Why can an orthogonal basis be more convenient than the “natural basis” 1, x, x^2,... for the purposes of calculation. How can we guarantee the accuracy of our representation in terms of a given number of basis functions? MATH MAJORS HELP!
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nickykat12
ANSWERS: 1
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When the basis is comprised of orthogonal functions, adding a new member to the expansion does not change coefficients of the already found shorter expansion. When the elements of the basis are not orthogonal, evry next element has non-zero prjections on previous elements (in general). As a result, if you want to add a new member to the expansion you will be adding "extra portions" to the elements of the basis that already have been included, which means that the already found coefficients of the expansion would have to be recalculated. In other words, with an orthogonal basis coefficients of new elements of the expansion are independent of the previously found coefficients. Therefore, the accuracy of your representation is determined by the total of the elements that have not been included yet but does not depend on what have been included already. This can be helpful since in some cases you know that the sum of non-included elements can be evaluated merely through _the first_ non-included element, not all of them.
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