scholarly journals Approximation by Boolean sums of linear operators: Telyakovskiǐ-type estimates

1990 ◽  
Vol 42 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Upper Bound ◽  
Algebraic Polynomial ◽  
Present Note ◽  
Type Inequality ◽  
Linear Operators ◽  
Polynomial Operators ◽  
Boolean Sums ◽  
General Conditions

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

scholarly journals New Telyakovskii-type estimates via the Boolean sum approach

1996 ◽  
Vol 54 (1) ◽  
pp. 131-146
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Present Note ◽  
Linear Operators ◽  
Convolution Type ◽  
The One ◽  
Boolean Sums ◽  
Boolean Sum

In the present note the magnitude of constants in Telyakovskii-type theorems is investigated. Our general approach to construct the linear operators yielding good constants is the one via Boolean sums. Explicit values for the constants in question are given for general convolution-type operators; the classical Fejér-Korovkin kernel is then used as an example for which one obtains rather small values. Furthermore, also an asymptotic assertion is derived which indicates the room left for improvement of the main results. This leads to a natural conjecture concluding this article.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

Complexity bounds on proofs

1981 ◽  
Vol 46 (2) ◽  
pp. 255-258 ◽  
Author(s):  
William S. Hatcher ◽  
Bernard R. Hodgson
Keyword(s):  
Upper Bound ◽  
Recent Article ◽  
Present Note ◽  
Strong Sense ◽  
Complexity Bounds ◽  
Mathematical Proofs ◽  
Close Analysis ◽  
Full Force

In a recent article in this Journal (see [3]), J.P. Jones states and proves a theorem which purports to give an “absolute epistemological upper bound on the complexity of mathematical proofs” for recursively axiomatizable theories. However, Jones' statement of this result is misleading, and in fact defective, as can be seen by a close analysis of it. Such an analysis is the object of the present note.The main point is that Jones' “epistemological bound” can in no way be considered a computational bound on the complexity of proofs. Not only is the “proof-theoretic interpretation of the number 243” contained in Jones' article objectionable but, more fundamentally, there is in a strong sense no way one can hope to recover anything like the full force suggested by Jones' original statement of the theorem.We wish to insist that our comments concern only the difficulties surrounding Jones' Corollary 1 on p. 338 of his article and not his ingenious construction of universal Diophantine representations of r.e. sets presented in the same article.

scholarly journals INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

2020 ◽  
Vol 6 (2) ◽  
pp. 87
Author(s):  
Tatiana M. Nikiforova
Keyword(s):  
Approximation Theory ◽  
Upper Bound ◽  
Algebraic Polynomial ◽  
Uniform Norm ◽  
Algebraic Polynomials

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

scholarly journals Asymptotically quasi-compact products of bounded linear operators

1973 ◽  
Vol 16 (3) ◽  
pp. 286-289 ◽  
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Fredholm Determinant ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

It is known (see, for instance, [1] p. 64, [6] p. 264) that, if A and B are bounded linear operators on a Banach space into itself (or, more generally, if A is a bounded linear operator on into a Banach space and B is a bounded linear operator on into), then AB and BA have the same spectrum except (possibly) for zero. In the present note, it is shown that AB is asymptotically quasi-compact if and only if BA is asymptotically quasi-compact, and that then any Fredholm determinant for AB is a Fredholm determinant for BA and vice versa.

scholarly journals A NOTE ON HILBERT TYPE INEQUALITY

1998 ◽  
Vol 29 (4) ◽  
pp. 293-298
Author(s):  
B. G. PACHPATTE
Keyword(s):  
Present Note ◽  
Type Inequality ◽  
Real Numbers ◽  
Integral Analogue ◽  
Hilbert Type

In the present note we establish a new Hilbert type inequality mvolving sequences of real numbers. An integral analogue of the main result is also given.

scholarly journals Lagrange-type operators associated with Uan

2014 ◽  
Vol 96 (110) ◽  
pp. 159-168 ◽  
Author(s):  
Heiner Gonska ◽  
Ioan Raşa ◽  
Elena-Dorina Stănilă
Keyword(s):  
Main Tool ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operators ◽  
Boolean Sums ◽  
Derivatives Of

We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein-Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the derivatives of the operator images are included. Our main tool is the eigenstructure of the members of the class.

scholarly journals Ordinal Pattern Based Entropies and the Kolmogorov–Sinai Entropy: An Update

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 63
Author(s):  
Tim Gutjahr ◽  
Karsten Keller
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Review Paper ◽  
Permutation Entropy ◽  
Theoretical Level ◽  
The Relationship ◽  
Ordinal Pattern ◽  
General Conditions ◽  
Strong Relationships

Different authors have shown strong relationships between ordinal pattern based entropies and the Kolmogorov–Sinai entropy, including equality of the latter one and the permutation entropy, the whole picture is however far from being complete. This paper is updating the picture by summarizing some results and discussing some mainly combinatorial aspects behind the dependence of Kolmogorov–Sinai entropy from ordinal pattern distributions on a theoretical level. The paper is more than a review paper. A new statement concerning the conditional permutation entropy will be given as well as a new proof for the fact that the permutation entropy is an upper bound for the Kolmogorov–Sinai entropy. As a main result, general conditions for the permutation entropy being a lower bound for the Kolmogorov–Sinai entropy will be stated. Additionally, a previously introduced method to analyze the relationship between permutation and Kolmogorov–Sinai entropies as well as its limitations will be investigated.

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