Given a Lebesgue integrable function f over [0, 2π], we consider the sequence of matrices {Y n T n [f ]} n , where T n [f ] is the n-by-n Toeplitz matrix generated by f and Y n is the flip permutation matrix, also called the anti-identity matrix. Because of the unitary character of Y n , the singular values of T n [f ] and Y n T n [f ] coincide. However, the eigenvalues are affected substantially by the action of the matrix Y n . Under the assumption that the Fourier coefficients are real, we prove that {Y n T n [f ]} n is distributed in the eigenvalue sense aswith g(θ) = |f (θ)|. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as φ 1 , under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.
In this article, we introduce Stancu type generalization of Baskakov–Durrmeyer operators by using inverse Pólya–Eggenberger distribution. We discuss some basic results and approximation properties. Moreover, we study the statistical convergence for these operators.
The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szegő theorem and the Avram-Parter theorem, in which the singular value symbol coincides with the generating function. More general versions of the theorem were later proved by Zamarashkin and Tyrtyshnikov, and Tilli. Considering (real) nonsymmetric Toeplitz matrix sequences, we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically, by using the notion of approximating class of sequences. In particular, under the assumption that the symbol is sparsely vanishing, we show that roughly half of the eigenvalues of the symmetrized Toeplitz matrix (i.e. a Hankel matrix) are negative/positive for sufficiently large dimension, i.e. the matrix sequence is * Corresponding author Email addresses: hon@maths.ox.ac.uk (Sean Hon), mamursaleen@uninsubria.it (Mohammad Ayman Mursaleen), stefano.serrac@uninsubria.it (Stefano Serra-Capizzano)
Our main purpose of this article is to study the convergence and other
related properties of q-Bernstein-Kantorovich operators including the
shifted knots of real positive numbers. We design the shifted knots of
Bernstein-Kantorovich operators generated by the basic q-calculus. More
precisely, we study the convergence properties of our new operators in the
space of continuous functions and Lebesgue space. We obtain the degree of
convergence with the help of modulus of continuity and integral modulus of
continuity. Furthermore, we establish the quantitative estimates of
Voronovskaja-type.
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