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A real number x is considered normal in an integer base b ⩾ 2 if its digit expansion in this base is "equitable", ensuring that for each k ⩾ 1 , every ordered sequence of k digits from { 0 , 1 , ... , b - 1 } occurs in the digit expansion of x with the same limiting frequency. Borel's classical result [4] asserts that Lebesgue-almost every x ∈ R is normal in every base b ⩾ 2 . This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set N (O , E) of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension [30] but zero Fourier dimension. The latter condition means that N (O , E) cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that N (O , E) supports a Rajchman measure μ , whose Fourier transform μ ^ (ξ) approaches 0 as | ξ | → ∞ by definiton, albeit slower than any negative power of | ξ | . Moreover, the decay rate of μ ^ is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt [38] and a construction of Lyons [24]. As a consequence, N (O , E) emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem [17] in the special case of N (O , E) . [ABSTRACT FROM AUTHOR]

The Chirp-Fourier transform is one of the most important tools of the modern signal processing. It has been widely used in the fields of ultrasound imaging, parameter estimation, and so on. The key to its application lies in the sampling and fast algorithms. In practical applications, nonuniform sampling can be caused by sampling equipment and other reasons. For the nonuniform sampling, we utilized function approximation and interpolation theory to construct different approximation forms of Chirp-Fourier transform kernel function, and proposed three fast nonuniform Chirp-Fourier transform algorithms. By analyzing the approximation error and the computational complexity of these algorithms, the effectiveness of the proposed algorithms was proved.

This study proposes a closed-form explicit, exact analytical model for bending deformation accounting for the interface nonlinearity in regular steel–concrete composite beams with partial shear connectors. A nonlinear shear load–slip equation that facilitates theoretical derivation and numerical simulation analysis was introduced to consider the nonlinear effects of partial shear interaction on composite structures. Applying the principle of minimum potential energy and the variational principle, a theoretical model was then proposed for the bending deformation and interface slip of a simply supported steel–concrete composite beam under varied load conditions. Using a trigonometric series, slip displacement and bending deflection functions were devised based on the undetermined coefficient method. A unified solution was also proposed for the beam's bending rigidity, taking into account nonlinear interface slip. The analytical solutions' applicability and accuracy were validated by comparing with existing experimental literature. Furthermore, a parametric study using the validated numerical nonlinear model demonstrated the influences of the stud spacing, the nonlinear interfacial effect, and the span-to-depth ratio of composite beams. This research can be referred to the accurate deformation estimation of composite beams. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Schrödinger–Pauli problem on a finite square domain. Because of the boundary conditions we have proposed as a solution a suitable trigonometric series. We have shown that these coefficients exist and can be univocally determined by solving a simple algebraic system. [ABSTRACT FROM AUTHOR]

Improving the rigidity of the process system in the cutting region of thin-walled parts is a challenging problem to ensure machining accuracy. For limited structural space, the use of ice support is an effective method. However, ice and workpiece constitute a completely new process system, which generates a complex process response under milling forces. Based on the Kirchhoff-Love thin-wall small-deformation assumption and the Winkler model for describing the inverse support of ice on cylindrical shell thin-walled parts (CSTWPs), a new prediction model is developed to predict the deflection of CSTWP under ice support. In the model, by introducing the displacement function in the form of annular triangular series, the analytical solution of the displacement is given for the cylindrical shell with nonsimply supported edges at both ends under ice support. A finite element model for milling CSTWP under ice support is developed, which takes into account the nonlinear behavior of ice and the complex mechanical behavior of the ice/workpiece interface. Based on this finite element model and the corresponding milling experiments, the accuracy and validity of the established model are verified. The work provides a theoretical basis for the prediction of the deformation of CSTWP under ice support. [ABSTRACT FROM AUTHOR]

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series. We find sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle. These conditions are expressed in terms of the Fourier coefficients and are weaker than Hardy's condition. We give a description of all Banach function spaces given over the circle and endowed with a norm been equivalent to a norm in a homogeneous Banach space. We study interpolation properties of such spaces and give new examples of them. We extend the classical Fejér theorem on the uniform Cesàro summability of the Fourier series on sets by means of a refined version of Cantor's theorem on the uniform continuity of a mapping between metric spaces. We also generalize the classical Hardy theorem on the uniform convergence of the Fourier series on sets. [ABSTRACT FROM AUTHOR]

In this paper we consider trigonometric series with p-bounded variation coefficients. We presented a sufficient condition for uniform convergance of such series in case p > 1 . This condition is significantly weaker than these obtained in the results on this subject known in the literature. [ABSTRACT FROM AUTHOR]

This article is devoted to the topic of absolute summation of series or Cesaro summation. The relevance of this article lies in the fact that a type of absolute summation with vector index which has not been previously studied is considered. In this article, a sufficient condition for the vector index absolute summation method was obtained in terms of the best approximation by «angle» of the functions from Lebesgue space. The theorem that gives a sufficient condition proves the conditions that are sufficient in different cases, which may depend on the parameters. From this proved theorem, a sufficient condition on the term mixed smoothness modulus of the function from Lebesgue space, which is easily obtained by a well-known inequality, is also presented.

We study how the well-known criterion for the uniform convergence of a sine series with monotone coefficients changes if, instead of monotonicity, one imposes the condition of -monotonicity with . Moreover, we obtain an addition to the well-known Kolmogorov theorem on the integrability of the sum of a cosine series with convex coefficients tending to zero. [ABSTRACT FROM AUTHOR]

In this note, we study the evaluations of Euler sums via trigonometric series. It is a commonly believed conjecture that for an even weight greater than seven, Euler sums cannot be evaluated in terms of the special values of the Riemann zeta function. For an even weight, we reduce the evaluations of Euler sums into the evaluations of double series and integrals of products of Clausen functions. We also re-evaluate Euler sums of odd weight using a new method based on trigonometric series. [ABSTRACT FROM AUTHOR]

A model predictive convex programming (MPCP) on $SE(3)$ parametrized by trigonometric series control is proposed in this paper, to solve the optimal control problem of spacecraft attitude orbit integration. Firstly, the geometric modeling of the spacecraft with six degrees of freedom for the attitude orbit integration is performed by the differential manifold $SE(3)$ , which can effectively avoid the problems of ambiguity, receding winding, and singularity that occurs in the conventional methods for rigid body attitude description. Then, based on differential geometric theories, such as the variational principle, the left-invariant principle of Lie group, and the topology of Lie algebraic space, MPCP is applied to $SE(3)$ . It can solve a class of optimization problems with process constraints and control input constraints during spacecraft flight. Furthermore, a control framework of trigonometric series is constructed, which is seamlessly integrated into MPCP to achieve smoother trajectory optimization control. Finally, the practicality and effectiveness of the proposed method are verified by numerical simulation.

Steel concrete beams have a reasonable structure in terms of using material and high load carrying capacity. This paper deals with an approximate solution based on a trigonometric series for the static of steel concrete beams. The displacement field is based on the higher-order theory using Reddy’s hypothesis. The governing equations are derived from variation principles. An approximate solution based on the representation of displacement fields by trigonometric series is developed to solve the static problem of steel concrete beams. In order to verify the accuracy of the present approximate solution, numerical results are compared with those of exact solutions using classical beam theory. The displacements and nominal stress distributions in the depth direction are obtained with various high of beams. The present approximate approach can accurately predict the displacements and stresses of steel concrete beams.

This paper presents a data-driven design and optimization of acoustic metamaterials with three-phase materials for highly tunable wave transmission. The geometry of representative unitcell is defined by the trigonometric series function to describe an arbitrary shape with symmetry, which enables the unitcell to achieve a large sub-wavelength bandgap. We propose a lightweight and efficient algorithm, 'decoupled gradient decent (DGD)', to search for the optimal design and uncover the 'best' shape features—the interface curvature—in tuning the wave transmission. As a result, the host composite can partly overlap the individual cell's bandgap and achieve a wide frequency gap that forbids wave transmission, namely a passive tunability. Another advantage of the trigonometric series designed shape is the high flexibility. A slight surface pressure obviously deforms the unitcell and shifts its band structure. Our simulation shows that a moderate pressure dramatically changes the frequency forbidding gap for both traversal and longitudinal wave transmissions, which indicates an active tunability. The surface deformation can be applied by either a mechanical pressure or external electric field if the composite uses a dielectric substrate. Therefore, this study opens a sandbox of manipulating wave transmission through the topology and structure optimization in applications such as seismic damping (Hz), noise insulating (kHz) and ultrasound imaging (MHz). [ABSTRACT FROM AUTHOR]

The properties of -monotone sequences are studied. A relationship between -monotonicity and the limiting rate of change of coefficients is revealed. Operations on sequences that do not lead out of the class are discussed. An analogue of the Lorentz theorem for trigonometric series with coefficients from the classes for is proved. [ABSTRACT FROM AUTHOR]

The average deformed fiber is a continuous and smooth function of the fourth order. The deflection and rotation of beams can be determined by various methods available in the literature. Thus, in this paper, the expression of the average deformed fiber is defined in advance, then it is considered an infinite sum of sinusoidal functions that are successively evaluated using sinusoidal trigonometric series if it is considered a periodic function, or using Fourier series if it is considered a non-periodic function. The method is examined by solving several beam problems. The results indicate that the method can be used with confidence for solving any bending beam problem. [ABSTRACT FROM AUTHOR]

The author's recent attempt to generalize the problem of L 1 convergence of trigonometric series to the non-periodic case is finalized here. It can now be said that any known condition which guarantees integrability of the Fourier transform of a function of bounded variation also leads to the L 1 convergence of its partial integrals provided that a universal necessary and sufficient condition is added. [ABSTRACT FROM AUTHOR]

In this paper, the fractal Toda oscillator is established by using the fractal variational theory and the exact analytical solution is obtained by the non-perturbative method. Also, trigonometric series and He’s frequency formula are applied to determine the approximate analytical solution. To illustrate the operability and effectiveness of these methods, the solution processes are described in detail, and the analytical solutions are compared with numerical ones, showing good agreement. It should be noted that the relative errors produced by He's frequency formula and the trigonometric series method are all small. The results show that the non-perturbative method and He's frequency formula are extremely effective and simple for fractal differential equations. Finally, the effect of fractal derivative order on the vibration characteristics is briefly explained graphically. The non-perturbative method provides additional advantages for quickly testing the frequency property of fractal nonlinearity vibration, which is of great significance in the study of the vibration characteristics of nonlinear oscillators.

This article is devoted to the topic of absolute summation of series or Cesaro summation. The relevance of this article lies in the fact that a type of absolute summation with vector index, which has not been previously studied, is considered. The necessary condition of absolute summation with vector index of double trigonometric series is obtained and several sufficient conditions have been proved. In order to prove this necessary condition, the lemma providing the necessary condition of absolute summation with vector index of the double number series is first time proved. Another lemma, which is auxiliary to the main result, has been proven for the special case of absolute summation with vector index. With the help of these two lemmas the necessary condition is obtained. The theorem that gives a sufficient condition proves the conditions that are sufficient in different cases, which may depend on the parameters. [ABSTRACT FROM AUTHOR]

Let F be a self-similar set on R associated to contractions fj(x)=rjx+bj, j∈A, for some finite A, such that F is not a singleton. We prove that if logri/logrj is irrational for some i≠j, then F is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of F. No separation conditions are assumed on F. We establish our result by showing that every self-similar measure μ on F is a Rajchman measure: the Fourier transform μˆ(ξ)→0 as |ξ|→∞. The rate of μˆ(ξ)→0 is also shown to be logarithmic if logri/logrj is diophantine for some i≠j. The proof is based on quantitative renewal theorems for stopping times of random walks on R. [ABSTRACT FROM AUTHOR]

Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor. Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip. Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length.

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space Lqϕ(Lq)(0,2π]2 is obtained.

An aircraft transition connects two flight conditions and is a fundamental component of aerospace operations. For achieving a seamless transition that accurately connects two steady flights, a hierarchical parameterization method is developed in this article for generating smooth flight trajectories satisfying hard boundary constraints on controls and control derivatives of all orders. The proposed work is based on a trigonometric series parameterization of control variables and hierarchically parameterizes the series coefficients with flat and logistic functions, leading to a new set of coefficients and a formulation that inherently guarantees the desired conditions. The new set of coefficients can be determined by optimization methods and can be integrated in existing methods with the advantage of streamlining the problem formulation. Simulation studies including comparisons demonstrate the effectiveness and performance of the proposed method. [ABSTRACT FROM AUTHOR]

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1-4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f∈Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ-summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1=...=βs=1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f∈Lq(I2) are obtained in the cases β1=1/q', -1

Abstract In this paper we introduce new classes of p-bounded variation sequences and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients belonging to these classes. This is a generalization of the results obtained by the first author [J. Inequal. Appl. 2010:1–19, 2010] and Dyachenko and Tikhonov [Stud. Math. 193(3):285–306, 2009].

Let be a Euclidean space of dimension . For a domain , we denote by the set of functions having zero integrals over all closed balls of radius r contained in G (if domain G does not contain such balls, we set ). Let E be a nonempty subset of . In this paper we study the following questions related to the extension problem. 1) Which conditions guarantee the extension of a continuous function defined on E to a continuous function of class defined on the whole ? 2) If the above extension exists, obtain growth estimates of the extended function at infinity. Theorem 1 of this paper shows that for a wide class of continuous functions on segment E defined in terms of the modulus of continuity, there exists an extension to a bounded function of class regardless of the length of segment E. A similar result is not true for open sets E with a diameter greater than 2r, even without conditions for extension growth. Theorem 1 also contains an estimate of the velocity decrease of the extended function at infinity in directions orthogonal to the segment E. [ABSTRACT FROM AUTHOR]

In this paper we characterize the functions that map the positive general monotone sequences to themselves, i.e., we obtain a necessary and a sufficient condition for the functions φ to satisfy ∑ k = n 2 n | a k + 1 − a k | ≤ C a n ⇒ ∑ k = n 2 n | ϕ (a k + 1 ) | ≤ C ′ ϕ (a n). for all { a n } n = 1 ∞ ⊂ ℝ + . [ABSTRACT FROM AUTHOR]

In practical scenarios, flight trajectories shall be smooth to comply with the required curvature behavior. This article describes a trajectory generation method that is able to produce smooth control commands with an arbitrary order of smoothness. The system inputs are formulated analytically as trigonometric series in which the coefficients are to be determined by optimization. Effective implementations of the proposed method are presented. These include a numerical benchmark that shows how the order of the series shapes the results as well as the comparison with the optimal solution, and the unmanned aerial vehicle transition trajectory generation using a convex optimization framework integrated with the proposed design. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical approach is presented for the forced vibration analysis of a multiple DOF system that is governed by ordinary differential equations with both time-periodic stiffness and external force that have different periods. Based on an equivalent dynamic system, a closed-form solution, i.e. a special trigonometric series, is presented for the forced vibration response of the system. Computation is realized by applying the harmonic balance operation, and the parametric vibration equation is converted into a set of infinite-order linear algebraic equations. Based on the physical property of the forced vibration response, all the coefficient vectors of the forced response are solved by the use of inverse matrix. The present approach can be used to predict the forced vibration response and its spectrum with better computation performance. It can help reveal some nonlinear phenomena in the parametric system. Therefore, it proves to be a useful tool for future research and engineering applications of parametric systems. [ABSTRACT FROM AUTHOR]

In this article, we have presented the necessary and suficient conditions for the power integrability with a weight of the sum of sine and cosine series whose coeficients belong to the RBV Sr,δ+,ω class. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a new class of sequences called GMS (p, ß, r), which is the generalization of a class considered by Tikhonov in [9] and Szal in [10]. Moreover, we obtained in this note sufficient and necessary conditions for the uniform convergence of sine and cosine series with (p, ß, r) -general monotone coefficients. [ABSTRACT FROM AUTHOR]

In this paper we have found the necessary and sufficient conditions for the power integrability with a weight of the sum of the sine and cosine series whose coefficients belong to a subclass of class. [ABSTRACT FROM AUTHOR]

A spatial system of two closed cylindrical shells is considered, having rigid fixation at one end and a free end at the other. Shells are located at some distance from each other and are interconnected by one intermediate link. One of the shells is under the action of a load, concentrated in the annular direction and uniformly distributed along one generatrix. To cal- culate the spatial system, the force method is used. Thus, the calculation of the spatial system is reduced to the calculation of individual closed cylindrical shells. From the condition of equality of radial displacements at the point of contact of the shells, the reactive pressure in the intermediate bond was determined. The magnitude of the reactive pressure in the bond varies depending on the location of the intermediate bond and the geometric parameters of the shells. Knowing the magnitude of the reactive pressure, one can determine the stress-strain state of each shell

In this paper, a closed-form solution is presented for the free vibration response of a 2-DOF parametric system by a technique with special trigonometric series. The technique extends the concept of modulation feedback to the problem of multiple degrees of freedom (M-DOF) parametric system using an equivalent dynamic system. The characteristic equation is derived by the harmonic balance in a parametric equation with the principal oscillation frequencies computed. The computation of modes and splitting modes is carried out by operation on a mixed matrix. The arbitrary constants in the general solution are determined by the initial conditions. A real-life example of a coupled inverted-pendulum system is provided to illustrate the applicability of the proposed approach. An approaching error is adopted to assess the computational errors both using the proposed technique and the Runge–Kutta algorithm. Of interest is that the time-periodic coefficient results in the changeability of the mode in the 2-DOF parametric system, while exciting the splitting mode in the harmonic combination resonance. The proposed technique is numerically demonstrated to possess not only high approaching accuracy, but is also capable of revealing some new phenomena in the parametric system, such as characteristics of mode and slitting mode, etc. It proves to be useful for future research and engineering applications of parametric systems. [ABSTRACT FROM AUTHOR]

A summation/integration method for fast summing trigonometric series is presented. The basic idea in this method is to transform the series to an integral with respect to some weight function on R+ and then to approximate such an integral by the appropriate quadrature formulas of Gaussian type. The construction of these quadrature rules, as well as the corresponding orthogonal polynomials on R+, are also considered. Finally, in order to illustrate the efficiency of the presented summation/integration method two numerical examples are included. [ABSTRACT FROM AUTHOR]