Second-order logarithmic methods for the summability of Fourier series

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

We prove the pointwise convergence of the Fourier series for radial functions in several variables, which in the case n = 1 n=1 is the Dirichlet-Jordan theorem itself. In our proof the method for the case of the indicator function of the ball is very useful.

The pointwise convergence of Fourier Series (II). Strong L1 case for the lacunary Carleson operator

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.

Denoting by S ∗ {S^\ast } the maximal partial sum operator of Fourier series, we prove that S ∗ ( f 1 , f 2 , … , f k , … ) = ( S ∗ f 1 , S ∗ f 2 , … , S ∗ f k , … ) {S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots ) is a bounded operator from L p ( l r ) {L^p}({l^r}) to itself, 1 > p , r > ∞ 1 > p,r > \infty . Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.

We offer a new approach to convergence of Fourier series on the unit sphere of the four-dimensional Euclidean space. The approach is via the quaternionic analysis setting with a crucial application of Fueter's theorem. Analogs to the Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.

In Chapter 29 we studied pointwise convergence of Fourier series. When solving a differential equation by Fourier methods we want the solution to be a super-position of trigonometric functions at all points of a given range. Thus, tacitly, pointwise convergence is the natural choice in this situation.

For a Schwartz function $f$ on the plane and a non-zero $v\in \mathbb {R}^2$ define the Hilbert transform of $f$ in the direction $v$ to be \begin{equation*} \operatorname H_vf(x)=\text {p.v.}\int _{\mathbb {R}} f(x-vy)\; \frac {dy}y. \end{equation*} Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le |\xi |\le 2$, and ${\boldsymbol \zeta }f=\zeta *f$. We prove that the maximal operator $\sup _{|v|=1}|\operatorname H_v{\boldsymbol \zeta } f|$ maps $L^2$ into weak $L^2$, and $L^p$ into $L^p$ for $p>2$. The $L^2$ estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.

In this chapter we shall deal rather summarily with some positive and negative results about the pointwise convergence of Fourier series. The reasons for not according this topic a fuller treatment are discussed in Chapter 1. The reader who is particularly attracted by these aspects may consult [Z], especially Chapters II, VIII, and XIII; [Ba], especially Chapters I, III–V, VII, IX; [HaR], especially Chapter IV; [I], pp. 23 ff., pp. 103 ff.; [A]; and the work of Carleson mentioned in 10.4.5.

We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$. Our approach works in the general context of maximally modulated Calder\'on-Zygmund operators.

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.

If F \mathcal {F} is a foliation of an open set Ω ⊂ R n \Omega \subset \mathbb {R}^n by smooth ( n − 1 ) (n-1) -dimensional surfaces, we define a class of functions B ( Ω , F ) \mathcal {B}(\Omega ,\mathcal {F}) , supported in Ω \Omega , that are, roughly speaking, smooth along F \mathcal {F} and of bounded variation transverse to F \mathcal {F} . We investigate geometrical conditions on F \mathcal {F} that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.

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.