Set Of Entire Functions Research Articles

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

Plancherel–Pólya Inequality for Entire Functions of Exponential Type in L2(ℝ)

We investigate the Plancherel–Polya inequality $$\sum {_{k \in \mathbb{Z}}} |f(k){|^2} \leqslant {c_2}\left( \sigma \right)||f||_{{L^2}\left( \mathbb{R} \right)}^2$$ on the set of entire functions f of exponential type at most σ whose restrictions to the real line belong to the space L2(ℝ). We prove that c2(σ) = [σ/π] for σ > 0 and describe the extremal functions.

Feature Sub Selection over High Dimensional Data based on Classification Models

Background: The main objective of this paper achieves feature selection in cluster categorical data sets. Although efforts have been made to fix the problem of clustering particular details via group outfits, with the results being competitive to traditional methods, it is noticed that these techniques unfortunately generate a final details partition based on imperfect details. The actual ensemble-information matrix provides only cluster-data point interaction, with many entries being left unknown. Feature choice includes determining a part of the most useful functions which makes suitable outcomes as the original entire set of functions. A function choice requirements may be analyzed from both the performance and performance opinions. Methodology: While the performance concerns the time required to find a part of functions, the performance is associated with the quality of the part of functions. Centered on these requirements, Fast clustering-based function selection algorithm (FAST) is suggested and experimentally analyzed in this paper. Findings: The FAST requirements works in two steps. In the starting point, functions are separated into groups by using graph-theoretic clustering methods. In the second phase, the most associate function that is highly relevant to target classes is selected from each group to form a part of functions. Improvement: The performance and performance of the FAST requirements are analyzed through a scientific study. The results, on 35 freely available real-world high-dimensional picture, small range, and written text information, show the FAST not only produces more compact subsets of features but also increases the activities of the four types of classifiers.

Lebesgue-type inequalities for the de la Vallée-poussin sums on sets of entire functions

For functions from the sets C ψ β L s , 1 ≤ s ≤ ∞, where ψ(k) > 0 and $ {\lim_{{k\to \infty }}}\frac{{\psi \left( {k+1} \right)}}{{\psi (k)}} $ , we obtain asymptotically sharp estimates for the norms of deviations of the de la Vallee-Poussin sums in the uniform metric represented in terms of the best approximations of the (ψ, β) -derivatives of functions of this kind by trigonometric polynomials in the metrics of the spaces L s . It is shown that the obtained estimates are sharp on some important functional subsets.

Classification of infinitely differentiable periodic functions

The set $ \mathcal{D}^\infty $ of infinitely differentiable periodic functions is studied in terms of generalized $ \overline \psi $ -derivatives defined by a pair $ \overline \psi = (\psi_1, \psi_2)$ of sequences ψ 1 and ψ 2. In particular, we establish that every function f from the set $ \mathcal{D}^\infty $ has at least one derivative whose parameters ψ 1 and ψ 2 decrease faster than any power function. At the same time, for an arbitrary function f ∈ $ \mathcal{D}^\infty $ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ 1 and ψ 2 have the same rate of decrease and for which the $ \overline \psi $ -derivative no longer exists. We also obtain new criteria for 2π-periodic functions real-valued on the real axis to belong to the set of functions analytic on the axis and to the set of entire functions.

Composite entire functions with no unbounded Fatou components

Let L be the set of all entire functions f such that for given ϵ > 0 , log L ( r , f ) > ( 1 − ϵ ) log M ( r , f ) for all r outside a set of logarithmic density zero. Let F = ⋃ K ⩾ 1 F K where F K is the set of all transcendental entire functions f such that log log M ( r , f ) ⩾ ( log r ) 1 K . If h = f N ○ f N − 1 ○ ⋯ ○ f 1 where f i ∈ F ∩ L ( i = 1 , … , N ) , then it is shown that h has no unbounded Fatou component.

URS' For Weierstrass Products without Exponential Factors

Let $ \\cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\\lim_{r \ o \\infty} \\prod_{|a_j|\\leq r}{(1- \\fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\\shadC} $ , let $ \\cal V $ be the set of $ f\\in {\\cal W} $ such that $ {\\shadC} [\\,f]\\subset {\\cal W} $ , and let $ {\\cal H} $ be the $ {\\shadC} $ -algebra of entire functions satisfying $ { {\\lim_{r\ o \\infty } } ({\\ln M(r,f) / r})=0} $ . Then $ \\cal H $ is included in $ {\\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\\in {\\shadN} ^* $ satisfy n > m S 3. Let $ a\\in {\\shadC}^* $ satisfies $ {a^n\ ot = \\fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\ eq (1+\\xi )^{n-m} (\\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\\cal V} $ . In particular this applies to functions such as sin x and cos x .

Natural examples of Π₅⁰-complete sets in analysis

The purpose of this paper is to show that given any non-negative real number α \alpha , the set of entire functions whose order is equal to α \alpha is Π 3 0 \boldsymbol {\Pi }_{3}^{0} -complete, and the set of all sequences of entire functions whose orders converge to α \alpha is Π 5 0 \boldsymbol {\Pi }_{5}^{0} -complete.

Completely regular growth on a prescribed set of rays and the limit set of entire functions

Some theorems are proved that generalize the well-known Valiron-Titchmarsh theorem of entire functions.

On reducibility of zero sets of entire functions of several variables

On reducibility of zero sets of entire functions of several variables

Spherical derivative and Picard sets of entire functions

Spherical derivative and Picard sets of entire functions

Further Results on Picard Sets of Entire Functions

Further Results on Picard Sets of Entire Functions

Budan’s theorem for a class of entire functions

It is easy to see that the even qualification can be omitted if all the zeros of f(z) are real, so that in this case we get exact information on every interval. In this note we describe a class of entire functions for which the natural generalization of Budan's rule likewise gives exact information about the number of zeros in a real interval. This class K is the set of entire functions f(z) which are real for real z, have negative real zeros and positive Taylor coefficients, and are of finite exponential type. Such functions can be written in any of the forms

Correction: Fundamental Sets of Entire Functions

Correction: Fundamental Sets of Entire Functions

Fundamental Sets of Entire Functions

Fundamental Sets of Entire Functions