Some Nuclear Hankel Operators

On Hankel operators associated with a class of non-Toeplitz operators

This is a survey (without proofs except for verifying a few new facts) of the slowest possible rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. A sequence of linear operators (L n ) is said to converge to a linear operator Larbitrarily slowly (resp., almost arbitrarily slowly) provided that (L n ) converges to L pointwise, and for each sequence of real numbers (ϕ(n)) converging to 0, there exists a point x = x ϕ such that \(\|{L}_{n}(x) - L(x)\| \geq \phi (n)\) for all n (resp., for infinitely many n). Two main “lethargy” theorems are prominent in this study, and they have numerous applications. The first lethargy theorem (Theorem 11.16) characterizes almost arbitrarily slow convergence. Applications of this lethargy theorem include the fact that a large class of polynomial operators (e.g., Bernstein, Hermite–Fejer, Landau, Fejer, and Jackson operators) all converge almost arbitrarily slowly to the identity operator. Also all the classical quadrature rules (e.g., the composite Trapezoidal Rule, composite Simpson’s Rule, and Gaussian quadrature) converge almost arbitrarily slowly to the integration functional. The second lethargy theorem (Theorem 11.21) gives useful sufficient conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, there is a “dichotomy” theorem (Theorem 11.27) which states that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. Some applications of the dichotomy theorem include generalizations and sharpening of (1) the von Neumann-Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

1. Main theorem. Let x = (xo, xi, • • • ) be a sequence of complex numbers and let | x he the norm | x | = (^Z | x„ |2)1/2 2:0. Let an asterisk denote complex conjugation. If \x < <» and |y| < oo, let [x, y] = [y, x] denote the sum of the series ^T,xnyn, that is, the scalar product of y by x*. For a given x with a finite norm |x|, let x(t) denote a function of class i2(0, 27r) having the Fourier series

Dirichlet-Taylor series are considered, where is a sequence of complex numbers, is a nondecreasing sequence of positive numbers, and is the number of times ?j occurs in the segment {?1,?,?j}. An adherence principle is established for these series. As applications of this principle, uniqueness theorems are proved for analytic functions which are asymptotically representable by partial sums of Dirichlet-Taylor series in strips. Bibliography: 16 items.

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Intermediate Hankel operators on the Fock space

Transceivers used for telecommunications transmit and receive specific modulation patterns that are represented as sequences of complex numbers. Classifying modulation patterns is challenging because noise and channel impairments affect the signals in complicated ways such that the received signal bears little resemblance to the transmitted signal. Although deep learning approaches have shown great promise over statistical methods in this problem space, deep learning frameworks continue to lag in support for complex-valued data. To address this gap, we study the implementation and use of complex convolutions in a series of convolutional neural network architectures. Replacement of data structure and convolution operations by their complex generalization in an architecture improves performance, with statistical significance, at recognizing modulation patterns in complex-valued signals with high SNR after being trained on low SNR signals. This suggests complex-valued convolutions enables networks to learn more meaningful representations. We investigate this hypothesis by comparing the features learned in each experiment by visualizing the inputs that results in one-hot modulation pattern classification for each network.

In this work, we consider reproducing kernel Hilbert spaces which have orthonormal bases of the form $f_n(z)=z^n \prod_{j=1}^J \left( 1 - a_{n}w_j z \right)$, where $w_1 ,w_2, \ldots w_J $ are distinct points on the circle $\mathbb{T}$ and $\{ a_n \}$ is a sequence of complex numbers with limit $1$. We provide general conditions based on a matrix recursion that guarantee such spaces contain a functional multiple of the Hardy space. Then we apply this general method to obtain strong results for finite bandwidth spaces when $\lim_{n\rightarrow \infty} n (1-a_n)=p$. In particular, we show that point evaluation can be extended boundedly to precisely $J$ additional points on $\mathbb{T}$ and we obtain an explicit functional decomposition of these spaces for $p>1/2$ in analogy with a previous result in the tridiagonal case due to Adams and McGuire. We also prove that multiplication by $z$ is a bounded operator on these spaces and that they contain the polynomials.

For an entire function f(z), a sequence of complex numbers {λn }, and an unbounded domain Ω, the completeness of the system {f(λ nz)} in (mean square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm

The main result of this paper is that if a sequence of complex numbers (an)n≥0 satisfies\(\sum\limits_{\mathop {k = 0}\limits_{k even} }^n {\left({_k^n } \right)a_k = O(n^r)} \)and % MathType!MTEF!2!1!+-\(\sum\limits_{\mathop {k = 0}\limits_{k odd} }^n {\left({_k^n } \right)a_k = O(n^r )} as n \to \infty,\) for some integerr≥0, thenan=0 for alln>r. As an application, we deduce a localized form of a theorem of Allan about nilpotent elements in Banach algebras, and this in turn leads to an invariant-subspace theorem. As a further application, we prove a variant of Carleman's theorem on the unique determination of probability distributions by their moments. The paper concludes with a quantitative form of the main result.

Let {z n } be a sequence of complex numbers. A sequence {z n } is bounded if there exists M >0 such that |z n | < M for all n ∈ ℕ.

Convergence of linear combinations of iterates of an inner function

The following result is proved. LetΛ=λn} be a sequence of complex numbers with ¦Reλ n¦≧δ¦λ n ¦, δ>0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef≠0 such thatf(λ)=0 and ¦f(iy)¦≦¦g(iy)¦,y∈R, if and only if there exists a constantM such that for all numbersr andR, 0<r<R<+∞, we have $$\max \left\{ {\mathop \sum \limits_{\mathop {r \leqq |\lambda _n | 0} } \operatorname{Re} \frac{1}{{\lambda _n }}} \right\} \leqq \frac{1}{{2\pi }}\mathop \smallint \limits_r^R \frac{{\ln |g(iy)g( - iy)|}}{{y^2 }}dy + M.$$

Let {zn} be a sequence of complex numbers with zn → ∞ (n → ∞). We consider entire functions φ which behave universal under translates (for short: T-universal functions), i. e. for every compact set K with connected complement and every function f, which is continuous on K and holomorphic in the interior of K, there exists a subsequence {nk} of natural numbers, such that