Notation let m max m 0 m 1 m n the osculating

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Notation: Let m = max { m 0 , m 1 , . . . , m n } . The osculating polynomial approximation of a function f C m [ a , b ] at x i , i = 0 , 1 , . . . , n is the polynomial (of lowest possible order) that agrees with { f ( x i ) , f ( x i ) , . . . , f ( m i ) ( x i ) } at x i [ a , b ] , i . The degree of the osculating polynomial is at most M = n + n summationdisplay i =0 m i . In the case where m i = 1, i the polynomial is called a Hermite Interpolatory Polynomial . Peter Blomgren, ( [email protected] ) #5 Interpolation and Polynomial Approximation — (24/40)
Polynomial Approximation: Practical Computations Polynomial Approximation, Higher Order Matching Beyond Hermite Interpolatory Polynomials Osculating Polynomials Hermite Interpolatory Polynomials Computing Hermite Interpolatory Polynomials Hermite Interpolatory Polynomials The Existence Statement If f C 1 [ a , b ] and { x 0 , x 1 , . . . , x n } ∈ [ a , b ] are distinct, the unique polynomial of least degree ( 2 n + 1) agreeing with f ( x ) and f ( x ) at { x 0 , x 1 , . . . , x n } is H 2n + 1 ( x ) = n summationdisplay j = 0 f ( x j ) H n , j ( x ) + n summationdisplay j = 0 f ( x j ) ˆ H n , j ( x ) , where H n , j ( x ) = bracketleftBig 1 2( x x j ) L n , j ( x j ) bracketrightBig L 2 n , j ( x ) ˆ H n , j ( x ) = ( x x j ) L 2 n , j ( x ) , and L n , j ( x ) are our old friends, the Lagrange coefficients : L n , j ( x ) = n productdisplay i =0 , i negationslash = j x x i x j x i . Further, if f C 2 n +2 [ a , b ], then for some ξ ( x ) [ a , b ] f ( x ) = H 2 n +1 ( x ) + producttext n i =0 ( x x i ) 2 (2 n + 2)! f (2 n +2) ( ξ ( x )) . Peter Blomgren, ( [email protected] ) #5 Interpolation and Polynomial Approximation — (25/40)
Polynomial Approximation: Practical Computations Polynomial Approximation, Higher Order Matching Beyond Hermite Interpolatory Polynomials Osculating Polynomials Hermite Interpolatory Polynomials Computing Hermite Interpolatory Polynomials That’s Hardly Obvious — Proof Needed! 1 of 2 Recall: L n , j ( x i ) = δ i , j = braceleftbigg 0 , if i negationslash = j 1 if i = j ( δ i , j is Kronecker’s delta). If follows that when i negationslash = j : H n , j ( x i ) = ˆ H n , j ( x i ) = 0. When i = j : braceleftBigg H n , j ( x j ) = bracketleftBig 1 2( x j x j ) L n , j ( x j ) bracketrightBig · 1 = 1 ˆ H n , j ( x j ) = ( x j x j ) L 2 n , j ( x j ) = 0 . Thus, H 2n + 1 ( x j ) = f ( x j ). H n , j ( x ) = [ 2 L n , j ( x j )] L 2 n , j ( x ) + [1 2( x x j ) L n , j ( x j )] · 2 L n , j ( x ) L n , j ( x ) = L n , j ( x ) bracketleftBig 2 L n , j ( x j ) L n , j ( x ) + [1 2( x x j ) L n , j ( x j )] · 2( x ) L n , j bracketrightBig Since L n , j ( x ) is a factor in H n , j ( x ): H n , j ( x i ) = 0 when i negationslash = j . Peter Blomgren, ( [email protected] ) #5 Interpolation and Polynomial Approximation — (26/40)
Polynomial Approximation: Practical Computations Polynomial Approximation, Higher Order Matching Beyond Hermite Interpolatory Polynomials Osculating Polynomials Hermite Interpolatory Polynomials Computing Hermite Interpolatory Polynomials Proof, continued... H n , j ( x j ) = [ 2 L n , j ( x j )] L 2 n , j ( x j ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright 1 + [1 2 ( x j x j ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright 0 L n , j ( x j )] · 2 L n , j ( x j ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright 1 L n , j ( x j ) = 2 L n , j ( x j ) + 1 · 2 · L n , j ( x j ) = 0 i.e. H n , j ( x i ) = 0 , i .

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