By Sheikh Zein Uddin, M.J. Rowlandson (translator)
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Extra resources for Tohfut-Ul-Mujahideen
Example text
This is known as the Schwarz inequality. 1. Suppose that 5. Weighted inner product. In many problems we are required to deal with a weighted inner product with respect to a positive weight Junction p(x), a ~ x ~ b. p). p(x)p(x) d:c This has the same properties of linearity and homogeneity as the ordinary inner product. We say that two functions CNCPN) 112 where (CI,"" eN) range over all real values. In other words, we are trying to find the best "mean square approximation" of the given function I(x), a ~ x ~ b, by means of linear combinations of the members of the orthogonal set. 2This means that the set of exceptional values can be included in a union of intervals whose total length is arbitrarily small. 3. 2. := IItpi11 2' c; = • The minimum distance is given by d2 . min = 11/112 _ ~ (/, tpi)2 ~ IItpill 2 • The Fourier coefficients Cb"" CN satisfy Bessel's inequality c~lltplW + ...