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Math 228B Theoretical Assignment 1 Winter 1999

Due: Monday, February 1, 1999

Problem 1     Prove that

$$L(\alpha u + \beta v) = \alpha L\: u + \beta L\: v$$

Problem 2     Show that

$$\vert\vert L u - L v \vert\vert \leq \vert\vert L\vert\vert\: \vert\vert u - v \vert\vert$$

(Hint: You should only need the definition of ||L|| given in class.)

Problem 3     Show that if

$$\vert\vert L\vert\vert \leq 1 + c \Delta t,$$

then the solution at some fixed time T₀ remains bounded as .

Problem 4     Let

$${\bf u}= ( u_0 , u_1 , ... , u_{M-1} )$$

be M discrete real numbers defined on a computational grid. and define b_k as usual

$$b_ k= \langle {\bf u}, {\bf w}^{(k)} \rangle .$$

Answer the following questions.

1.

Consider the function f(x) defined on the interval [0, D] defined by

$$f(x) = \sum_{k=-{M / 2}+1}^{M / 2} b_k {\bf W}^{(k)} (x) .$$

Is f a real valued function?

2.

Now consider the function F(x) defined on the interval [0, D] defined by

$$F(x) = \sum_{k=0}^{M-1} b_k {\bf W}^{(k)} (x) .$$

Is F a real valued function?

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