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(c) For I=[a, b] an interval, define P3(I)= {v: v is a polynomial of degree<3 on I, ie, v has the form v(x)=a3x3+a2x²+a1x+ao, xel where a¡ER}. Show that veP3(I) is uniquely determined by the values v(a), v'(a), v(b), v'(b). Find the corresponding basis functions (the basis function corresponding to the value v(a) is the cubic polynomial v such that v(a)=1, v'(a)=0, v(b)=v'(b)=0, etc). (d) Starting from (c) construct a finite-dimensional subspace Wh of W consisting of piecewise cubic functions. Specify suitable parameters to describe the functions in Wh and determine the corresponding basis functions.

(c) For I=[a, b] an interval, define P3(I)= {v: v is a polynomial of degree<3 on I, ie, v has the form v(x)=a3x3+a2x²+a1x+ao, xel where a¡ER}. Show that veP3(I) is uniquely determined by the values v(a), v'(a), v(b), v'(b). Find the corresponding basis functions (the basis function corresponding to the value v(a) is the cubic polynomial v such that v(a)=1, v'(a)=0, v(b)=v'(b)=0, etc). (d) Starting from (c) construct a finite-dimensional subspace Wh of W consisting of piecewise cubic functions. Specify suitable parameters to describe the functions in Wh and determine the corresponding basis functions.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.2: Complex Solutions Of Equations
Problem 7ECP: Find the cubic polynomial function f with real coefficients that has 1 and 2+i as zeros, and f2=2.
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(c) For I=[a, b] an interval, define P3(I)= {v: v is a polynomial of degree<3 on I, ie, v has the form v(x)=a3x3+a2x²+a1x+ao, xel where a¡ER}. Show that veP3(I) is uniquely determined by the values v(a), v'(a), v(b), v'(b). Find the corresponding basis functions (the basis function corresponding to the value v(a) is the cubic polynomial v such that v(a)=1, v'(a)=0, v(b)=v'(b)=0, etc). (d) Starting from (c) construct a finite-dimensional subspace Wh of W consisting of piecewise cubic functions. Specify suitable parameters to describe the functions in Wh and determine the corresponding basis functions.
Transcribed Image Text:(c) For I=[a, b] an interval, define P3(I)= {v: v is a polynomial of degree<3 on I, ie, v has the form v(x)=a3x3+a2x²+a1x+ao, xel where a¡ER}. Show that veP3(I) is uniquely determined by the values v(a), v'(a), v(b), v'(b). Find the corresponding basis functions (the basis function corresponding to the value v(a) is the cubic polynomial v such that v(a)=1, v'(a)=0, v(b)=v'(b)=0, etc). (d) Starting from (c) construct a finite-dimensional subspace Wh of W consisting of piecewise cubic functions. Specify suitable parameters to describe the functions in Wh and determine the corresponding basis functions.
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