(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)
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.
Related questions
Question
![(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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe48896e0-7fec-4e7d-9f45-fe2419175997%2Fae088fb2-ca75-4efa-bedf-973991e55183%2Fchw47y_processed.jpeg&w=3840&q=75)
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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