17.7. EXERCISES 419
10. Using the above problem or standard techniques of calculus, show that{√2√π
sin(nx)
}∞
n=1
are orthonormal with respect to the inner product
⟨ f ,g⟩=∫
π
0f (x)g(x)dx
Hint: If you want to use the above problem, show that sin(nx) is a solution to theboundary value problem
y′′+n2y = 0, y(0) = y(π) = 0
11. Find S5 f (x) where f (x) = x on [−π,π] . Then graph both S5 f (x) and f (x) if youhave access to a system which will do a good job of it.
12. Find S5 f (x) where f (x) = |x| on [−π,π] . Then graph both S5 f (x) and f (x) if youhave access to a system which will do a good job of it.
13. Find S5 f (x) where f (x) = x2 on [−π,π] . Then graph both S5 f (x) and f (x) if youhave access to a system which will do a good job of it.
14. Let V be the set of real polynomials defined on [0,1] which have degree at most 2.Make this into a real inner product space by defining
⟨ f ,g⟩ ≡ f (0)g(0)+ f (1/2)g(1/2)+ f (1)g(1)
Find an orthonormal basis and explain why this is an inner product.
15. Consider Rn with the following definition.
⟨x,y⟩ ≡n
∑k=1
xkykk
Does this define an inner product? If so, explain why and state the Cauchy Schwarzinequality in terms of sums.
16. From the above, for f a piecewise continuous function,
Sn f (x) =1
2π
n
∑k=−n
eikx(∫
π
−π
f (y)e−ikydy).
Show this can be written in the form
Sn f (x) =∫
π
−π
f (y)Dn (x− y)dy
where
Dn (t) =1
2π
n
∑k=−n
eikt