Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences

In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, the authors consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.

LEBESGUE’S CRITERION FOR INTEGRABILITY

Lebesgue integral was defined constructively by using outer measure, measurable set, measurable function and Lebesgue measure concept. Then it was constructed by using simple function. The requirement of a Lebesgue integrable function to be Riemann integrable is bounded function and the set of discontinuities of that function has measure zero.

Ostrowski and Trapezoid Type Inequalities for the Generalized k-g-Fractional Integrals of Functions with Bounded Variation

Let g be a strictly increasing function on a , b , having a continuous derivative g′ on a , b . For the Lebesgue integrable function f : a , b → C , we define the k-g-left-sided fractional integral of f by S k , g , a + f x = ∫ a x k g x - g t g ′ t f t d t , x ∈ a , b and the k-g-right-sided fractional integral of f by S k , g , b - f x = ∫ x b k g t - g x g ′ t f t d t , x ∈ [ a , b ) , where the kernel k is defined either on 0 , ∞ or on 0 , ∞ with complex values and integrable on any finite subinterval. In this paper we establish some Ostrowski and trapezoid type inequalities for the k-g-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

On the Nørlund Summability of a Class of Fourier Series

1. Our aim in this paper is to determine a necessary and sufficient condition for N∅rlund summability of Fourier series and to include a wider class of classical results. A Fourier series, of a Lebesgue-integrable function, is said to be summable at a point by N∅rlund method (N, pn), as defined by Hardy [1], if pn → Σpn → ∞, and the point is in a certain subset of the Lebesgue set. The following main results are known.

On the Nörlund summability of Fourier series

1. The purpose of this note is to prove a result which includes certain classical theorems generally thought of as being unconnected; in explicit terms, a result about the Fourier series of a periodic Lebesgue-integrable function showing that the series is summable at a point by a Nörlund method (N, pn) defined as usual ((2), p. 64) if pn ↓ 0, Σpn = ∞ and the point is in a certain subset of the Lebesgue set. More precisely, the purpose is to prove Theorem I on the Nörlund summability of Fourier series and to derive from it the well-known Theorems A, B which follow and the recent extension of Theorem A in Theorem A' which appears later and is due to Sahney (8).