I want to implement the Gram Schimdt procedure to the space of functions in the following form \begin{align} R(y,x)=\left\{ \begin{array}{ll} f(x,y), & y\leq x\\ \\ g(x,y), & y>x \end{array} \right. \end{align} the inner product is defined as \begin{align} \langle u ,v\rangle= u(a) v(a)+\int_a^b u^{'}(x) v^{'}(x)dx, \qquad \|u\|^2=\langle u ,u\rangle \end{align} I couldn't implement it in Matlab and I am an amateur in Mathematica, so are there anybody to help me?
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1I did implement Gram-Schmidt for functions in this answer. It may be not the best implementation, but you can use it as a starting point. – Leonid Shifrin Jan 20 '14 at 09:31
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How can I define function R(y,x) in Mathematica? – D. N. S Jan 20 '14 at 09:39
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@D.N.S you can try rr[x_, y_] := Piecewise[{{ff[x, y], y <= x}, {gg[x, y], x > y}}]; Orthogonalize[{{1, 1}}, rr, Method -> "GramSchmidt"] – s.s.o Jan 20 '14 at 09:44