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x, y We let in .NET Generation gs1 datamatrix barcode in .NET x, y We let




How to generate, print barcode using .NET, Java sdk library control with example project source code free download:
x, y We let using barcode integration for none control to generate, create none image in none applications. bar code = y x = n=1. xn y n ,. xn = n x,. yn = n y,. Ax = n=1. n n n x be the generator none none of a diagonal semigroup A with eigenvalues n and eigenvectors n (see Examples 3.3.3 and 3.

3.5), and let X 1 = D (A). By Example 3.

3.5, the vectors x X 1 are characterized by the fact that their coordinates xn in the basis n satisfy x. 2 X1. n=1. (1 + n . 2 ). xn 2 < , B(X 1 ; X ) by n=1. xn = n x,. x X 1. Let us de ne the operator x= n=1. (1 + n . 2 )1/2 n n x = (1 + n . 2 )1/2 xn n . (4.9.1).

Then is closed and self-adjoint as an operator in X , and it has a bounded inverse n=1. (1 + n . 2 ) 1/2 n n x = n=1. (1 + n . 2 ) 1/2 xn n . The generators Obviously, X , and is an isometric isomorphism of X 1 onto X , i.e., it maps X 1 onto x = x X1 ,. x X 1. Thus X 1 is a Hilbert space with inner product x, y x, y n=1. (1 + n . 2 )xn y n ,. xn = n x,. yn = n y. In particular, n = (1 + n . 2 )1/2 ,. and { n } is an none for none orthogonal (but not orthonormal) basis in X 1 . The standard n=1 formula for the expansion of a vector in terms of this basis in X 1 is x= =. n=1 n=1 x, n X 1 n = n 2 1 X x, n X n n=1 . (1 + n . 2 ) x, n (1 + n . 2 ). n n x. Thus, we get the same result if we expand x X 1 with respect to the orthonormal basis { n } in X , or with respect to the same orthogonal (but not orthonormal) n=1 basis in X 1 . We proceed to construct the space X 1 in the same way as we did in Section 3.6, but with A replaced by , i.

e., we let X 1 be the completion of X with the weaker norm x. 2 X 1. n=1. (1 + n . 2 ) 1 . xn 2 ,. xn = n x. This norm is induced by the inner product x, y X 1. n=1. (1 + 2 ) 1 xn y n . n In particular, X none for none 1 is a Hilbert space, and can be extended to an isometric isomorphism of X onto X 1 . The sequence { n } is still an orthogonal basis n=1 in X 1 , but it is not orthonormal since n. X 1. = (1 + n . 2 ) 1/2 .. Thus, vectors x X 1 are characterized by the fact that they can be written in the form x = xn n , where n=1 x. 2 X 1. n=1. (1 + n . 2 ) 1 . xn 2 < ,. x X 1 . 4.9 Diagonal and normal systems Here the coef cients xn are given by xn = x, n X 1 = (1 + n . 2 ) x, n n 2 1 X X 1 ,. and they are eq ual to n x whenever x X . As X is a Hilbert space, we can identify X with X (every y X is of the form x y x = x, y X = xn y n for some y X ). It is also possible n=1 to identify the dual of X 1 with X 1 and the dual of X 1 with X 1 , but it is more convenient to identify the dual of X 1 with X 1 , arguing as follows.

Every y (X 1 ) is of the form x x, y1 X 1 for some y1 X 1 . For x X we can write this as (since 1 is self-adjoint in X ). x, y X 1. = x,. 2 . = x, y where y = y1 X 1 . Thus, to every y (X 1 ) there corresponds a unique y X 1 such that y x = x, y and this induces a duality pairing y x = x, y. (X 1 ,X 1 ) n=1 X,. x X,. xn y n ,. x X 1 ,. y X1 between X 1 and X 1 . The use of the same expression y x to represent both x, y X and x, y (X 1 ,X 1 ) , depending on the context, is possible since the actual formula for its computation, y x = xn y n , is the same in both cases. (The n=1 construction presented above is comparable to the one in Remark 3.

6.1 if we identify X with X and use the fact that is self-adjoint, hence D ( ) = D ( ) .) If the diagonal semigroup A is part of an L p .

Reg-well-posed li none none near system A B on (Y, X, U ), then we can develop representations based on the eigenC D vectors and eigenvalues of the generator A of A for all the different operators appearing in the theory. In our representation for the input operator B we need the notion of a left-sided Laplace transform, introduced in De nition 3.12.

1. Theorem 4.9.

1 Let = A B be an L p . Reg-well-posed li near system on C D (Y, X, U ), where A is a diagonal semigroup on the separable Hilbert space X generated by the operator A with eigenvalues n and eigenvectors n , n = 1, 2, 3, . . .

Let B be the control operator, C the observation operator, C&D the combined observation/feedthrough operator, and D the transfer function of , and de ne Cn = C n and Bn = n B, n = 1, 2, 3, . . .

Denote the growth bound of A by A, and let > A and (A). Then the following representation formulas are valid (all the sums converge in the strong topology in the given spaces):.
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