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Math 118 :: Winter 2009 :: Homework 2 :: Due February 12 1. In this exercise we work with x ∈ Rn . Find the values of α > 0 for which f (x) = (1 + kxk)−α is integrable on Rn , in the sense of Lebesgue. [Hint: the polar change of coordinates reads dx = rn−1 drdΩ, where Ω are the angular variables. The measure of the (n − 1)-dimensional sphere is finite.] 2. In the notes and in class we justified the formula |fˆ(k)| ≤ Ck −1 for BV functions which are also of class C 1 . Prove the same result for general BV functions, which are not necessarily C 1 . [Hint: use the formula involving a limit and an integral.] 3. In this exercise, we show that proper smoothing of sinc by a moving average makes it integrable, only for some particular choice of the length over which the average is done. Let f (x) = sin x ? χ[−a,a] (x), x where ? denotes convolution. (a) Compute the Fourier transform fˆ(k) of f . (b) For which values of a is fˆ(k) continuous? Think graphically. ˆ (c) If a is one of those values, argue that dfdk(k) is of bounded variation. (d) In that case, how fast does f decay, and is that fast enough to get Lebesgue integrability? (e) What becomes of (c) and (d) in case a is not “one of those values”? Justify. 4. Consider f (x) = 1/x for x ∈ [1, a] for some a > 1. (a) Consider N + 1 equispaced points xj in [1, a], with j = 0, . . . , N . Find a formula for xj . (b) Consider N Y πN +1 (x) = (x − xj ). j=0 If the equispaced grid were extended outside of [1, a], one would have the expression xN +1 = a + a−1 N for the first gridpoint outside [1, a], to the right of a. Prove that πN +1 (xN +1 ) > |πN +1 (x)| for all x ∈ [1, a]. [Hint: Compare distances to the zeros of πN +1 . Watch the sign of πN +1 (x), there is an absolute value above.] (c) Find an expression for πN +1 (xN +1 ), involving a factorial. This gives us a good upper bound on maxx∈[1,a] |πN +1 (x)|. (d) Find a formula for the n-th derivative of f (x) = 1/x. (e) Use your results so far to give a bound on the error of polynomial interpolation of f (x) from its samples f (xj ), kf − pN kL∞ [1,a] , where pN is the Lagrange interpolant. (f) For which values of a does this bound on the error tend to zero as N → ∞? [Hint: For understanding how large a factorial is, it is often convenient to use Stirling’s bounds √ √ 2 n nn e−n < n! < 3 n nn e−n when n ≥ 1.]