Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by b \in \BMO. We also prove some norm inequalities for $S_n$ when $\nu$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$

Normkonvergenz von Fourierreihen in rearrangement invarianten Banachräumen

AbstractThis paper studies rearrangement invariant Banach spaces of 2π-periodic functions with respect to norm convergence of Fourier series. The main result is that norm convergence takes place if and only if the space is an interpolation space of (Lp′(T), Lp(T)), 1 < p < 2, 1p′ + 1p = 1, and Lp′(T) is dense in it (compare Satz 2.8). Since norm convergence and continuity of the conjugation operator are closely connected (compare Satz 2.2), this is achieved by a careful examination of this operator similar to that of D. W. Boyd for the Hilbert transform on the whole real axis. Finally, there are applications to Orlicz and Lorentz spaces

The 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 \emph{largest} Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{1,\infty}$. In particular, we disprove a conjecture stated by Konyagin in his 2006 ICM address. 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.Comment: 82 pages, no figures. We have added the following items: 1) Section 5 presenting a suggestive example; 2) Section 6 explaining the fundamental role of the so called grand maximal counting function; 3) Section 12 presenting a careful analysis of the Lacey-Thiele discretized Carleson model and of the Walsh-Carleson operator. Accepted for publication in J. Eur. Math. Soc. (JEMS

Convergence of the FourierLaplace series in the spaces with mixed norm

Solution of some boundary value problems and initial problems in unique ball leads to the convergence and summability problems of Fourier series of given function by eigenfunctions of Laplace operator on a sphere - spherical harmonics. Such a series are called as Fourier-Laplace series on sphere. There are a number of works devoted investigation of these expansions in different topologies and for the functions from the various functional spaces. In this paper we study convergence and summability problems of the Fourier Laplace series on the unique sphere in the spaces with the mixed norm

Quantum gate synthesis by small perturbation of a free particle in a box with electric field

A quantum unitary gate is realized in this paper by perturbing a free charged particle in a one-dimensional box with a time- and position-varying electric field. The perturbed Hamiltonian is composed of a free particle Hamiltonian plus a perturbing electric potential such that the Schr$\ddot{o}$dinger evolution in time $T$, the unitary evolution operator of the unperturbed system after truncation to a finite number of energy levels, approximates a given unitary gate such as the quantum Fourier transform gate. The idea is to truncate the half-wave Fourier sine series to $M$ terms in the spatial variable $\mathbf x$ before extending the potential as a Dyson series in the interaction picture to compute the evolution operator matrix elements up to the linear and quadratic integral functionals of $\mathbf V_n(t)'$s. As a result, we used the Dyson series with the Frobenius norm to reduce the difference between the derived gate energy and the given gate energy, and we determined the temporal performance criterion by plotting the noise-to-signal energy ratio (NSER). A mathematical explanation for a quantum gate's magnetic control has also been provided. In addition, we provide a mathematical explanation for a quantum gate that uses magnetic control.Comment: 16 page

Maximal estimates for the (C, alpha) means of d-dimensional Walsh-Fourier series

The d-dimensional dyadic martingale Hardy spaces H(p) are introduced and it is proved that the maximal operator of the (C, alpha) (alpha = (alpha(1),..., alpha(d))) means of a Walsh-Fourier series is bounded from H(p) to L(p) (1/(alpha(k) + 1) < p < infinity) and is of weak type(L(1), L(1)), provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the (C, alpha) means of a function f is an element of L(1) converge a.e. to the function in question. Moreover, we prove that the (C; alpha) means are uniformly bounded on H(p) whenever 1/(alpha(k) + 1)< p < infinity. Thus, in case f is an element of H(p), the (C; alpha) means converge to f in H(p) norm. The same results are proved for the conjugate (C; alpha) means, too

On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update

Frequency-Domain Analysis of Linear Time-Periodic Systems

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature