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A Characterization of Compactness in Banach Spaces with Continuous Linear Abstract Representations of the Rotation Group of a Circle. Abdullah Çavu¸s and Mehmet Kunt cavus@ktu.edu.tr, mkunt@ktu.edu.tr Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey Let H be a complex Banach space, T be the unit circle {z ∈ C : |z| = 1}, SO(2) be the group of all rotations of T, GL(H) be group of all invertible bounded linear operators on H, α : SO(2) → GL(H) be a continuous linear representation, x ∈ H. For all n ∈ Z, n-th Fourier coefficient of x with respect to the α is defined by Pn(x) = 1 � e 2π T −int α(t)(x)dt and the Fourier series of x with respect to the α is defined by +∞� n=−∞ Pn(x). (1) The convergence of this series and some properties of Pn(x) are investigated in [5]. In this work, a characterization of compactness in Banach space H is given by means of Fourier coefficients Pn(x). One of the main results is as follows: Theorem :Suppose that dimHn < +∞ for all n ∈ Z. Then a closed subset A ⊂ H is compact if and only if for any ε > 0 there exists a natural number N(ε) such that� n n+1 σn(x) − x� < ε for all x ∈ A and n ≥ N(ε). Where, for all n ∈ N ∪ {0}, σn(.) : H → H is a linear bounded operator which is defined by σn(x) = 1 n + 1 n� Sk(x) for all x ∈ H, Sk(x) is the k-th partial sum of (1) for all k ∈ N ∪ {0} and for all n ∈ Z. References k=0 Hn := {x ∈ H : α(t)(x) = e int x, ∀t ∈ T} [1] Edwards R. E., Fourier Series : A Modern Introduction, Springer-Verlag, Berlin/Heydelberg/New York, 1982. [2] Kislyakov S. V., Classical themes of Fourier analysis, Commutative harmonic analysis I, General survey, Classical aspects, Encycl. Math. Sci., 15, 113-165 1991. [3] Schechter M., Principles of Functional Analysis, Graduate Studies in Mathematics, vol. 36, Prov- idence, R. I. American Mathematical Society, (AMS), 2001. [4] Khadjiev Dj., Çavu¸s A., The imbedding theorem for continuous linear representation of the rotation group of a circle in Banach spaces, Dokl. Acad. Nauk of Uzbekistan, N 7, 8-11, 2000. Page 28
[5] Khadjiev Dj., Çavu¸s A., Fourier series in Banach spaces, Inverse and Ill-Posed Problems Series, Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis, Proceedings of the Inter- national Conference, Samarcand, Uzbekistan, Editor-in-Chief : M. M. Lavrent’ev, VSP, Utrecht-Boston, 71-80, 2003. [6] Khadjiev Dj., The widest continuous integral, J. Math. Anal. Appl. 326, 1101-1115, 2007. Acknowledgement. This work was supported by the Commission of Scientific Research Projects of Karadeniz Tech- nical University, Project number: 2010.111.3.1. Page 29
Page 3 and 4: TABLE OF CONTENTS P a g e | i Shado
Page 5 and 6: P a g e | iii Sturm Liouville Probl
Page 7 and 8: P a g e | v Real Time 3D Palmprint
Page 9 and 10: ��
Page 11 and 12: Abstract On the Classifications of
Page 13 and 14: A Note on the Numerical Solution of
Page 15 and 16: Abstract A Third-Order of Accuracy
Page 17 and 18: Boundary Value Problem for a Third
Page 19 and 20: [18] S. G. Krein, Linear Differenti
Page 21 and 22: Positivity of Two-dimensional Ellip
Page 23 and 24: On the Numerical Solution of Ultra
Page 25 and 26: Abstract Optimal Control Problem fo
Page 27 and 28: On Stability Of Hyperbolic- Ellipti
Page 29 and 30: Abstract FUZZY CONTINUOUS DYNAMICAL
Page 31 and 32: [10] D.L. DeAngelis, R.A. Goldstein
Page 33 and 34: Bright and dark soliton solutions f
Page 35: [22] H., Triki, A.M.,Wazwaz, Bright
Page 39 and 40: Paths of Minimal Length on Suborbit
Page 41 and 42: On A SUBCLASS OF UNIVALENT FUNCTION
Page 43 and 44: Approximation by Certain Linear Pos
Page 45 and 46: PARABOLIC PROBLEMS WITH PARAMETER O
Page 47 and 48: On the Numerical Solution of a Diff
Page 49 and 50: [3] Y. Ma, and Y. Ge, Applied Mathe
Page 51 and 52: A Fuzzy Approach to Multi Objective
Page 53 and 54: Derivation and numerical study of r
Page 55 and 56: [11] G.A. Baker, P. Graves-Morris,
Page 57 and 58: Three Models based Fusion Approach
Page 59 and 60: The normal inverse Gaussian distrib
Page 61 and 62: Compact and Fredholm Operators on M
Page 63 and 64: Abstract Extended Eigenvalues of Di
Page 65 and 66: A New General Inequality for double
Page 67 and 68: Exact Solutions of the Schrödinger
Page 69 and 70: [9] Titeux I., Yakubov Y., Complete
Page 71 and 72: Positivity of Elliptic Difference O
Page 73 and 74: Transient and Cycle Structure of El
Page 75 and 76: Abstract Characterizations of Slant
Page 77 and 78: Page 69 [20] Struik, D. J., Differe
Page 79 and 80: Applied Mathematics Analysis of the
Page 81 and 82: Multibody Railway Vehicle Dynamics
Page 83 and 84: Existence of Global Solutions for a
Page 85 and 86: On the stability of the steady-stat
Page 87 and 88:
On the Numerical Solution of Diffus
Page 89 and 90:
Using expanding method of (G ′ /G
Page 91 and 92:
Weighted Bernstein Inequality for T
Page 93 and 94:
Abstract An Application on Suborbit
Page 95 and 96:
On The First Fundamental Theorem fo
Page 97 and 98:
Abstract Almost Convergence and Gen
Page 99 and 100:
Forecast results in contrast with o
Page 101 and 102:
Page 93 [15] M. El-Shahed, A. Salem
Page 103 and 104:
yields a system of equations which
Page 105 and 106:
Abstract Application of the Trial E
Page 107 and 108:
Oscillation Theorems for Second-Ord
Page 109 and 110:
Abstract On Hadamard Type Integral
Page 111 and 112:
A geometrical approach of an optima
Page 113 and 114:
A Perturbation Solution Procedure f
Page 115 and 116:
Solution of Differential Equations
Page 117 and 118:
Aproximation Properties of a Genera
Page 119 and 120:
Blow up of a solution for a system
Page 121 and 122:
A Note on Some Elementary Geometric
Page 123 and 124:
Abstract Solving Crossmatching Puzz
Page 125 and 126:
Polygonal Approximation of Digital
Page 127 and 128:
Panoramic Image Mosaicing Using Mul
Page 129 and 130:
[9] Naimark, M. A. Linear Differant
Page 131 and 132:
Abstract Chaos in cubic-quintic non
Page 133 and 134:
Semismooth Newton method for gradie
Page 135 and 136:
Multiple solutions for quasilinear
Page 137 and 138:
The Modified Bi-quintic B-spline ba
Page 139 and 140:
The Modified Kudryashov Method for
Page 141 and 142:
Study of an inverse problem that mo
Page 143 and 144:
Abstract Finding Global minima with
Page 145 and 146:
PARAMETER DEPENDENT NOVIER-STOKES L
Page 147 and 148:
7. D. J. Fyfe, The use of cubic spl
Page 149 and 150:
Page 141 An error correction method
Page 151 and 152:
Classification of exact solutions f
Page 153 and 154:
Abstract Numerical Solution of a Hy
Page 155 and 156:
generalized hyperbolic tangent func
Page 157 and 158:
EQUIVALENCE OF AFFINE CURVES Yasemi
Page 159 and 160:
Modified trial equation method for
Page 161 and 162:
Normal Extensions of a Singular Dif
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