International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 07 | July 2024
p-ISSN: 2395-0072
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A RELATIONSHIP BETWEEN THE CONVOLUTION AND TRANSLATION OPERATORS FOR CONVOLUTABLE FRECHET SPACES OF DISTRIBUTIONS NISHU GUPTA Associate Professor, Department of Mathematics, Maharaja Agrasen Institute of Technology, Delhi, India ---------------------------------------------------------------------***--------------------------------------------------------------------2.2. Definition
Abstract - In this paper, a relationship between the
convolution and translation operators is established for Convolutable Frechet Spaces of Distributions (CFD-spaces).
implies Tx f Tx f in E for each f E .
Key Words: Fourier analysis, Banach spaces, Frechet spaces, Locally convex spaces, circle group and homogeneous space.
0
3. SOME USEFUL RESULTS
1. INTRODUCTION
Using Theorems 3.25 and 3.27 of [4] we can state the following.
If E is L (1 p ) or C , f E and g L ,then it is 1
p
x x0 in G
A CFD-space E is said to be homogenous if
shown in [1, Vol. I, 3.19] that g f is the limit in E of finite
3.1. Theorem. Let X be a Frechet space and be a
linear combinations of translates of f . Also, a relationship between the convolution and translation operators is established in [2] and [5] for Convolutable Banach Spaces of Distributions and for Frechet Spaces of Distributions (FDspaces). In this paper, we extend that result to homogeneous Convolutable Frechet Spaces of Distributions (CFD-spaces).
complex Borel measure on a compact Hausdorff space Q. If
f : Q X is continuous, then the integral f d Q
exists (and lies in X) such that
F
2. DEFINITIONS AND NOTATIONS
f d F( f )d F X Q
Q
.
All the notations and conventions used in [3] and [6] will be continued in this paper. In particular, G R / 2 Z will denote the circle group and D will denote the space of all distributions on G. For the convenience of the reader, we repeat the following definitions given in [3].
3.2. Theorem. If p is a continuous seminorm on a Frechet
2.1 Definition
F ( y ) p( y ) and | F ( x ) | p( x ) for all x X
space X and y
exists F X such that
(See Theorem 3.3 of [4]). In particular,
(2.1) M , f E f E , where M denotes the set of all (Radon) measures.
p
Q
Q
The above two results will be useful to find the relationship between the convolution and translation operators for homogeneous CFD-spaces.
space and E will denote its strong* dual (see [7], Ch. 10).
Impact Factor value: 8.226
Q
Q
Throughout the paper E , if not specified, will denote a CFD-
|
f d F f d F ( f ) d p( f ) d | |
E is a closed subspace of C .
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Q
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:
(2.2) C
f d is defined as above, then there
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