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Chemical Engineering Hand Written_Notes_Part_20

# Chemical Engineering Hand Written_Notes_Part_20 - 42 2...

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42 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS (2) Hermite Polynomials: X L 2 ( −∞ , ) , i.e. space of continuous functions over ( −∞ , ) with 2 norm de f ned on it and (4.57) h x ( t ) , y ( t ) i = Z −∞ x ( t ) y ( t ) dt Apply Gram-Schmidt to the following set of vectors in L 2 ( −∞ , ) (3) Laguerre Polynomials: X L 2 (0 , ) , i.e. space of continuous functions over (0 , ) w i th2normde f ned on it and (4.58) h x ( t ) , y ( t ) i = Z 0 x ( t ) y ( t ) dt Apply Gram-Schmidt to the following set of vectors in L 2 (0 , ) x (1) ( t )=e x p ( t 2 ); x (2) ( t )= tx (1) ( t ); (4.59) x (3) ( t )= t 2 x (1) ( t ); ...... x ( k ) ( t )= t k 1 x (1) ( t ); .... (4.60) The f rst few Laguerre polynomials are as follows L 0 ( t )=1 ; L 1 ( t )=1 t ; L 2 ( t )=1 2 t +(1 / 2) t 2

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Chemical Engineering Hand Written_Notes_Part_20 - 42 2...

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