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  1. E

    Function + tangent line = 0

    Thank you for your reply! Yes, f(x) is continuous. And indeed f(x) + g(x,y) is monotone. What I meant to ask was if there is a way to explicitly find that value for x at which f+g=0 other than the "brute force" way of inverting the expression? Or perhaps, more generally - does an equation of...
  2. E

    Function + tangent line = 0

    Say we have two functions with the following properties: f(x) is negative and monotonically approaches zero as x increases. g(x,y) is a linear function in x and is, for any given y, tangent to f(x) at some point x_0(y) that depends on the choice of y in a known way. Additionally, for any...
  3. E

    Approximating function by trigonometric polynomial

    Thank you mfb for your reply! Yes, that was my original idea as well. If g is the approximation in the RHS of (1) , then I reasoned that the optimal result should be when (f-g) \perp f . However, (f-g, f) is a linear function in the coefficients a_n so there are no extrema (I am...
  4. E

    Approximating function by trigonometric polynomial

    Hi! Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1) The best approximation theorem says that in a function space equipped with the inner product (f,g) = \frac{1}{2...
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