0, we say that f is positive homogeneous of degree k

1. Homework

Definition 1.1. A function f : V → W between vector spaces over R is said to be homo- geneous of degree k ∈ N if

f (λv) = λ^kf (v)

for any nonzero real number λ for any v ∈ V. If the above equation holds for any λ > 0, we say that f is positive homogeneous of degree k.

(1) Let α = (α1, · · · , αn) be an n-tuple of nonnegative integers. For each vector x = (x₁, · · · , x_n) ∈ Rⁿ, we denote x^α = x^α₁¹x^α₂²· · · x^α_nⁿ and |α| = a₁ + · · · + α_n. A polynomial function on Rⁿ is a function of the form P (x) =P

αaαx^α where aα = 0 for all but finitely many α. The degree of P (x) is defined to be deg P (x) = max{|α| : a_α6= 0}.

(a) Let P_k be the space of homogeneous polynomials of degree k on Rⁿ together with the zero polynomial. Prove that P_k forms a vector subspace of C(Rⁿ, R).

(b) Let Hk= {P (x) ∈ Pk: ∆P = 0}. Prove that Hk is a vector subspace of Pk. (c) Let H_k = {P |_Sⁿ⁻¹ : P ∈ H_k}. Here P |_Sn−1 is the restriction of P to Sⁿ⁻¹ =

{x ∈ Rⁿ : kxk = 1}. Prove that Hk is a vector subspace of C(Sⁿ⁻¹, R) such that the restriction map res : H_k → H_k sending P to its restriction to Sⁿ⁻¹ is a linear isomorphism.

(d) (Difficult) Let r : Rⁿ→ R be the function r(x) = kxk. Prove that we have the following direct sum decomposition of vector spaces P_k= H_k⊕ r²P_k−2.

(e) Let P (x) = P

αaαx^α be a polynomial function on Rⁿ of degree N. For each 0 ≤ j ≤ N, we set P_j(x) = P

|α|=ja_αx^α. Show that P_j(x) is homogeneous of degree j for 0 ≤ j ≤ N. Furthermore, if Qj(x) is a polynomial function homogeneous of degree j for each 0 ≤ j ≤ N so that P (x) =PN

j=0Qj(x), show that Q_j(x) = P_j(x) for all 0 ≤ j ≤ N.¹ This implies that we have direct sum decomposition

P =M

k≥0

P_k

where P is the space of polynomials on Rⁿ.

(f) Let H = {P ∈ P : ∆P = 0} be the space of harmonic polynomials on Rⁿ. Show that H is a vector subspace of P.

(g) Let P (x) and Pj(x) as above. Show that P (x) is harmonic if and only if Pj(x) are harmonic for all 0 ≤ j ≤ N. This implies that we have a direct sum decom- position of vector spaces

H =M

k≥0

H_k.

(2) Let f : Rⁿ\ {0} → R be a C¹-function. Show that f is positive homogenous of degree k if and only if²

(1.1) x · ∇f (x) = kf (x).

1Hint: consider P (tx) for any t 6= 0.

2Hint: consider f(λx) = λ^kf (x). Differentiate this function with respect to λ.

1

2

When n = 2, (1.1) is equivalent to x∂f

∂x+ y∂f

∂y = kf (x, y).

(3) Classify the critical points of the following functions on R². (a) f (x, y) = x⁴+ y⁴− x²− y²+ 1.

(b) f (x, y) = x²+ 2xy + 2y²+ 4x.

(c) f (x, y) = x³− y³+ 3x²+ 3y²− 9x.

(d) f (x, y) = x²− xy + y⁴.

(4) Let f : R³\ {0} → R be the function f (x) = 1

kxk, x ∈ R³\ {0}.

Write down the 3rd Taylor polynomial of f at a ∈ R³\ {0}. Prove or disprove that the function f is analytic at every point of R³\ {0}. If the function is analytic at some point a in R³\ {0}, can you find its Taylor expansion at a (in a neighborhood of a?)