Abstract
Müntz’s theorem asserts, for example, that the linear span of the even powers \(1, x^2, x^4,\dots \) is dense in \(C([0,1])\). We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating \(f(x)=x\) to accuracy \(\varepsilon = 10^{-6}\) in this basis requires powers larger than \(x^{280{,}000}\) and coefficients larger than \(10^{107{,}000}\). We present a theorem establishing exponential growth of coefficients with respect to \(1/\varepsilon \).
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1 Introduction
The Müntz approximation theorem, conjectured by Bernstein [4] and proved by Müntz [12], is a beautiful result. Suppose we are interested in continuous functions on \([0,1]\), i.e., \(f\in C([0,1])\), and we want to approximate them by linear combinations of the monomials
where \(\{a_k^{}\}\) is a set of exponents (not necessarily integers) satisfying
Certainly this is possible if \(a_k^{} = k\) for each k, by the Weierstrass approximation theorem [14], but what if the set of powers is sparser? For example, is the span of the even powers
dense in \(C([0,1])\)? The theorem characterizes all suitable sets of exponents:
Theorem 1
(Müntz approximation theorem) The linear span of the family \(\{x^{a_k^{}}\}\) is dense in \(C([0,1])\) if and only if
Thus the set (1) easily qualifies, as do many other collections of exponents, such as the primes:
For discussions of the theorem with proofs, see [2, 6, 11]. After 1914, Müntz’s theorem was generalized by Szász and others.
As a numerical analyst, I work with algorithms based on expanding functions in nonorthogonal bases, a powerful technique in certain contexts [1, 10]. This led me to consider Müntz’s theorem from a computational angle, and what emerged is startling. To make the point, it is enough to consider a particular case of what might be regarded as the most basic nontrivial Müntz approximation. The name “E” alludes to the use of even powers.
Problem E. Given \(\varepsilon >0\), find an integer \(n\ge 0\) and coefficients \(c_0^{}, \dots , c_n^{}\) such that
We shall prove
Theorem 2
If \(\varepsilon < 1/2\), then any solution of Problem E has
and
Actually, I believe the following sharper bounds hold:
For accuracy \(\varepsilon = 10^{-6}\), my estimate is that one needs \(n> 140{,}000\) and \(\max _k^{} |c_k^{}|> 10^{107{,}000}\). In such an expansion, the enormous coefficients have oscillating signs, so that they cancel almost exactly (namely to one part in \(10^{107{,}000}\)). On a computer in floating-point arithmetic, all information will be lost unless one works in a precision of more than 107, 000 digits. (The usual precision is 16 digits.)
2 Proof
Problem E is equivalent to the more familiar problem of approximation of |x| on \([-1,1]\):
Since Lebesgue first used approximations of |x| for a proof of the Weierstrass approximation theorem at age 23 in 1898, a great deal has been learned about this problem, as recounted in chapter 25 of [14]. In particular, Bernstein’s 1914 paper [5] was a landmark contribution. Among many other things, Bernstein proved that \(\varepsilon \) satisfies
for any \(n\ge 1\), which implies, since \(4(1+\sqrt{2}) \approx 9.66\),
Since \(\varepsilon < 1/2\) in (9) implies \(n\ge 1\), this establishes condition (5) of Theorem 2.
To establish condition (6), we make use of (11). Given \(\varepsilon \), let n and \(\{c_k^{}\}\) define a solution (4) of Problem E. If we split the series into roughly the first quarter and the last three-quarters,then by (11), the first part can approximate |x| no more closely
than \(4\varepsilon \). More to our purpose, by a linear scaling, it can approximate |x| over the subinterval \([-1/2,1/2]\) no more closely than \(2\varepsilon \). Therefore, since the sum of the two series in (12) has accuracy better than \(\varepsilon \), the second series must have maximal size at least \(\varepsilon \) over \([-1/2,1/2]\). Since \(|x^{2k}| \le 2^{-2k}\) for \(x\in [-1/2,1/2]\), this implies that there must be some huge coefficients. Specifically, summing a power series involving powers of 4 shows that the second series of (12) is bounded by
Therefore, we must have
that is,
This implies (6), completing the proof of Theorem 2. \(\square \)
3 Numerical Estimates
The theorem and proof just given were all about lower bounds, but now let us look at more accurate (though unrigorous) estimates. Bernstein [5] also proved that the best degree \(2n\) maximum-norm approximation errors \(\varepsilon \) satisfy
for some \(\beta \), and in 1985, Varga and Carpenter [16] gave the numerical estimate
To achieve \(\varepsilon \le 10^{-1}\), \(10^{-2}\), \(10^{-3}\), and \(10^{-4}\), respectively, this suggests (rounding up to the next even numbers) that we will need degrees \(2n\) of approximately 4, 28, 282, and 2802. It turns out that the actual minimal degrees (as computed with the Chebfun minimax command [7, 8]) are exactly these four numbers. For accuracy \(10^{-6}\), for example, though this is beyond Chebfun, it seems clear that the required degree will be close to \(n=280{,}170\).
Thus we see again that an approximation (4) requires degrees of order \(O(1/\varepsilon )\), but why are the coefficients so large? The explanation is that the monomials \(1,x^2, x^4, \dots , x^{2n}\) are an exponentially ill-behaved basis for the space of even degree 2n polynomials on \([-1,1]\). Numerical analysts quantify this observation by noting that the condition number of this set of functions is of the approximate order
[3, 9]. With \(2n=280{,}170\) for accuracy \(10^{-6}\), this suggests the expansion coefficients will need to be of order about \(10^{107{,}000}\). Our best empirical approximation based on calculations for n up to 300 is
Table 1 summarizes our computations and estimates for accuracies \(\varepsilon = 10^{-1}, \dots , 10^{-8}.\)
Figure 1 illustrates graphically where the big coefficients come from. For \(2n = 28, 56, \dots , 140\), it plots the coefficients \(|c_k^{}|\) for \(k = 0, 1, \dots , n\) in a monomial expansion of the best approximations.
4 A Remark About Mathematics
Theorem 2 is startling and interesting. From the usual mathematical point of view, however, it is not much more than that. After all, Müntz’s theorem remains valid and beautiful. From the usual mathematical perspective, Müntz’s theorem expresses a fundamental truth, and Theorem 2, however interesting, is an engineering footnote.
As I have discussed in the context of other problems [13, 15], I believe this usual perspective is too comfortable. Theorem 2 implies that typical sets of powers deemed useful by Müntz’s theorem would in fact be useless in any actual application. If it is not the business of mathematicians to notice and analyze such an effect, then whose business is it?
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Acknowledgements
I am grateful for helpful suggestions to Jose María Almira, Anthony Austin, Michael Ganzburg, Daan Huybrechs, Christian Lubich, Doron Lubinsky, Yuji Nakatsukasa, Allan Pinkus, and Endre Süli.
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Trefethen, L.N. Spectacularly Large Expansion Coefficients in Müntz’s Theorem. La Matematica 2, 31–36 (2023). https://doi.org/10.1007/s44007-022-00039-6
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DOI: https://doi.org/10.1007/s44007-022-00039-6