3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by C[1,2]. We equip C[1,2] with the L² norm, 1/2 ||S|| = (S{* (S(x))² dr), vƒ € C[1,2]. Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi that minimizes ||p- ƒ ||². Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that 이(a,b) = 0 and (1,6) = 0.
3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by C[1,2]. We equip C[1,2] with the L² norm, 1/2 ||S|| = (S{* (S(x))² dr), vƒ € C[1,2]. Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi that minimizes ||p- ƒ ||². Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that 이(a,b) = 0 and (1,6) = 0.
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![3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by
C[1,2]. We equip C[1,2] with the L² norm,
1/2
||S|| = (S{* (S(x))² dr), vƒ € C[1,2].
Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi
that minimizes ||p- ƒ ||².
Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that
이(a,b) = 0 and (1,6) = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9328926a-b343-4d47-8a00-cbd9e6e36ac2%2Fa5ddf495-ccae-4d7f-9116-6849336658f9%2Fwehd0pl_processed.png&w=3840&q=75)
Transcribed Image Text:3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by
C[1,2]. We equip C[1,2] with the L² norm,
1/2
||S|| = (S{* (S(x))² dr), vƒ € C[1,2].
Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi
that minimizes ||p- ƒ ||².
Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that
이(a,b) = 0 and (1,6) = 0.
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