OFFS_1-11

OFFS_1-11 - Tutorial OFFS Orthogonal Functions and Fourier...

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Unformatted text preview: Tutorial OFFS: Orthogonal Functions and Fourier Series 3. Solutions to Exercises Exercise 1. Refer to Eqns. 19 - 21 in the tutorial, TF: ‘Trigg’ Functions . C n = ¸= 2 : Exercise 2. One might say, Ifthere exists no vector , V , except the null vector, ; , with the property that ^ x i ¢ V = 0 for all i; then the basis f ^ x i g is said to be complete. Exercise 3. Suppose that f ( x ) = C , a non-zero constant. Then, Z x + ¸ x C ½ cos 2 n¼x ¸ or sin 2 n¼x ¸ ¾ dx = C Z x + ¸ x ½ cos 2 n¼x ¸ or sin 2 n¼x ¸ ¾ dx = for any n 6 = 0 : Thus, one needs to add to the set of functions a constant, say Á = 1 ; so that Z x + ¸ x CÁ dx 6 = 0 : Exercise 4. We have Z b a Á ¤ m ( x ) g ( x ) ! ( x ) dx = Z b a Á ¤ m ( x )[ f ( x ) ¡ ©( x )] ! ( x ) dx = a m ¡ Z b a Á ¤ m ( x ) X n =1 a n Á n ( x ) ! ( x ) dx = a m ¡ X n =1 a n Z b a Á ¤ m ( x ) Á n ( x ) ! ( x ) dx = a m ¡ X n =1 a n ± mn = a m ¡ a m = 0 ; where we have used, respectively, Eqns. 3, 2 and 1....
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OFFS_1-11 - Tutorial OFFS Orthogonal Functions and Fourier...

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