Inner Product Spaces. Another notation sometimes used is X.?

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1 Inner Product Spaces X In 8 we have an ÞÞÞ? 8@ 8Þ Another notation sometimes used is @ 8 8 The inner product in?@ ß in 8 has several essential properties ( see Theorem, p. 33) that we have used repearedly: a)?@ b)?@ ßA c) c?@ ß -?@ ß d)?? ß 0 and?? ß =0 if and only if?! Þ We defined the length of a vector? by?? ß@, and the distance between two vectors? ß@ as llß that is, Þ Earlier in the course, we used the essential properties of vectors in as the starting point to define more general vector spaces ZÐp. 90). In the same spirit, we now use the properties a)-d) to describe, in any vector space, how an inner product should behave. For a vector space Z with real scalars: any rule ß that creates a scalar from each pair of vectors in Z and satisfies a)-d) will be called an inner product in ZÞ ( Properties a)-d) are modified slightly when complex scalars are allowed.) A vector space Z with an inner product defined is called an inner product space. ecause such an inner product acts just like the inner product from 8, many of the same theorems remain true. You can look at a little of this material in the textbook: Section 6.7. Here is a little more detail involving one specific example Let GÒ ß Ó be vector space of all continuous real valued functions defined on the interval Ò ß ÓÀ call it G, for short. For vectors (functions) 0ß in Gß define an inner product by 0ß 0ÐÑÐÑ. This definition satisfies all the essential properties for an inner product listed above: a) 0ß ß0 because 0ÐÑÐÑ. ÐÑ0ÐÑ. b) 0 ß2 0ß2 ß2 because Ð0ÐÑ ÐÑÑ2ÐÑ. 0ÐÑ2ÐÑ. ÐÑ2ÐÑ. 8 c) c0ß - 0ß d) 0ß0 0 and 0ß0! if and only if 0! Ð the constant function!ñ You should check c) and d). The last part of d) is the only one that uses that the functions 0 Gare continuous. Continuing to parallel to our definitions in 8, we define the norm (or length ) of 0 by

2 and the distance between 0 and as ll0ll 0ß 0 0 ÐÑ. ll 0 ll Ð0ÐÑ ÐÑÑ. and we say that 0 and are orthogonal Ð0 ¼ Ñ if 0ß 0ÐÑÐÑ.! For example: on Ò ß Ó, the functions sin and cos are orthogonal because Ðsin ÑÐcos Ñ. sinðñ. cosðñl! Many of the techniques we developed using inner products still work. For example: If Ö0 ß 0 ß ÞÞÞß 08 is a basis for a subspace [ of G, we can convert the basis into an orthogonal basis using the same Gram Schmidt process. ¼ For a subspace [, we can define [ Ö0À 0ß! for all in [ If [ is a subspace of G with an orthogonal basis Ö ß ÞÞÞß and 0 G 8 then we can uniquely write 00 s where 0 [ s ¼ and [. 0sis called the projection of 0 on [ and 0s is given by the formula s 0ß 0ß ß ß ÞÞÞ 8 8 Then 0 s is the function in [ closest to 0, that is 00ll s 0llfor all in [ different from 0s Note: Unlike 8, G is not finite dimensional. ut for the results just listed, that doesn't matter. What does matter is that the subspace [ is finite dimensional. A calculation in G À find the polynomial of degree Ÿ 5 in G that is closest to the function sin. % We use the subspace [Span Ößßßßß} ÞThe polynomial we want is ; sin è proj [ sinþ This polynomial ; is the closest in [ to sin closest in the sense of the distance we defined: Î ll ; sin ll l;ðñ sin l. is as small as possible. In that sense, ; is the best available approximation in [ for sin. It's easy to compute standard basis Proj sin if we choose an orthonormal basis for [Þ So we convert Y the

3 ß Öß ß ß ÞÞÞß Ö/ ß / ß / ß ÞÞÞß / ' into an orthonormal basis We use the Gram Schmidt process (and normalize, to get a vector of length at each step). Note: the integrations below were done using Matlab. Although every integration needed to find the / 3 's is very easy, the constants that arise are messy and pile up fast; they can easily lead to errors when the computation is done by hand. Try to compute at least e ß /ß/ for yourself (with or without computer assistance) to be sure you understand what's going on. The steps are the same as for the usual Gram Schmidt process in 8 Þ We start with the first basis is not a unit vector in G because ll@ ll llll. Þ So we normalize and use ll llll For 4 ß ÞÞÞß ' in turn we use the Gram Schmidt formula, normalizing at each step to get a unit 4 ß/ 4 ß/ / 2 4ß/ 4 / 4 ß/ ß/ / ß/ / ß/ / / ll. ll Ðsince llll.!ñ ' Î. Ð' Ñ Î

4 ß/ ß/ ß/ ß/ / ll Ð /.Ñ/ Ð /.Ñ/ ll Ð /.Ñ/ Ð /.Ñ/ ll ' Ð '.Ñ Ð.Ñ ll Ð.Ñ Ð ' '.Ñ ll ÞÞÞ ) Continuing in this way gives: Notice that each integration requires nothing harder than 8.. ut already the constants are becoming a headache. / ÞÞÞ % / ÞÞÞ / ÞÞÞ ' ) % Ð Ñ ) ( ) Ð Ñ ( ) )! ) % % ' Ð ( Ñ ß and finally * ) %'* %! Ð ( * Ñ Then / ß ÞÞÞß /' are orthogonal polynomials of degree Ÿ à we use them as our orthogonal basis for [Þ With them, we can compute ;ÐÑProj sin sin /, / sin /, / ÞÞÞsin ß/ / [ ' ' ' ' Ð Ðsin Ñ /.Ñ/ Ð Ðsin Ñ/.Ñ/ ÞÞÞ Ð Ðsin Ñ/.Ñ/ ' Ð Ð Ñ.Ñ Ð ' sin Ðsin Ñ.Ñ ) % Ð Ñ ) % Ð Ñ ) ) Ð Ðsin Ñ.Ñ ÞÞÞ three more terms corresponding to /ß/ß % and /' (Notice that the integrations involved are now more challenging because they involve terms like 8 sin.. ut they are manageable, with enough patience, using integration by parts.)

5 When all the smoke clears, Matlab gives % % ' ;ÐÑ ) Ð %' )Ñ Ð(!! %'! Ñ! ) ' % Ð (' (% ÑÑ (*) If we convert the exact coefficients in (*) to approximate decimal coefficients we have ;ÐÑ!Þ*)()'(%'(!Þ(%!'%!Þ!!'%(*(' In the sense of the distance ll ll that we defined in G, ; is the closest polynomial in [ to sin. th For comparison: there is better known 5 degree polynomial approximation for sin : use the th 5 degree Taylor polynomial: X ÐÑ 0Ð!Ñ 0 Ð!Ñ ÞÞÞ. ww w 0 Ð!Ñ 0 ÐÑ x x For 0ÐÑ sin ß sin X ÐÑ With approximate decimal coefficients, Ð@Ñ x x X ÐÑ!Þ''''''''''''(!Þ!!) ecause XÐÑ is constructed using derivatives of 0at 0, it turns to be the better approximation for sin near!, but it produces a larger and larger error as gets further and further from!. We can see this in the figure on the next page.

6 The figure below shows the graphs of sin, X ÐÑ and ;ÐÑ on the interval Ò ß ÓÞ Across the whole interval Ò ß ÓÀ the graphs of sin and ;ÐÑare so close that you can't see any difference between them at this scale. You can see that XÐÑ and sin also are indistinguishable near! but get fairly far apart as gets nearer to the endpoints Þ The table on the following page illustrates three things (the third is not clear from the graphs above): i) As we move away from! toward, the approximation from X ÐÑ is not as good as the ;ÐÑ approximation ii) Linear algebra gives us the ;ÐÑ approximation. It has the advantage that it gives us a good approximation for sin over the whole interval Ò ß ÓÞ iii) The Taylor polynomial XÐÑ is actually a better approximation than ;ÐÑfor sin th when is near! ( in some sense, XÐÑ is acutally the best of all 5 degree polynomials to approximate sin near!).

7 All table values are rounded to 4 significant digits so, for example, is not exactly! lerror l lerrorl sin X ÐÑ ;ÐÑ l sin X ÐÑ l l sin ;ÐÑ l à large X!.060 à smallish ; à error à error à near à over ã à the whole à interval ã Î Ã very à but error à small à for ; Ãlerrorl à near! à near! à is larger à using à than for ÃX à T approx à approx ã ã Î Ã smallish ; ã à error à large X à over à error à the whole à near à interval

8 One more example, without many details We are still working in the vector space G interval Ò ß Ó. of continuous functions on the Conside the subspace [ Span Öß sin ß sin ß ÞÞÞß sin 8ß cos ß ÞÞÞß cos 8 for some 8Þ You can check that these functions form an orthogonal basis for [À. For example, to show sin ¼ cos À sin ß cos Ð sin ÑÐ cos Ñ. Since sin Ðα Ñ sin α cos cos α sin and sin Ðα Ñ sin α cos cos α sin then sinðα Ñsin Ðα Ñ sin α cos If we let α and ß we sinðñ sin Ð Ñ sin ÐÑ cos ÐÑ so Ð sinðñ sin ÐÑ Ñ sin ÐÑ cos ÐÑ Therefore Ð sin ÑÐ cos Ñ. sinðñ sin ÐÑ. Ð cos cos.ñl! A function in [ is a linear combination of the basis elements. If 0is in G, we can compute proj 00 s the function in [ closest to 0 [ 0ÐÑß 0ÐÑß cos 0ÐÑß cos 8 ß cos ßcos cos 8ßcos 8 cos ÞÞÞ cos 8 For the denominators: 0ÐÑß sin 0ÐÑß sin 8 sin ßsin sin 8ßsin 8 sin ÞÞÞ sin 8 ß. cos 5ßcos 5> cos 5. ( why? ) sin 5ßsin 5> sin 5. ( why? )

9 So proj [ 00 s Ð 0ÐÑ.Ñ Ð 0ÐÑ cos.ñ cos ÞÞÞ Ð 0ÐÑ cos 8.Ñ cos 8 Ð 0ÐÑ sin.ñ sin ÞÞÞ Ð 0ÐÑ sin 8.Ñsin 8 +! + cos ÞÞÞ+ 8 cos 8, sin ÞÞÞ, 8sin 8 So proj [ 00 s +! + 5cos 5 sin 5 (*) where + 8 5! 8 5 0ÐÑ. and + 5 Ð 0ÐÑ cos 5.Ñ, Ð 0ÐÑ sin 8.Ñ 5 for Ÿ5Ÿ8 for Ÿ5Ÿ8 The coefficients used in writing proj [ 0 are called the Fourier coefficients of 0 th and the trigonmetric series (*) is the 8 Fourier approximation for 0. If 8Ä ß then ll0proj [ 0ll Ä!, that is, the approximation error Ä!Ðas measured by our distance function ll 0 proj [ 0 ll Þ ÐThis is called mean square convergence. Ñ Mean square convergence is not the same as saying: for each Ò ß Óß 0ÐÑ s Ä 0ÐÑ ( this is called pointwise convergence ) and it's a much harder problem to determine for which 's this is true. However, pointwise converge is true, for example, at any point where 0 is differentiable.