Let f(x, y) = x^2 + y^2. Find the limit of f(x, y) as (x, y) approaches (0, 0).
Some advanced exercises utilize Fourier theory to prove that any continuous function on a closed interval can be uniformly approximated by polynomials. Critical Tips for Students Mathematical Analysis Apostol Solutions Chapter 11
Step 3 – Conclude: By the Riemann-Stieltjes condition, (f \in \mathcalR(\alpha)). By symmetry or by integration by parts (once integrability of one is known), (\alpha \in \mathcalR(f)). Let f(x, y) = x^2 + y^2
Unlike elementary calculus, these problems often require verifying pointwise versus uniform convergence . You must frequently determine if a function satisfies the Dirichlet conditions (e.g., being piecewise smooth) before concluding that its Fourier series converges to the average of its left and right limits. Critical Tips for Students Step 3 – Conclude:
Show that if (\sum_n=1^\infty (|a_n|+|b_n|) < \infty), then the Fourier series converges uniformly to a continuous function.