Abstract
We showed that for each function $f(t)$, which is not equal to zero
almost everywhere in the Lebesgue measurable set, functionals
$I\left(f,z\right)=\int_A{{|f(t)|}^z dt}$ as functions of a
complex variable $z=p+iy$ are continuous on the domain and
analytic on a set of all inner points of this domain. The functions
$I(f,p)$ as functions of a real variable $ p $ are strictly convex
downward and log-convex on the domain.
We proved that functionals $n(f,p)$ as functions of a real variable
$p$ are analytic at all inner points of the interval, in which the
function $n(f,p)\neq 0$ except the point $p=0$,
continuous and strictly increasing on this interval.
Key words: Functional, approximation, analyticity, monotonicity, continuity.
Full Text
Article Information
| Title | Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p
dt\right)^{\frac{1}{p}}$ as functions of variable $p$ |
| Source | Methods Funct. Anal. Topology, Vol. 25 (2019), no. 4, 339-359 |
| MathSciNet |
MR4049689 |
| Milestones | Received 17/11/2018; Revised 07/10/2019 |
| Copyright | The Author(s) 2019 (CC BY-SA) |
Authors Information
D. M. Bushev
Lesya Ukrainka Eastern European National University, 13 Volya Avenue, 43025, Lutsk, Ukraine
I. V. Kal'chuk
Lesya Ukrainka Eastern European National University, 13 Volya Avenue, 43025, Lutsk, Ukraine
Citation Example
D. M. Bushev and I. V. Kal'chuk, Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p
dt\right)^{\frac{1}{p}}$ as functions of variable $p$, Methods Funct. Anal. Topology 25
(2019), no. 4, 339-359.
BibTex
@article {MFAT1242,
AUTHOR = {D. M. Bushev and I. V. Kal'chuk},
TITLE = {Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p
dt\right)^{\frac{1}{p}}$ as functions of variable $p$},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {4},
PAGES = {339-359},
ISSN = {1029-3531},
MRNUMBER = {MR4049689},
URL = {http://mfat.imath.kiev.ua/article/?id=1242},
}
