A Generalization of Abel and Dirichlet Criteria
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A GENERALIZATION OF ABEL AND DIRICHLET CRITERIA V. Yu. Slyusarchuk
UDC 517.382 + 517.52
We obtain vector analogs of the Abel and Dirichlet criteria.
1. Abel and Dirichlet Criteria The following statements play an important role in mathematical analysis: Theorem 1 (Abel criterion). Suppose that: (i) a function f : [a, +1) ! R is integrable on [a, +1) and (ii) a function g : [a, +1) ! R is monotone and bounded. Then the improper integral +1 Z f (t)g(t) dt
(1)
a
is convergent. Theorem 2 (Dirichlet criterion). ments [a, b], b > a, and
Suppose that: (i) a function f : [a, +1) ! R is integrable on seg� b � �Z � � � � sup � f (t) dt�� < +1; b>a � � a
(ii) a function g : [a, +1) ! R monotonically approaches 0 as t ! +1. Then the improper integral (1) converges. X1 Theorem 3 (Abel criterion). Suppose that: (i) a numerical series an is convergent; (ii) a numerical n=1 sequence (bn )n≥1 is monotone and bounded. Then the numerical series 1 X
a n bn
(2)
n=1
is convergent.
X1 Theorem 4 (Dirichlet criterion). Suppose that: (i) partial sums of the numerical series an are collecn=1 tively bounded; (ii) a numerical sequence (bn )n≥1 is monotone and limn!1 bn = 0. Then the numerical series (2) is convergent. In the first two criteria and in what follows, a is an arbitrary element of the set of real numbers R. For the substantiation of the presented criteria, see, e.g., [1]. National University of Water Management and Utilization of Natural Resources, Rivne, Ukraine; e-mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 527–539, April, 2020. Original article submitted November 22, 2019. 0041-5995/20/7204–0607
© 2020
Springer Science+Business Media, LLC
607
V. Y U. S LYUSARCHUK
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2. Main Object of Investigations The aim of the present paper is to establish general statements on the convergence of improper vector integrals and series whose special cases are the Abel and Dirichlet criteria. In these statements, we consider functions with values in Banach spaces that are not ordered spaces. This does not enable us to use the condition of monotonicity of the functions, which is essential for the Abel and Dirichlet criteria. Instead of these functions, we use functions of bounded variation, which extends the domain of applicability of assertions similar to the theorems presented in the previous section. 3. Main Notation, Definitions, and Auxiliary Results Let C be the set of all complex numbers, let X, Y, and Z be Banach spaces over the field of real or complex numbers with the norms k · kX , k · kY , and k · kZ , respectively, and let L(X, Y ) be a Banach space of linear continuous operators A : X ! Y with the norm kAkL(X,Y ) = sup kAxkY . kxkX =1
An ordered pair (X, Y ) of Banach spaces X and Y is called proper if, for any elements x 2 X and y 2 Y, the rule of action of the vector x upon the vector y (the product of vectors denoted by xy ) is defined, and every product xy is an element of a Banach space L that depends on X and Y and the operation of multiplication
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