APPROXIMATIONS IN GENERALIZED MORREY SPACES
Morrey spaces were first introduced by C.B. Morrey in 1938 while investigating elliptic partial differential equations. Morrey spaces can be considered as a generalization of Lebesgue spaces. While every function in Lebesgue spaces can be approximated by smooth functions, this property does not a...
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| Format: | Dissertations |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/83375 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | Morrey spaces were first introduced by C.B. Morrey in 1938 while investigating
elliptic partial differential equations. Morrey spaces can be considered as a generalization
of Lebesgue spaces. While every function in Lebesgue spaces can be
approximated by smooth functions, this property does not apply in Morrey spaces.
C. T. Zorko in 1986 gave a function in Morrey spaces that smooth or continuous
functions could not approximate. A subset of a Morrey space with a specific
condition is defined so that each member can be approximated by compactly
supported smooth functions. The subset, later, is called Zorko space and can be
described as {f 2 Lp,!(Rn) : ||f(· + y) ? f(·)|| ! 0 as |y|!0}. The difference
of function of first order utilized in defining Zorko spaces is a suitable tool to
describe smoothness spaces. Many studies have been conducted to investigate the
connection between smoothness spaces and spaces defined using the differences of
function. For instance, Besov space as a smoothness space can be described using
the difference of function of the first order in the form of f(.+h)?f(.) and higher
order differences with sufficiently large smoothness exponents. The consequences
of employing higher-order differences of function in Zorko spaces to the approximation
property by smooth functions are now of interest to investigate further.
This dissertation discusses a set of all functions f in Morrey spaces and generalized
Morrey spaces with the property kf(· + h) + f(·?h) ? 2f(·)k ! 0 as |h| ! 0.
The new set is a modification of Zorko space using the second-order difference of
function. The dissertation presents research results concerning the relation between
the Zorko space and its modification, as well as approximation properties by smooth
functions in modified Zorko spaces. The inclusion property between Zorko spaces
and modified Zorko spaces is studied and some examples of functions related to the
spaces are provided. Each member in the modified Zorko spaces can be approximated
by functions in the form of convolution with compactly supported smooth
functions. By employing the inclusion relation between the Zorko space and the
diamond space, it turns out that the modified Zorko space is identical to the classical
one in Morrey spaces and generalized Morrey spaces. |
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