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