Several sets of orthogonal functions have become standard bases for approximating functions. 0000005444 00000 n 433 0 obj<> endobj xref /BS /UCR2 /Default 5 0 obj /ca 1 433 26 /OPM 0 The results of these examples will be very useful for the rest of this chapter and most of the next chapter. /Alternate /DeviceRGB 0000001474 00000 n 0000028040 00000 n In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions. 0000002115 00000 n 0000027414 00000 n 0000013532 00000 n %PDF-1.4 9 0 obj %PDF-1.4 %���� /Border [0 0 0] /AIS false �ꇆ��n���Q�t�}MA�0�al������S�x ��k�&�^���>�0|>_�'��,�G! endobj 0000003375 00000 n >> <]>> We will also work a couple of examples showing intervals on which cos( n pi x / L) and sin( n pi x / L) are mutually orthogonal. Orthogonal Functions contd. >> /A 6 0 R /Length 2571 /S /GoTo /Rect [71.804695 711.493469 332.707489 729.758057] /BM /Normal /TK true 0000027645 00000 n "F$H:R��!z��F�Qd?r9�\A&�G���rQ��h������E��]�a�4z�Bg�����E#H �*B=��0H�I��p�p�0MxJ$�D1��D, V���ĭ����KĻ�Y�dE�"E��I2���E�B�G��t�4MzN�����r!YK� ���?%_&�#���(��0J:EAi��Q�(�()ӔWT6U@���P+���!�~��m���D�e�Դ�!��h�Ӧh/��']B/����ҏӿ�?a0n�hF!��X���8����܌k�c&5S�����6�l��Ia�2c�K�M�A�!�E�#��ƒ�d�V��(�k��e���l ����}�}�C�q�9 0000001659 00000 n Orthogonal Functions and Fourier Series. << /N 3 8 0 obj For then ⁡ ⁡ = ⁡ ((−)) − ⁡ ((+)), and the integral of the product of the two sine functions vanishes. 0000000016 00000 n Thus the vector concepts like the inner product and orthogonality of vectors can be extended to func-tions. 435 0 obj<>stream W�$Tdr�&m;�X�x�C�R'��$�:���H(�*. /W 0 /op false 12 0 obj ORTHOGONAL SETSWe are primarily interested in infinite sets of orthogonal 0000003153 00000 n 0000002517 00000 n Orthogonal functions A function can be considered to be a generalization of a vector. 0000014844 00000 n /SA false stream << >> 0000005815 00000 n /Type /ExtGState Analogy between functions of time and vectors 2. For example, the sine functions sin nx and sin mx are orthogonal on the interval ∈ (−,) when ≠ and n and m are positive integers. /TR2 /Default 0000006272 00000 n We have Zπ −π sin(3x) cos(3x)dx = 0 since sin(3x) cos(3x) is odd and the interval [−π,π] is symmetric about 0. 0000007054 00000 n 0 Orthogonal Functions and Fourier Series March 17, 2008 Today’s Topics 1. /Type /Annot >> endobj �V��)g�B�0�i�W��8#�8wթ��8_�٥ʨQ����Q�j@�&�A)/��g�>'K�� �t�;\�� ӥ$պF�ZUn����(4T�%)뫔�0C&�����Z��i���8��bx��E���B�;�����P���ӓ̹�A�om?�W= 0000006528 00000 n /Subtype /Link n�3ܣ�k�Gݯz=��[=��=�B�0FX'�+������t���G�,�}���/���Hh8�m�W�2p[����AiA��N�#8$X�?�A�KHI�{!7�. Fourier series Take Away Periodic complex exponentials have properties analogous to vectors in n dimensional spaces. University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don Fussell Vector Spaces Set of vectors Closed under the following operations Vector addition: v 1 + v 2 = v 3 Scalar multiplication: s v 1 = v 2 Linear combinations: Scalars come from some field F e.g. •Example: f(x) = sin(3x), g(x) = cos(3x). 0000007475 00000 n 0000005044 00000 n /CA 1 %���� N'��)�].�u�J�r� << H���yTSw�oɞ����c [���5la�QIBH�ADED���2�mtFOE�.�c��}���0��8�׎�8G�Ng�����9�w���߽��� �'����0 �֠�J��b� << 0000003117 00000 n 6 0 obj /Type /ExtGState 0000003040 00000 n �x������- �����[��� 0����}��y)7ta�����>j���T�7���@���tܛ�`q�2��ʀ��&���6�Z�L�Ą?�_��yxg)˔z���çL�U���*�u�Sk�Se�O4?׸�c����.� � �� R� ߁��-��2�5������ ��S�>ӣV����d�`r��n~��Y�&�+`��;�A4�� ���A9� =�-�t��l�`;��~p���� �Gp| ��[`L��`� "A�YA�+��Cb(��R�,� *�T�2B-� 0000013729 00000 n A familiar example is Fourier series, where the function is a periodic function on the interval ( L=2;L=2). /Filter /FlateDecode 0000005682 00000 n >> /SM 0.02 /D [7 0 R /Fit] << 4 0 obj [5 0 R] >> startxref Periodic signals can be represented as a sum of sinusoidal functions. ��w�G� xR^���[�oƜch�g�`>b���$���*~� �:����E���b��~���,m,�-��ݖ,�Y��¬�*�6X�[ݱF�=�3�뭷Y��~dó ���t���i�z�f�6�~`{�v���.�Ng����#{�}�}��������j������c1X6���fm���;'_9 �r�:�8�q�:��˜�O:ϸ8������u��Jq���nv=���M����m����R 4 � 0000000833 00000 n %%EOF endobj /OP false 0000001970 00000 n /S /S << 0000006768 00000 n /TR2 /Default x�b```b``5d`e`�X��π �@1V�p!� ��`CF����.�F�G�k%�I\��� �!z�WC(��Aj߻8��L�-.�tx_TX��4e��͠)k5�L�֪z1� �ER|�5�s~��2r).x�u�����} /HT /Default endobj /Type /Border For example, the functions f 1(x) x2 and f 2(x) x3 are orthogonal on the interval [ 1, 1], since Unlike in vector analysis, in which the word orthogonal is a synonym for perpendic- ular, in this present context the term orthogonal and condition (1) have no geometric significance. 2y�.-;!���K�Z� ���^�i�"L��0���-�� @8(��r�;q��7�L��y��&�Q��q�4�j���|�9�� Orthogonal Functions -Orthogonal Functions -DDefinitionefinition ... another Example ... f (x) =x2, 0