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L 2 space associated with u . This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N ⩽ r . We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E n (f (r)) , the error of best approximation of f (r) in L 2 (u) . [ABSTRACT FROM AUTHOR]

The orthogonal structure in two variables on the quartic curve y 2 = a x 4 + b x 2 + c is considered. For an even weight function on the curve, we show that orthogonal polynomials can be expressed in terms of two families of orthogonal polynomials in one variable. We establish relationships between the partial Fourier sum on the curve and partial Fourier sums in one variable. We also investigate the quadrature rule and polynomial interpolation on the curve. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new weighted maximal operators of the Fejér means of the Walsh–Fourier series. We prove that for some “optimal” weights, these new operators indeed are bounded from the martingale Hardy space H p (G) to the Lebesgue space L p (G) , for 0 < p < 1 / 2 . Moreover, we also prove sharpness of this result. As a consequence, we obtain some new and well-known results. [ABSTRACT FROM AUTHOR]

We introduce some new weighted maximal operators of the Fejér means of the Walsh–Fourier series. We prove that for some "optimal" weights these new operators are bounded from the martingale Hardy space H p (G) to the space weak- L p (G) , for 0 < p < 1 / 2 . Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results. [ABSTRACT FROM AUTHOR]

Using the Fourier–Gegenbauer series, we prove several identities that generalize known results. In particular, it is proved, that ∑ n = 0 ∞ 1 4 n ( 2 n n ) z − 2 n ( z − 1 / 2 n ) ( z n ) 3 = tan ⁡ (π z) π for all complex numbers z such that ℜ (z) > − 1 2 and z ∉ 1 2 + Z . In addition, for all nonnegative integers m, we obtain the Ramanujan-type series 2 π = ∑ n = m ∞ (− 1) n − m (1 + 4 n) 64 n ( 2 n − 2 m n − m ) ( 2 n n ) ( 2 m + 2 n m + n ) . 2 4 m + 5 π 2 = ∑ n = 0 ∞ (1 + 2 m + 4 n) (2 m + 1) (2 m + 3) (m + n + 1) 2 (2 n − 1) 2 256 n ( 2 n n ) 2 ( 2 m + 2 n m + n ) 2. Which are known in the case m=0. [ABSTRACT FROM AUTHOR]

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results. [ABSTRACT FROM AUTHOR]