International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 10 Issue: 07 | July 2023
p-ISSN: 2395-0072
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ON CONVOLUTABLE FRECHET SPACES OF DISTRIBUTIONS NISHU GUPTA Associate Professor, Department of Mathematics, Maharaja Agrasen Institute of Technology, Delhi, India ---------------------------------------------------------------------***--------------------------------------------------------------------(2.1) M , f E f E , where M denotes the set
Abstract - In this paper, many results of Fourier analysis
which are known for convolutable Banach spaces of distributions (BCD-spaces) and Frechet spaces of distributions (FD-spaces) have been generalized to convolutable Frechet spaces of distributions (CFD-spaces). Also, we discuss the dual space of a CFD-space and obtain some useful results about homogeneous CFD-spaces.
of all (Radon) measures.
It is obvious that every BCD-space is a CFD-space (see the definition of BCD-space in [7]). But C is a CFD-space which is not a BCD-space as it is not a Banach space.
Key Words: Fourier analysis, Banach spaces, Frechet spaces, circle group, dual space and homogeneous space.
Throughout the paper, E, if not specified, will denote a CFD-space and E* will denote its strong* dual (see [8], Ch. 10).
1. INTRODUCTION
2.2. We now give an example of a non-empty Frechet space
In [7] some results of Fourier analysis, which are known for Lp (1 p ) , C and M etc., were obtained for convolutable Banach spaces of distributions (BCD-spaces). But those results cannot be applied to some important spaces like C (the space of all infinitely differentiable functions). Also, in [6] some results of Fourier analysis were obtained for Frechet spaces of distributions (FD-spaces). But those results cannot be applied to some important spaces like Hardy’s spaces H p (1 p ) . To overcome these deficiencies, in this paper we define the convolutable Frechet spaces of distributions (CFD-spaces).
E continuously embedded in D which satisfies the assumption (2.1) but not (2.2). Let E be the set of all f C such that f where E
f E
2.3. Now, we give an example of a non-empty Frechet space E continuously embedded in D, such that (2.2) is satisfied but not (2.1). Let M d (G) denote the set of purely discontinuous measures on G i.e., the set of all those measures μ on G for which there exists a countable subset A of G such that ( AC ) 0 . Then
2. DEFINITIONS, NOTATIONS EXAMPLES AND PRELIMINARY RESULTS
M d (G) is the required space (see [7]). The above examples show the independence of both the assumptions taken in the definition of a CFD-space.
We refer to [1], [5] and [8] for all the standard definitions, notations and assumptions. In particular, all our distributions are assumed to be defined on the circle group G R / 2 Z , and the space of all distributions is denoted by D.
2.4. Theorem. Let E be a CFD-space. Then the transformation S : M E E defined by
S ( , f ) f for each M and f E ,
2.1 Definition
is continuous on M E . Further, for each continuous seminorm p on E , there exists a continuous seminorm q on E such that p( f ) || || 1 q( f ) for each M and for
A Frechet space E is called a convolutable Frechet space of distributions, briefly a CFD-space, if it can be continuously embedded in (D, strong*), and if, regarded as a subset of D; it satisfies the following properties:
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|| D k f || ( k 1) k 0
Then E is the required space as every Banach space is a Frechet space (see [7]).
In Section 2, we define CFD-spaces and state some preliminary results dealing with CFD-spaces. In Section 3, we define homogeneous CFD-spaces and obtain some important results about homogeneous CFD-spaces. In Section 4, we discuss the dual space of a CFD-space and obtain some useful results.
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E is a closed subspace of C .
(2.2) C
each f E .
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