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By Borel E.

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This is the famous Gibbs phenomenon, which has to do with how poorly a trigonometric series converges in the vicinity of a jump. T h e natural conjecture would be that the overshoots vanish as J + cm,but surprisingly, this does not take place (actually, that overshoots are proportional t o a jump). Note that while cosine approximations are not perfect for some corner functions, understanding how these partial sums perform may help us to " r e a d messages of these approximations and guess about underlying functions.

5), SJ is the optimal trigonometric polynomial of degree J for approximation of a square integrable function f under the Lqnorm. Thus, all known results about optimal approximation of functions from specific function classes may be applied to ISB. 15). It is discussed in DeVore and Lorentz (1993, p. 3). 46 2. 3. For any integer r exists a finite constant c such that sup ISBj(f) 2 0, real or < cJ-'O, P t (0,1] and finite L there := r + or. 18) StLip,,,~ There are two important function spaces defined via Fourier coefficients.

08(x). ) 5. Delta. 5) where the function is very large. Such a function resembles many practical situations where a short but abrupt deviation from a normal process occurs. The Delta mimics the theoretical delta function, which has zero width and is integrated to 1. oz(x). 6. Angle. 5, I; = 0,&1,. . , where the derivative changes sign. 5 < x 1. 7. Monotone. This function is smooth over the interval, but its 1-periodic continuation has a jump a t all integers x. We shall see that this makes approximation of such a function challenging due to boundary effects.

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Lecons ser les Fonctions Entieres by Borel E.

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