Convergence Of Fourier Series Research Articles

On Uniformly Converging Fourier Series of Functions from Space LP

On Uniformly Converging Fourier Series of Functions from Space LP

A mixed problem for an inhomogeneous wave equation with a summable potential

A mixed problem for an inhomogeneous wave equation with fixed ends in the case of a summable potential is studied. Using the Krylov method for acceleration of the convergence of Fourier series, a classical solution under minimal conditions on the initial data and a generalized solution in the case of quadratic summable initial data and perturbing function are obtained.

Lipschitz functions on SU(2) have uniformly convergent Fourier series

We demonstrate that the Fourier partial sums of a Lipschitz continuous function on the two-dimensional special unitary group converge uniformly to the function.

On the absolute convergence of Fourier series with respect to general orthonormal systems

Abstract The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.

Pointwise convergence of Fourier series (I). On a conjecture of Konyagin

We provide a near-complete classification of the Lorentz spaces $\Lambda\_{\varphi}$ for which the sequence ${S\_{n}}{n \in \mathbb N}$ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function $\varphi$, we identify $\Lambda{\varphi}:= L$ log log $L$ log log log log $L$ as the largest Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{1,\infty}$. As a consequence, we disprove a conjecture stated by Konyagin in his 2006 ICM address; provide a negative answer to an open question related to the Halo conjecture. Our proof relies on a newly introduced concept of a "Cantor Multi-tower Embedding," a special geometric configuration of tiles that can arise within the time-frequency tile decomposition of the Carleson operator. This geometric structure plays an important role in the behavior of Fourier series near $L^1$, being responsible for the unboundedness of the weak-$L^1$ norm of a "grand maximal counting function“ associated with the mass levels.

A Generalized Fejér’s Theorem for Locally Compact Groups

The classical Fejer’s theorem is a criterion for pointwise convergence of Fourier series on the unit circle. We generalize it to locally compact groups.

Strong boundedness, strong convergence and generalized variation

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $L^{p}$ spaces, $1\leq p < \infty$. We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation.

Estimates and trace formulas for unitary and resolvent comparable perturbations

We prove estimates for Schatten norms of multiple operator integrals with symbols arising from Fourier series and, as a consequence, derive higher order trace formulas for functions given by absolutely convergent Fourier series of unitary operators and for rational functions bounded on the real line of resolvent comparable self-adjoint operators. These results substantially extend the main results of [12,14,13], where the estimates were proved for multiple operator integrals with symbols arising from polynomials, the trace formulas in the unitary case for analytic functions on the unit disc, and the trace formulas in the resolvent comparable self-adjoint case for rational functions bounded and analytic in the lower half-plane. Our results rely crucially on an algebraic approach to symbols of multiple operator integrals developed in this paper to replace the prior restrictive analytic approach of [12].

Majorants of the Dirichlet kernels and the Dini pointwise tests for generalized Haar systems

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classicalDini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained.

Digital inversive vectors can achieve polynomial tractability for the weighted star discrepancy and for multivariate integration

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not grow exponentially fast with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications, true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of Hölder continuous, absolute convergent Fourier and cosine series, where the proposed method is successful.

On T-Strong Convergence of Numerical Sequences and Fourier Series

In this paper, we generalize the notion of \(\varLambda \)-strong convergence of numerical sequences defined by Moricz (Acta Math Hung 54(3–4):319–327, 1989) to T-strong convergence, using a lower triangular matrix \(T=(a_{n,k})\) with nondecreasing monotone rows of positive numbers tending to \(\infty \) i.e., \(a_{n,k}\le a_{n,k+1} \forall n\) and \(\lim _{k\rightarrow \infty } a_{n,k}=\infty \,\, \forall n\). We also establish a relationship between ordinary convergence and T-strong convergence. We further show that this concept can also be applied to the strong convergence of Fourier series under C-metric and \(L_{p}\)-metric.

Сходимость рядов Фурье по полиномам Якоби в весовом пространстве Лебега с переменным показателем

Сходимость рядов Фурье по полиномам Якоби в весовом пространстве Лебега с переменным показателем

On the absolute convergence of Fourier series and generalisation of Lipschitz spaces, defined by differences of fractional order

We obtain the necessary and sufficient conditions in terms of Fourier coefficients of $2\pi$-periodic functions $f$ with absolutely convergent Fourier series, for $f$ to belong to the generalized Lipschitz classes $H^{\omega, \alpha}_{\mathbb{C}}$, and to have the fractional derivative of order $\alpha$ ($0 &lt; \alpha &lt; 1$).

On the Application of Spectral Filters in a Fourier Option Pricing Technique

When Fourier techniques are applied to specific problems from computational finance with nonsmooth functions, the so-called Gibbs phenomenon may become apparent. This seriously affects the efficiency and accuracy of the numerical results. For example, the variance gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier series. We apply spectral filters to achieve faster convergence. Filtering is carried out in Fourier space; the series coefficients are pre-multiplied by a decreasing function. This does not add any significant computational costs. Tests with different filters show how the algebraic index of convergence is improved.

On the divergence of Fourier series in the spaces ϕ(L) containing L

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than Lp = Lp[−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ(L) of Kolmogorov’s theorem on the convergence of Fourier series in Lp, p ∈ (0, 1), cannot be improved.

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Undecidability through Fourier series

In computability theory a variety of combinatorial systems are encountered (word problems, production systems) that exhibit undecidability properties. Here we seek such structures in the realm of Analysis, more specifically in the area of Fourier Analysis. The starting point is that sufficiently strongly convergent Fourier series give rise to predicates in the sense of first order predicate calculus by associating to any s-ary Fourier series the predicate “the Fourier coefficient with index (n1,…,ns) is non-zero”. We introduce production systems, viewed as counterparts of the combinatorial ones, that generate all recursively enumerable predicates in this way using as tools only elementary operations and functions from classical Analysis. The problem arises how simple such a system may be. It turns out that there is a connection between this question and an as yet unproved conjecture by R. Büchi. This is discussed in the second half of the paper.

Convergence of Fourier series with respect to general orthonormal systems

Convergence of Fourier series with respect to general orthonormal systems

Convergence of Fourier series on the systems of rational functions on the real axis

The systems of rational functions {Φn(z)}, n ∈ ℤ; that are orthonormalized on the real axis ℝ and are defined by the fixed set of points a := {ak}k = 0∞, (Im ak > 0) and b := {bk}k = 1∞, (Im bk 1; and the pointwise convergence of Fourier series on the systems {Φn(t)}, n ∈ ℤ; are studied under the certain restrictons on the sequences of poles of these systems. Some analogs of the classical Jordan–Dirichlet and Dini–Lipschitz criteria of convergence of Fourier series in a trigonometric system are constructed.

Convergence of Fourier Series with Respect to General Orthonormal Systems

Convergence of Fourier Series with Respect to General Orthonormal Systems