ReviewM_00W - Review Problems for the Midterm Examination...

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Review Problems for the Midterm Examination February 14, 2000 1. Define an inner product of a linear space V that is a set of all continuous functions defined on an interval () 0,1 , and show that it satisfies the required properties of the inner product. Orthogonalize three functions () () () 2 12 3 1, ,and xx x φφ φ == = with respect the inner product you have defined 2. State the required properties of a norm . in a linear space V. What is the natural norm? Show that the following two norms satisfy the required properties of a norm: [] 0,1 max x ff x = 1 2 0 x d x = where V is a set of all continuous functions defined on 0,1 . Show that the inequality 1 2 0 0,1 max x fx d x fx . State your idea whether or not a positive constant 0 α > exists that satisfies the inequality 1 2 0 0,1 max x x . 3. Suppose that a data set {} , 1,. .., 1 i fi n =+ is given at a set of sampling points ,1 , . . . i xi n . Define the least squares method to approximate a function by using a set of linearly independent functions , 1,. .., 1 k xk m . Also find the necessary condition of the least squares method. For the data
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ReviewM_00W - Review Problems for the Midterm Examination...

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