On Nonuniqueness in Nonlinear L_{2}–Approximation by Immo Diener* Institut für Numerische und Angewandte Mathematik Lotzestraße 16–18, D–3400 Göttingen

Abstract: It is shown that under weak assumptions nonlinear L_{2}–approximation problems generally have unbounded numbers of local best approximations. This includes the rational and the exponential family of approximating functions. In addition, for a certain class of approximating families, we construct functions with three global best approximations. The results apply for instance to exponential and rational approximating families with one nonlinear parameter. Finally, we extend results of Spieß and Braess for the finiteness of the number of local best approximations by rational functions.

1. General nonlinear L_{2}–approximation

In this section we show that general nonlinear L_{2}–approximation problems have unbounded numbers of local best approximations. This extends results of Wolfe [7] for the special case of ordinary rational functions. We consider the Hilbert space H := L_{2}[−1, 1]. Let S be an open subset of R^{M} and A a twice Fréchet-differentiable map from S to H. Thus, elements of H are to be approximated by elements of A(S) = {A(x)|x ∈ S}. The first and second Fréchet-derivative of a transformation g at a point x will be denoted by g′(x, ·) and g′′(x, ·, ·) respectively. For a function f out of H the functional N_{f}(x) from S to R is defined by N_{f}(x) := ∥A(x) − f∥^{2}. Then this functional is twice differentiable with respect to x.The span of vectors x_{1}, …, x_{n} is denoted by ⟨x_{1}, …, x_{n}⟩.

In general a local minimum of N_{f}(·) at x_{0} does not imply that A(x_{0}) is a local best approximation to f with respect to A(S). For this, some further conditions must be met, especially A(x_{0}) must be normal. This condition is even sufficient in most cases. The interested reader may consult Wolfes paper [7] for further details. He defines normality as follows:

Definition: A point A(x) ∈ A(S) is called normal if A^{−1} exists on a neighbourhood of A(x), is continuous at A(x), and A′(x, ·) is one to one. ♡

Now we want to show how for a given mapping A with certain properties one can get a result of the following general structure:

1.1: The number of isolated local best approximations to a function f ∈ H cannot be bounded independently of f: For any natural number q there are distinct parameters P_{ν} ∈ S, ν = 1, …, q and a function f ∈ H such that A(P_{1}), …, A(P_{ν}) are isolated, local best approximations to f with respect to the approximating family A(S). ♡

We need a lemma of J.M. Wolfe:

1.2 Lemma [7]: Let M_{i}, i = 1, 2, … be a sequence of finite dimensional subspaces of a Hilbert space H such that

M_{i} ∩ (∑_{j=1}^{i−1} M_{j}) = ⟨0⟩  (1.3)

* This paper summarizes the authors thesis [4] for all i = 2, 3, … Let r_{i} ∈ M_{i} be given for i = 1, 2, … where r_{i} ≠ 0. Then for each n ∈ N,

L_{n} := ⋂_{i=1}^{n}(r_{i} + M_{i}^{⊥})

is nonvoid. ♡

We use this to show

1.4 Lemma: Let P_{ν} ∈ S, ν = 1, …, q be parameters, satisfying A(P_{ν}) ≠ 0 for all ν = 1, …, q. Let M_{ν} be linear finite dimensional subspaces of L_{2}[−1, 1]. Suppose that A′(P_{ν}, ·) is injective and

A(P_{ν}), A′(P_{ν}, h), A′′(P_{ν}, h, h) ∈ M_{ν}  (1.5)

for ν = 1, …, q,and all h ∈ R^{M}. Then, if the M_{ν} satisfy (1.3), there is a function f ∈ L_{2}[−1, 1] such that the functional N_{f}(·) has isolated local minima at the points P_{1}, …, P_{q}. ♡

Proof: Under the above hypotheses L_{q} := ⋂_{i=1}^{q}(A(P_{i}) + M_{i}^{⊥}) is not empty. Let f be a function in L_{q}. Then for any h ∈ R^{M} and any ν = 1, …, q we obtain

\frac{1}{2}N′_{f}(P_{ν}, h) = (A(P_{ν}) − f, A′(P_{ν}, h)) = 0,

since f has a representation f = A(P_{ν}) − g_{ν}, with (g_{ν}, A′(P_{ν}, h)) = 0. Furthermore, for any h ∈ R^{M} and any ν = 1, …, q, we have

\frac{1}{2}N′′_{f}(P_{ν}, h, h) = ∥A′(P_{ν}, h)∥^{2} + (A(P_{ν}) − f, A′′(P_{ν}, h, h)).

The last term vanishes since f can be written in the form f = A(P_{ν}) + ĝ_{ν}, with a function ĝ_{ν} that satisfies (ĝ_{ν}, A′′(P_{ν}, h, h)) = 0, and the first term is positive definite since A′(P_{ν}) is injective. ♡

To apply this lemma one has to find suitable subspaces M_{ν}. One obvious way is to consider the subspaces generated by A(P_{ν}), A′(P_{ν}, h), and A′′(P_{ν}, h, h). So we define for P_{ν} := (x_{1}^{ν}, …, x_{M}^{ν}) ∈ S the spaces M_{ν} by

M_{ν} := ⟨A(P_{ν}), \frac{∂A}{∂x_{μ}} (P_{ν}), \frac{∂^{2}A}{∂x_{μ}∂x\underline{μ}} (P_{ν})|μ, \underline{μ} = 1, …, M⟩.

These M_{ν} obviously satisfy (1.5) and we have

d_{ν} := dimM_{ν} ≤ 1 + M + \binom{M}{2} + M = \binom{M + 2}{2}.

Let {ω_{j}^{ν}}_{j=1}^{d_{ν}} be a basis for M_{ν}. Then (1.3) will be satisfied if the functions

ω_{j}^{ν}; ν = 1, …, q; j = 1, …, d_{ν}  (1.6)

are linearly independent over R.

So in general, to prove 1.1 for a given A it is sufficient to find for any q certain parameters P_{ν} ∈ S, ν = 1, …, q, such that A(P_{ν}) is normal and the functions (1.6) are linearly independent over R. This can be done in many cases. We exhibit some examples:

1.7 Example: Set S := R × R^{+} × (0, 2π), A(x_{1}, x_{2}, x_{3}) := x_{1} sin(x_{2}t + x_{3}). One obtains that A′(x) is injective iff x_{1} ≠ 0. For such parameters we have d_{ν} = 5 and the functions

ω_{1}^{ν} = sin(x_{2}^{ν}t + x_{3}^{ν}), ω_{2}^{ν} = cos(x_{2}^{ν}t + x_{3}^{ν}), ω_{3}^{ν} = tω_{1}^{ν}, ω_{4}^{ν} = tω_{2}^{ν}, ω_{5}^{ν} = t^{2}ω_{1}^{ν}

constitute a basis for M_{ν}. If x_{1}^{ν} ≠ 0 for ν = 1, …, q and x_{1}^{ν} ≠ x\underline{ν} ; 2 for ν ≠ \underline{ν}, these functions are linearly independent over R.

1.8 Example: Set S := R and,A(x_{1}) = e^{x_{1}t} + x_{1}t. A′(x_{1}) is injective for all x_{1} ∈ S and we have d_{ν} = 3. A basis is given by

ω_{1}^{ν} = te^{x_{1}^{ν}t} + t, ω_{2}^{ν} = t^{2}e^{x_{1}^{ν}t}, ω_{3}^{ν} = A(x_{1}^{ν}).

Thus, the functions (1.6) are linearly independent, iff all x_{1}^{ν}, ν = 1, …, q are distinct.

1.9 Example: Set S := R^{2} and A(x_{1}, x_{2}) := e^{x_{1}t} + x_{2}t. Then A′(x) is injective, iff x_{1} ≠ 0. We obtain d_{ν} = 4 and the functions

ω_{1}^{ν} = e^{x_{1}^{ν}t}, ω_{2}^{ν} = te^{x_{1}^{ν}t}, ω_{3}^{ν} = t^{2}e^{x_{1}^{ν}t}, ω_{4}^{ν} = t

constitute a basis for the M_{ν}. But we have M_{ν} ∩M\underline{ν} ⊃ ⟨t⟩. So (1.3) is not satisfied and the above construction is not applicable. However, the difficulty in this case (and in similar ones) can be overcome by using the same trick Braess used in [1] to prove his generalization of Wolfes theorem 6 in [7]. This shall not be further exemplified here.

1.10 Remark: Lemma 1.4 equally applies to the discrete case, i.e if we take R^{N} as the Hilbert space H. In this case q can of cause not be arbitrarily large, because (1.3) can only hold if q is smaller than the dimension of H.

1.11 Example: As an example we report without proof a result one obtains by applying 1.4 to discrete approximation in the family R_{m}^{n} of polynomial rational functions with the Euclidean norm:

Let D be a discrete set of N nodes and for 0 ≤ n < m suppose N^{∗} ≤ \frac{N+m−n−1}{3m}. For ν = 1, …, N^{∗} let r_{ν} = \frac{p_{ν}}{q_{ν}} be rational functions in R_{m}^{n}(D). Suppose the r_{ν} are normal and all q_{ν} are of degree m. Suppose further, that q_{ν} and q\underline{ν} have no common factors unless ν = \underline{ν}. Then there exists a function f ∈ C(D) such that the r_{ν} for ν = 1, …, N^{∗} are isolated local best approximations to f with respect to the Euclidean norm.

1.12 Remark: Note that the function f in (1.11) may not have a best approximation in R_{m}^{n}(D).

1.13 Remark: If P_{ν}, ν = 1, …, q are points from S such that the functions (1.6) are linearly independent, one can repeat the argument in the Hilbert space

H := ⨁_{ν=1}^{q} M_{ν} ⊂ L_{2}[−1, 1].

Then f can be choosen from H. It then has the form

f = ∑_{1≤ν≤q} α_{ν}A(P_{ν}) + ∑_{{1≤μ≤M \atop 1≤ν≤q}} β_{νμ} \frac{∂A}{∂x_{μ}} (P_{ν}) + ∑_{{1≤μ,\underline{μ}≤M \atop 1≤ν≤q}} γ_{νμ\underline{μ}} \frac{∂^{2}A}{∂x_{μ}∂x\underline{μ}} (P_{ν})

with certain real coefficients α_{ν}, β_{νμ}, γ_{νμ\underline{μ}}. ♡

2. The rational and exponential case

It would be desirable to formulate 1.1 more precisely, but in general this turns out to be rather technical and tedious. So we confine ourselves to a special case which includes general rational and exponential approximation.

Notation: Suppose X and Y = {ψ_{1}, …, ψ_{m}}, respectively, are n- and m-sets from C[−1, 1] and 0 is not in X. We say Y is linearly independent over X, iff

∑_{i=1}^{m} α_{i}ϕ_{i}ψ_{i} ≡ 0 on [−1, 1], and ϕ_{i} ∈ X, α_{i} ∈ R

implies α_{i} = 0 for i = 1, …, m.

2.1 Theorem: Let I be a compact real interval and let s be a fixed real number. Let the kernel γ̂ be defined by

γ̂(x) := \{ \begin{matrix}(1 + (1 − s)x)^{1/(1−s)} & for s ≠ 1; \\ e^{x}, & for s = 1. \end{matrix}

Let ψ_{0} and φ_{0} denote the constant function 1 on I, and let {ψ_{ν}}_{ν=1}^{m} , and {φ_{μ}}_{μ=1}^{n} be two systems of linearly independent functions from C(I). Define two affine–linear functions l_{1} and l_{2} from R^{n} and R^{m}, respectively, to C[I] by

\begin{aligned}l_{1}(x) & = ∑_{μ=1}^{n} x_{μ}φ_{μ} + c_{1}φ_{0} \\ l_{2}(y) & = ∑_{ν=1}^{m} y_{ν}ψ_{ν} + c_{2}ψ_{0} \end{aligned}

Using B_{s} := {y ∈ R^{m}|1 + (1 − s)l_{2}(y) > 0}, we consider the open and convex subset S := R^{nN} × B_{s}^{N} of R^{N(n+m)} for a fixed natural number N. For y ∈ B_{s}, set γ(y) := γ̂(l_{2}(y)) and for each parameter P := (x^{1^{T}} , …, x^{N^{T}} , y^{1^{T}} , …, y^{N^{T}}) ∈ S define the approximating function by

A(P) := ∑_{k=1}^{N} l_{1}(x^{k})γ(y^{k}).

Let the linear space L be defined by

L := ⟨φ_{μ}ψ_{ν}ψ\underline{ν}|μ = 0, …, n; ν, \underline{ν} = 0, …, m⟩.

Suppose, that for ω = 1, …, q ∈ N, certain points P_{ω} := (x_{ω}^{1}, …, x_{ω}^{N}, y_{ω}^{1}, …, y_{ω}^{N}) ∈ S are given such that for ω = 1, …, q the functions A′(P_{ω}) are injective, A(P_{ω}) ≠ 0, and the functions γ^{2s−1}(y_{ω}^{k}) for ω = 1, …, q und k = 1, …, N are linearly independent over L.

Then there exists a function f from

⨁_{ω,k} L · γ^{2s−1}(y_{ω}^{k}) ⊂ L_{2}(I),

such that the the functional N_{f}(·) has isolated local minima at the points P_{1}, …, P_{q}. ♡

Before we prove this theorem, we shall consider some examples.

2.2 Example (Approximation with rational functions): We show how to get Wolfes theorem [7, Theorem 6] from 2.1. We set s = 2, c_{1}, c_{2} = 0, N = 1, n = n_{Wolfe} + 1, m = m_{Wolfe}, φ_{μ} = t^{μ−1}, ψ_{ν} = −t^{ν}. We get B_{s} = {y ∈ R^{m}| ∑_{ν=1}^{m} y_{ν}t^{ν} + 1 > 0} and for P = (x_{1}, …, x_{n}, y_{1}, …, y_{m}) we obtain

A(P) = \frac{x_{1} + x_{2}t + ⋯ + x_{n}t^{n−1}}{1 + y_{1}t + ⋯ + y_{m}t^{m}} ∈ R_{m}^{n−1}

For L we get

L = ⟨t^{μ−1}t^{ν}t\underline{ν}|μ = 1, …, n; ν, \underline{ν} = 0, …, m⟩ = P_{2m+n−1}

and we have γ^{2s−1}(y) = (1 + y_{1}t + ⋯ + y_{m}t^{m})^{−3}. The prepositions of 2.1 are surely satisfied if A(P_{ω}) is normal for ω = 1, …, q (then A′(P_{ω}) is injective), the denominators are relatively prime and have full degree m, and if L contains no polynomial of degree ≥ 3m. This means that 2m + n − 1 < 3m. These are exactly the premises that Wolfe needed for his theorem. Moreover, we can say something about the form of f: If the Q_{i} are the given denominator polynomials, then f can be chosen from ⨁_{i=1}^{q} \frac{P_{2m+n−1}}{Q_{i}^{3}} .

2.3 Example (Exponential approximation): We set s = n = m = 1, φ_{1} ≡ 1, ψ = id_{I}, c_{1} = c_{2} = 0 to get S = R^{N} × R^{N}, L = P_{2} and

A(P) = ∑_{k=1}^{N} x^{k}e^{y^{k}t}.

If all x_{μ} ≠ 0 and all y_{ν} are distinct, then A′(P) is injective. The functions γ^{2s−1}(y_{ω}^{k}) = e^{y_{ω}^{k_{t}}} are linearly independent over L = P_{2}, if the parameters y_{ω}^{k} are distinct for k = 1, …, N and ω = 1, …, q.

2.4 Example: Set I = [a, b], 0 < a < b < ∞, s = m = n = N = 1, φ_{1} ≡ 1, ψ_{1} = ln t, S = R × R, L = ⟨1, ln t, (ln t)^{2}⟩ and

A(x, y) = xt^{y}

A′(x, y) is injective, iff x does not vanish. The functions γ^{2s−1}(y_{ω}^{k}) = t^{y_{ω}} are linearly independent over L, iff all y_{ω} are pairwise distinct.

2.5 Proof of 2.1: We consider the linear subspaces

M_{ω} := ⨁_{k=1}^{N} L · γ^{2s−1}(y_{ω}^{k})

of the Hilbert space H := ⨁_{ω=1}^{q} M_{ω}. It follows from the linear independence of the functions γ^{2s−1}(y_{ω}^{k}) over L that these sums are direct and the M_{ω} satisfy condition (1.3). In order to apply 1.4, we need to show that

A(P_{ω}), A′(P_{ω}, h), A′′(P_{ω}, h, h) ∈ M_{ω}

for all h ∈ R^{N(n+m)}. We have

M_{ω} = ⟨φ_{μ}, ψ_{ν}, ψ\underline{ν}γ^{2s−1}(y_{ω}^{k})|1 ≤ k ≤ N, 0 ≤ μ ≤ n, 0 ≤ ν, \underline{ν} ≤ m⟩.

Then 0 ≠ A(P_{ω}) ∈ M_{ω}. Corresponding to P := (x^{1}, …, x^{N}, y^{1}, …, y^{N}) we also partition h according to h := (u^{1}, …, u^{N}, v^{1}, …, v^{N}) and define

\begin{aligned}l̂_{1}(x) & := l_{1}(x) − c_{1}φ_{0} \\ l̂_{2}(y) & := l_{2}(y) − c_{2}ψ_{0}. \end{aligned}

Then

A(P + h) − A(P) = ∑_{k=1}^{N} l_{1}(x^{k} + u^{k})γ(y^{k} + v^{k}) − ∑_{k=1}^{N} l_{1}(x^{k})γ(y^{k})

= ∑_{k=1}^{N}(l_{1}(x^{k}) + l̂_{1}(u_{k}))γ̂(l_{2}(y^{k}) + l̂_{2}(v^{k})) − ∑_{k=1}^{N} l_{1}(x^{k})γ(y^{k})

= ∑_{k=1}^{N}(l_{1}(x^{k}) + l̂_{1}(u_{k}))(γ(y^{k}) + γ̂′(l_{2}(y^{k}))l̂_{2}(v^{k})

+ \frac{1}{2}γ̂′′(l_{2}(y^{k}))l̂_{2}(v^{k})^{2} + O(∥v^{k}∥^{3})) − ∑_{k=1}^{N} l_{1}(x^{k})γ(y^{k})

= ∑_{k=1}^{N} (l_{1}(x^{k})γ̂′(l_{2}(y^{k}))l̂_{2}(v^{k}) + l̂_{1}(u^{k})γ(y^{k}))

+ ∑_{k=1}^{N} (γ̂′(l_{2}(y^{k}))l̂_{2}(v^{k})l̂_{1}(u^{k}) + \frac{1}{2}l_{1}(x^{k})γ̂′′(l_{2}(y^{k}))l̂_{2}(v^{k})^{2})

+ O(∥h∥^{2}).

Because of the identities

\frac{d}{dx}γ̂(x) = γ̂^{s}(x), \frac{d^{2}}{dx^{2}} γ̂(x) = sγ̂^{2s−1}(x)

we obtain

\begin{aligned}A′(P, h) & = ∑_{k=1}^{N} (l_{1}(x^{k})γ(y^{k})l̂_{2}(v^{k}) + γ(y^{k})l̂_{1}(u^{k})) \\ A′′(P, h, h) & = ∑_{k=1}^{N} (γ^{s}(y^{k})l̂_{1}(u^{k})l̂_{2}(v^{k}) + \frac{s}{2}γ^{2s−1}(y^{k})l̂_{2}(v^{k})^{2}) . \end{aligned}

From this the theorem follows immediately. ♡

3. Nonuniqueness of global best approximations

Several authors have published examples for functions with more than one global best L_{2}–approximation (g.b.a.). In the approximation with ordinary rational functions on a compact real interval, one uses a symmetry argument and the fact, that best approximations are normal [2]. We shall give a more general argument that works for nonsymmetric cases too, and gives three best approximations in symmetric cases.

Consider the L_{2}–approximation in R_{m}^{n}[−1, 1]. Local best approximations to functions f ∉ R_{m−1}^{n−1} are always normal [3] and the subset of normal functions in R_{m}^{n} is not connected for n ≥ 0.

3.1 Theorem: Let C_{0} and C_{1} be two connected componnents of R_{m}^{n}\R_{m−1}^{n−1}. Suppose f_{0} and f_{1} are functions in L_{2}[−1, 1] such that f_{0} has a g.b.a in C_{0} and f_{1} has a g.b.a. in C_{1}. Let H : [0, 1] → L_{2}[−1, 1]\R_{m−1}^{n−1}[−1, 1] be a continuous curve from f_{0} to f_{1}. Then there exists λ^{∗} ∈ [0, 1], such that H(λ^{∗}) has two global best approximations in R_{m}^{n}. ♡

Proof: Let C_{2} be the set of normal functions from R_{m}^{n} that are neither in C_{0} nor in C_{1}. Let

E(λ) := inf_{r∈R_{m}^{n}} ∥H(λ) − r∥

be the distance from H(λ) to R_{m}^{n} and

E_{i}(λ) := inf_{r∈C_{i}} ∥H(λ) − r∥ , i = 0, 1, 2

be the distance from H(λ) to C_{i}. The functions E, E_{i} are continuous on [0, 1] and for all λ ∈ [0, 1] we have

min{E_{0}(λ), E_{1}(λ), E_{2}(λ)} = E(λ).

For λ = 0 the minimum is attained at E_{0} and for λ = 1 at E_{1}. Thus there exists λ^{∗} ∈ [0, 1] with E_{j}(λ^{∗}) = E_{k}(λ^{∗}) = E(λ^{∗}) and j ≠ k. Since best approximations are normal, the corresponding best approximations must lie in C_{j} and C_{k}. ♡

Using this result we can eliminate the restriction to odd r in [2,Theorem 4.1]:

3.2 Theorem: Let n ≥ 0, m ≥ 1. To posess only one local best approximation in R_{m}^{n}[−1, 1] is no generic property in L_{2}[−1, 1]. ♡

For the rest of this chapter we use I := [−1, 1], S := (−1, 1). Let u : S × I → R^{+} be a continnuous kernel function, positive on S × I, such that zero is the only function orthogonal to every function in u(S, ·). Let ϕ_{i}, f : I → R^{N}, i = 1, 2, …, N, be linearly independent functions on I such that ϕ_{1}(t)f(t) ≢ 0 on I. We define the approximating family by a mapping A : S × R^{N} → C[I] where

A(b, x) := u(b, ·) ∑_{i=1}^{N} x_{i}ϕ_{i}.

For simplicity (although we do not need that much) we require that b ↦ A(b, ·) is injective and that any continuous function f on I has a best L_{2}–approximation in A(S × R^{N}).

3.3 Theorem: Under the assumptions stated above, define for any λ ∈ R^{N} a continuous function f_{λ} by

f_{λ} := \frac{1}{1 + ∥λ∥}(f − ∑_{i=1}^{N} λ_{i}ϕ_{i}).

Then

(i) There exists a vector λ̃ ∈ R^{N} such that fλ̃ has two global best approximations in A(S × R^{N}). (ii) If u(b, t) ≡ u(−b, −t) and the functions f, ϕ_{i} are even, then there exists a vector λ̃ ∈ R^{N} such that fλ̃

has three global best approximations in A(S × R^{N}). ♡

Proof:

1. Zero is not a best approximation to any f ≢ 0 because otherwise ω(t) := ∥tu(b, ·)ϕ_{1} − f∥^{2} would, for all b ∈ S, have a minimum at t = 0. But then ϕ_{1}f would be orthogonal to all u(b, ·), b ∈ S, contradicting the assumption that ϕ_{1}f does not vanish identically and that zero is the only function orthogonal to all u(b, ·), b ∈ S.

2. For b ∈ S let λ^{∗}(b) be the solution of the linear system (u(b, ·)ϕ_{j}, f_{λ^{∗}(b)}) = 0 , j = 1, …, N. Then λ^{∗} exists, is uniquely determined and depends continuously on b, because the matrix of this system is a Gramian matrix of the linearly independent functions ϕ_{i} with respect to the positive weight function u(b, ·). For any b ∈ S the minimum of ∥A(b, x) − f_{λ^{∗}(b)}∥^{2} where x ranges over R^{N} is attained for x = 0.

3. Suppose that for any λ ∈ R^{N} there exists exactly one b^{∗}(λ) ∈ S, such that A(b^{∗}(λ), x^{∗}) is the g.b.a. to f_{λ}. Then b^{∗} depends continuously on λ. But then b^{∗}(R^{N}) is contained in a compact subinterval of S. To see this, assume that there exists a sequence {λ^{ν}} of vectors in R^{N} such that b^{∗}(λ^{ν}) converges to +1 or −1 as ν goes to infinity. By choosing an appropriate subsequence if necessary, we may suppose that λ^{ν}/(1 + ∥λ^{ν}∥) converges to a vector λ̂ ∈ R^{N}. If ∥λ^{ν}∥ converges to q ∈ R ∪ {∞}, then f_{λ^{ν}} converges to the continuous function

f_{∞} := \frac{f}{1 + q} + ∑_{i=1}^{N} λ̂_{i}ϕ_{i}

which does not vanish identically on I. Thus f_{∞} has a unique g.b.a. A(b^{∞}, x^{∞}) with b^{∞} ∈ S and x^{∞} ≠ 0.

4. Thus, if the g.b.a. to any f_{λ} is unique, the mapping b^{∗} ∘ λ^{∗} is continuous and maps S into a compact subset of S. It follows that b^{∗} ∘ λ^{∗} has a fixed point b̂ ∈ S. By construction zero would be a g.b.a. to f_{λ^{∗}(b̂)}. This function, however, does not vanish identically, and this is a contradiction to part 1. of this proof. Then b^{∗} does not depend continuously on λ and so there exists a λ̃ ∈ R^{N} such that fλ̃ has two g.b.a. in A(S × R^{N}).

5. To prove (ii) we note that under the symmetry assumptions made, A(b, x) is a g.b.a. to f_{λ} if and only if A(−b, x) is also a g.b.a. to f_{λ}. Now we repeat the arguments 1. to 4. but restrict b to the interval [0, 1). It follows that there is a λ̃ ∈ R^{N}, such that fλ̃ has two g.b.a. in A([0, 1) × R^{N}). One of them might have b^{∗}(λ̃) = 0, but the other one can be paired with a g.b.a. in A((−1, 0) × R^{N}). ♡

Obviously we can take for u(b, t) the kernels e^{bt} and (1 − bt)^{−1} to obtain results for exponential or rational approximation and thus answering questions of Braess [1,2].

3.4 Example: The following example has already been considered by Lamprecht [5]; he did however not give a rigorous proof. We consider L_{2}–approximation in R_{1}^{0}[−1, 1] and choose f(t) ≡ t^{2}, ϕ_{1}(t) ≡ 1. The theorem implies that there exists λ̃ ∈ R, such that fλ̃(t) = t^{2} − λ̃ has three g.b.a. in R_{1}^{0}[−1, 1]. A closer analysis shows that λ̃ ∈ (\frac{1}{3}, 1). ♡

4. The number of critical points in rational L_{2}–approximation

It is still an open problem whether for some f ∈ L_{2}[−1, 1] the function ∥f − ·∥^{2} has infinitely many (local) minima in R_{m}^{n}. Finiteness was reported for the case m=1 in [7], but this information was based on a linguistic translation error of a result by Spieß [6]. We parametrize R_{1}^{0}[−1, 1] by a · u(b, ·) where u(b, t) := \sqrt{(1 − b^{2})/2}(1 + bt)^{−1} is the normalized kernel. Spieß proved that for f ∈ L_{2}[−1, 1] the critical points of N_{f}(a, b) := ∥a · u(b, ·) − f(·)∥^{2} cannot accumulate at points (a^{∗}, b^{∗}) with b^{∗} ∈ (−1, 1).

Later Braess [1] showed that the number of critical points in R_{1}^{0} is finite if f is continuous on [−1, 1] and f(1) · f(−1) is nonzero. In this section a much stronger result is obtained.

With the above parametrisation (a, b) is critical iff

a − (u(b), f) = 0 and a · (u′(b), f) = 0.

We can restrict the computations to the case a ≠ 0 and thus have to prove that (u′(b), f) has only finitely many zeros for b ∈ (−1, 1). We obtain

\frac{∂u}{∂b}(b, t) = \frac{−1}{\sqrt{2}\sqrt{1 − b^{2}}} \frac{b + t}{(1 + bt)^{2}} .

Thus the possible values for b are the zeros of F : (−1, 1) → R, defined by

F (b) := ∫_{−1}^{1} f(t)\frac{t + b}{(1 + bt)^{2}} dt

.

4.1 Lemma: Let f be a continuous function on [−1, 1]. Suppose f(−1) ≠ 0 or there exists a constant δ ∈ (0, 1) such that f is monotonous in the interval [−1, −1 + δ]. Then F : (−1, 1) → R, defined by

F (b) := ∫_{−1}^{1} f(t)\frac{t + b}{(1 + bt)^{2}} dt

has only finitely many zeros in the interval (0, 1). ♡

Proof: We may suppose f(−1) = 0 because the other case is covered by the result of Braess in [1] cited above. F is analytic on (−1, 1). Thus, if the zeros of F had an accumulation point in (−1, 1), F would be identically zero. This cannot happen for f ∈ L_{2}[−1, 1] as has been shown by Spieß in [6]. Suppose the zeros accumulate at 1. Then, for each n = 0, 1, … a sequence b_{i}^{n}, i = 0, 1, … exists such that b_{i}^{n} → 1 for i → ∞ and F^{(n)}(b_{i}^{n}) = 0 for all i, n ≥ 0 Let K(b, t) denote the kernel

K(b, t) := \frac{t + b}{(1 + bt)^{2}} .

For n ≥ 0 induction gives

K_{n}(b, t) := \frac{∂^{n}}{∂b^{n}} K(b, t) = (−1)^{n} \frac{n! t^{n−1}((n + 1)t^{2} + bt − n)}{(1 + bt)^{n+2}}

We now split F^{(n)}(b) into two terms

F^{(n)}(b) = ( ∫_{−1}^{−1+δ}+ ∫_{−1+δ}^{1} ) (f(t)K_{n}(b, t) dt)

and estimate roughly:

∥ ∫_{−1+δ}^{1} f(t)K_{n}(b, t) dt∥_{∞,[0,1]} ≤ ∥f∥_{∞} · ∥ ∫_{−1+δ}^{1} K_{n}(b, t) dt∥_{∞,[0,1]}

≤ ∥f∥_{∞} · 2 · sup_{b∈(0,1)} sup_{t∈[−1+δ,1]} |K_{n}(b, t)|

≤ 2∥f∥_{∞}|K_{n}(1, −1 + δ)|

≤ 2\frac{(n + 1)!}{δ^{n+2}}∥f∥_{∞}

Thus we have

F^{(n)}(b) ≥ −2∥f∥_{∞} \frac{(n + 1)!}{δ^{n+2}} + ∫_{−1}^{−1+δ}f(t)K_{n}(b, t) dt.

Now, K_{n}(b, −1) = −\frac{n!}{(1−b)^{n+1}} is less than zero and K_{n}(b, ·) has exactly one zero in (−1, 0), namely

z_{n}(b) := −\frac{1}{2n + 2} (b + \sqrt{(2n + 1)^{2} − (1 − b)}) .

We find z_{n}(b) → −1 for b → 1 or n → ∞. We may suppose, that f increases monotonously on [−1, −1 + δ]. Thus for z_{n}(b) ∈ (−1, −1 + δ) we get

∫_{−1}^{−1+δ}f(t)K_{n}(b, t) dt ≥ f(z_{n}(b)) ∫_{−1}^{−1+δ}K_{n}(b, t) dt.

The integral can asymptotically be estimated by

∫_{−1}^{−1+δ}K_{n}(b, t) dt = \frac{∂^{n}}{∂b^{n}} \{ln(1 − b + δb) − ln(1 − b) − \frac{δb(1 + b)}{1 − b + δb}\} = O((1 − b)^{−n})

for fixed n ≥ 1 and b → 1. So we obtain

F^{(n)}(b) ≥ −2∥f∥_{∞} \frac{(n + 1)!}{δ^{n+2}} + f(z_{n}(b))O((1 − b)^{−n})

for b → 1. Since F^{(n)}(b_{i}^{n}) = 0 we must have

f(z_{n}(b_{i}^{n}))O((1 − b_{i}^{n})^{−n}) < 2∥f∥_{∞} \frac{(n + 1)!}{δ^{n+2}}

for i → ∞. This implies f(z_{n}(b)) ≤ O((1 − b)^{−n}) for b → 1. Because of z_{n}(b) > −1 + (1 − b)/(2n + 2) this means we have f(t) = O((1 + t)^{n}) for t → −1. So f(t) vanishes faster than any polynomial for t → −1. But then

c_{n} := lim_{b→1} ∫_{−1}^{−1+δ}f(t)K_{n}(b, t) dt

exists for all n and this limit must be smaller than 2∥f∥_{∞} \frac{(n+1)!}{δ^{n+2}}. We now show that under these circumstances F can be analytically continued beyond b = 1 and we again obtain a contradiction to the result of Spieß. We expand F into a series around b = 1 and show that this series has a positive radius of convergence:

F (b) := ∑_{n=0}^{∞} \frac{(b − 1)^{n}}{n!} ∫_{−1}^{+1}K_{n}(1, t)f(t) dt.

The coefficients can be estimated by

| ∫_{−1}^{+1}K_{n}(1, t)f(t) dt| = | ∫_{−1}^{−1+δ}+ ∫_{−1+δ}^{+1} | ≤ 4∥f∥_{∞} \frac{(n + 1)!}{δ^{n+2}}.

Thus the series has radius of convergence at least δ. ♡

Likewise F has only finitely many zeros in (−1, 0), provided f is monotonous at the right end of [−1, 1]. Thus we obtain

4.2 Theorem: Let f be a continuous function on [−1, 1]. Suppose that for x = +1 and x = −1 either f(x) ≠ 0 or there exists a neighbourhood U(x) of x such that f is monotonous in U(x) ∩ [−1, 1]. Then the function ∥f − ·∥^{2} has only finitely many critical points in R_{1}^{0}.

4. References

[1] D. Braess: On Rational L_{2}–Approximation, Journal of Approximation Theory 18(1976)136–151.

[2] D. Braess: Über das Anzahlproblem bei der rationalen L_{2}–Approximation, in: Numerische Methoden der Approximationstheorie, Band 3, ISNM 30(1976)109–121.

[3] E.W. Cheney, A.A. Goldstein: Mean–Square Approximation by Generalized Rational Functions, Math. Zeitschrift 95(1967)232–241.

[4] I. Diener: Zur Mehrdeutigkeit bei der nichtlinearen L_{2}–Approximation,Dissertation, Göttingen 1983.

[5] G. Lamprecht: Mehrdeutigkeit bei der Approximation in der L_{p}–Norm mit Hilfe rationaler Funktionen,Computing 5(1970)349–355.

[6] J. Spieß: Eindeutigkeitssätze bei der nichtlinearen Approximation in strikt-konvexen Räumen, Dissertation, Hamburg 1969.

[7] J.M. Wolfe: On the Unicity of Nonlinear Approximation in Smooth Spaces,Journal of Approximation Theory 12(1974)165–181.