Chapter 12 - Fourier Series and Integrals

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is Lp-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We obtain sharp conditions for the absolute uniform convergence of Fourier series in the eigenfunctions of the Schrodinger operator with Kato potential in a bounded domain for functions lying in the domains of generalized fractional powers of the original Schrodinger operator or in generalized Besov classes with a sharp exponent.

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We writebeing the complex Fourier series of f (θ).

A computationally useful expression for the modified Green' function in a rectangular region is derived. This can be used to obtain solutions of Laplace' equation or Poisson' equation for the electrical potential associated with any configuration of current sources that satisfies the condition of charge balance, and represents a generalization of a well–known result due to Newman applicable to an infinite channel. The starting point for the analysis is the very slowly converging Fourier–series expansion of the modified Green' function. This Fourier series can be separated into a Fourier series in one variable that can be summed in closed form, and a double Fourier series that can be partly summed with respect to one index and transformed by application of the Poisson summation formula and integration by residues. The result of applying the Poisson formula can be expressed either as a rapidly converging series of logarithmic functions, or as a rapidly converging Fourier series which can be integrated analytically with respect to the coordinates of the source point. The use of the modified Green' function is demonstrated first by determining potential distributions that arise from specification of uniform current density on two or more of the boundary surfaces. The potential in a cell with two electrodes embedded in opposite sides of the rectangle is then calculated, and circumstances in which this potential approaches the potential in an infinite channel are determined.

For a separable finite diffuse measure space {mathcal {M}} and an orthonormal basis {varphi _n} of L^2({mathcal {M}}) consisting of bounded functions varphi _nin L^infty ({mathcal {M}}), we find a measurable subset Esubset {mathcal {M}} of arbitrarily small complement |{mathcal {M}}{setminus } E|<epsilon , such that every measurable function fin L^1({mathcal {M}}) has an approximant gin L^1({mathcal {M}}) with g=f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of {mathcal {M}}=G/H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

A note on Appell sequences, Mellin transforms and Fourier series

On Weighted Integrability of Trigonometric Series and L 1 -Convergence of Fourier Series

Introduction to Circuit Analysis and Design

We consider the properties of systems $\Phi_1$ orthogonal with respect to a weighted discrete-continuous Sobolev inner product of the form $\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\displaystyle\int_a^b f'(t)g'(t)w(t)dt$. The completeness of systems $\Phi_1$ in the Sobolev space $W^1_{L^2_w}$ and the relation of $\Phi_1$ to systems orthogonal in weighted Lebesgue spaces $L^2_u$ are studied. We also analyze properties of the Fourier series with respect to systems $\Phi_1$. In particular, conditions for the uniform convergence of Fourier series to functions from $W^1_{L^2}$ are obtained.

In the first part of our theses we give sufficient conditions for the absolute convergence of the double Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to double Fourier series. In the second part we give sufficient conditions for the absolute convergence of the double Walsh-Fourier series of a function. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity and s-bounded fluctuation.

Purpose. To compare the efficiency of using finite elements with variable and averaged mechanical and geometric parameters and to investigate the convergence of results obtained by the semi-analytical finite element method (SAFEM) using Fourier series and polynomials with the results obtained by the finite element method (FEM). The methods. The construction and development of an algorithm for studying the stress-strain state of spatial bodies with variable and averaged mechanical and geometric parameters were carried out based on SAFEM. Findings. Solvability indicators of SAFEM were obtained for calculating nodal reactions and stiffness matrix coefficients of finite elements with variable and averaged mechanical and geometric parameters. Numerical convergence studies of results obtained using SAFEM with Fourier series, polynomials, and the finite element method were conducted for a benchmark example, which was the Boussinesq problem for a half-space subjected to a concentrated force. The results indicate that the convergence of the investigated coordinate functions in the considered problem is of the first order. The originality. The obtained solvability indicators of SAFEM for calculating nodal reactions and stiffness matrix coefficients of finite elements with variable and averaged mechanical and geometric parameters allow for the study of various classes of problems. Numerical convergence studies using Fourier series, polynomials, and the finite element method were conducted for the benchmark example. Practical implementation. The practical significance lies in the development of a methodology for determining the stress-strain state of relevant spatial elements of structures with variable and averaged mechanical and geometric parameters subjected to arbitrarily distributed spatial loads.

Many results introduced for the absolutely convergence of Fourier series in terms of absolutely continuous functions. Here we study the convergence of Fourier series in terms of $p$-integrable functions series.

In the present article , We have extended the result of S. Tikhonov[8] and used y-general monotone coefficients to prove the L^1 -convergence of the trigonometric series. Also,"||g-Sn||≡o(1)" considered as , the necessary and sufficient condition for the convergence in terms of coefficients.

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.

In this paper, we show a complete characterization of the uniform boundedness of the partial sum operator in a discrete Sobolev space with Jacobi measure. As a consequence, we obtain the convergence of the Fourier series. Moreover it is showed that this Sobolev space is the first category which implies that it is not possible to apply the Banach–Steinhaus theorem.