Chemical Engineering Hand Written_Notes_Part_24

Chemical Engineering Hand Written_Notes_Part_24 - 50 2....

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50 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS Theorem 1 . ( Weierstrass Approximation Theorem): Consider space C [ a, b ] , the set of all continuous functions over interval [ a, b ] , together with ∞− norm de f ned on it as (5.30) k f ( t ) k = max t [ a, b ] | f ( t ) | Given any ε> 0 , for every f ( t ) C [ a, b ] there exists a polynomial p ( t ) such that | f ( t ) p ( t ) | for all t [ a, b ] . This fundamental result forms the basis of many numerical techniques. Example 36 . We routinely approximate unknown continuous functions us- ing polynomial approximation. For example, speci f cheato fapuresubstanceat constant pressure ( C p ) is approximated as a polynomial function of temperature C p ' a + bT + cT 2 (5.31) or C p ' a + bT + cT 2 + dT 3 (5.32) While developing such empirical correlations over some temperature interval [ T 1 , T 2 ] , it is su cient to know that ( C p ) is some continuous function of tem-
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Chemical Engineering Hand Written_Notes_Part_24 - 50 2....

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