Possibilties of chebyshev polynomial waveshaping
For $-1 &lt;= x &lt;= 1$, let's compare Chebyshev polynomials of the first kind, $T_n(x)$, and the basis functions of the Fourier cosine series, $F_n(x)$:

 $F_n(x)=\cos(n \pi x)$

 $T_n(x)=\cos(n\ \text{acos}\ x)$

Writing $T_n(x_T) = F_n(x_F)$ and solving for $x_F$ gives $x_F = (\text{acos}\ x_T) / pi$, revealing that the Chebyshev polynomial series is simply an argument-warped version of the Fourier cosine series that maps the range $0 &lt;= x_F &lt;= 1$ to $-1 &lt;= x_T &lt;= 1$:



This exclusion of $x_F &lt; 0$ makes irrelevant the constraint that the Fourier cosine series is only applicable to even functions, functions symmetric around $x_F = 0$, meaning that $f(x_F) = f(-x_F)$. The mapping function is well-behaved in the range of interest, so if the Fourier cosine series can represent any even function, then the Chebyshev polynomial series can represent any function.

Not knowing the amplitude of the input sinusoid, it is not possible to control the proportional amplitudes of the generated harmonics by the choice of the waveshaper. For example if the waveshaper is $f(x) = x^2 + x$:

$$f(\cos x) = \frac{1}{2} + \cos x + \frac{1}{2} \cos 2x\\
f(\frac{1}{2} \cos x) = \frac{1}{8} + \frac{1}{2} \cos x + \frac{1}{8} \cos 2x$$

The first one has the amplitude of the second harmonic at -6 dB compared to the fundamental, and the second one at -12 dB. Chebyshev polynomials are no miracle cure. The above polynomial $x^2 + x$ was a weighted sum of the first three Chebyshev polynomials. Any polynomial can be written as a weighted sum of Chebyshev polynomials, and vice versa.

The problem persists for pure Chebyshev polynomials, for example $f(x) = 4 x^3 - 3x$:

$$f(\cos x) = \cos 3x\\
f(\frac{1}{2} \cos x) = \frac{1}{8} \cos 3x - \frac{9}{8} \cos x$$

A Chebyshev polynomial series does have the advantage of better numerical stability than a simple power series, and if the amplitude of the input sinusoid is controlled somehow, it can be used to control the level of harmonics in the way you describe.