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ON THE STABILITY OF SOME QUADRATIC FUNCTIONAL EQUATION
"... Abstract. In this paper we establish the general solution of the functional equation which is closely associated with the quadratic functional equation and we investigate the HyersUlamRassias stability of this equation in Banach spaces. 1. Introduction and ..."
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Abstract. In this paper we establish the general solution of the functional equation which is closely associated with the quadratic functional equation and we investigate the HyersUlamRassias stability of this equation in Banach spaces. 1. Introduction and
APPROXIMATION OF MIXED TYPE FUNCTIONAL EQUATIONS IN p–BANACH SPACES
"... Abstract. In this paper, we investigate the generalized HyersUlam stability of the functional equation n∑ i=1 f(xi − 1 n n∑ j=1 xj) = n∑ i=1 f(xi) − nf ( 1 n n∑ i=1 xi) (n ≥ 2), in p–Banach spaces. 1. Introduction and ..."
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Abstract. In this paper, we investigate the generalized HyersUlam stability of the functional equation n∑ i=1 f(xi − 1 n n∑ j=1 xj) = n∑ i=1 f(xi) − nf ( 1 n n∑ i=1 xi) (n ≥ 2), in p–Banach spaces. 1. Introduction and
FUZZY STABILITY OF ADDITIVE–QUADRATIC FUNCTIONAL EQUATIONS
, 2009
"... In this paper we investigate the generalized Hyers Ulam stability of the functional equation in fuzzy Banach spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) in ..."
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In this paper we investigate the generalized Hyers Ulam stability of the functional equation in fuzzy Banach spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) in
A NOTE ON LENGTH AND ANGLE
"... Our slogan is that defining length comparable regardless of direction is the same as defining angle. We temper this idealism with only one proviso, to wit the parallelogram law. An abstract vector space does not come equipped with notions of length and angle other than that one may compare vectors ‘ ..."
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Our slogan is that defining length comparable regardless of direction is the same as defining angle. We temper this idealism with only one proviso, to wit the parallelogram law. An abstract vector space does not come equipped with notions of length and angle other than that one may compare vectors ‘in the same direction’: It always seems to make sense to say that the vector αv has α  times the length of the vector v, where α is a scalar and   is some absolute value on the field of scalars. No doubt, we also think of a nonzero vector as having nonzero length, but that plays only a concluding rôle below. Throughout, V is a vector space over a field K. We intend to restrict ourselves to subfields of R, and eventually of C. Nevertheless, to maintain the generality of our remarks, we do our best not to use properties peculiar to such fields until absolutely necessary. The field K will come accompanied with an absolute value  , that is, a positive definite map K − → R preserving multiplication in K and obeying the triangle inequality. In mildly technical language, Definition. Length is the composition of a map V−→K respecting multiplication by scalars, and an absolute value on K. In different words, length is a map to the ordered semigroup R≥0: v ↦ → ‖v ‖ with the homogeneity property (1) ‖αv ‖ = α‖v ‖ , α ∈ K. Comparability of lengths arises from the ordering on R. In principle, we could assume that the absolute value  , and hence length ‖ ‖, takes its values in some more general ordered ring.
URL: www.emis.de/journals/AFA/ SOME GEOMETRIC CONSTANTS OF ABSOLUTE NORMALIZED NORMS ON R 2
"... Abstract. We consider the Banach space X = (R2, ‖ · ‖) with a normalized, absolute norm. Our aim in this paper is to calculate the modified NeumannJordan constant C ′ NJ (X) and the Zbăganu constant CZ(X). 1. Introduction and ..."
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Abstract. We consider the Banach space X = (R2, ‖ · ‖) with a normalized, absolute norm. Our aim in this paper is to calculate the modified NeumannJordan constant C ′ NJ (X) and the Zbăganu constant CZ(X). 1. Introduction and
Reproducing Kernels Preserving Algebraic Structure: A Duality Approach
"... Abstract — From the classical reproducing kernel theory of function spaces it is wellknown that there is an inverse relationship between innerproducts and kernels. In applications, such as linear system theory and machine learning, these kernels are often highly structured. In order to exploit al ..."
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Abstract — From the classical reproducing kernel theory of function spaces it is wellknown that there is an inverse relationship between innerproducts and kernels. In applications, such as linear system theory and machine learning, these kernels are often highly structured. In order to exploit algebraic structure, it is common to choose basis functions and fall back to matrix representations. However, the basis has to be chosen in a way that is compatible with the algebraic structure, which is itself a nontrivial task. We therefore choose a different approach and use standard duality theory where additional algebraic structures form no obstacle. This is demonstrated by examples from linear system theory, namely two variable polynomials given by Bézoutians and quadratic differential forms.
Dedicated to the memory of Professor Kenichi Miyazaki By
, 2000
"... Abstract. Banach spaces which are isomorphic to a subspace (or a quotient space) of Lp are classified by means of the Jordanvon Neumann constant. 1. ..."
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Abstract. Banach spaces which are isomorphic to a subspace (or a quotient space) of Lp are classified by means of the Jordanvon Neumann constant. 1.
Gordon and Breach Science Publishers imprint. Printed in Malaysia.
, 1997
"... J. of lnequal. & Appl., 1998, Vol. 2, pp. 8997 Reprints available directly from the publisher Photocopying permitted by license only ..."
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J. of lnequal. & Appl., 1998, Vol. 2, pp. 8997 Reprints available directly from the publisher Photocopying permitted by license only