Fourier Series Expansion of the Function
To determine if the function \( f(x) = (t^2 - 1)^{\frac{1}{4}} \) can be expressed as a Fourier series within the interval \([-π, π]\), we need to consider a few key aspects of Fourier series.

1. Function Characteristics
- **Periodicity**: Fourier series are useful for periodic functions. If \( f(x) \) can be defined periodically, we can expand it as a Fourier series.
- **Continuity**: The function needs to be continuous or piecewise continuous over the defined interval. Discontinuities can affect the convergence of the series.

2. Fourier Series Formulation
- The Fourier series of a function \( f(x) \) in the interval \([-L, L]\) is given by:
\[
f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n \frac{\pi x}{L}) + b_n \sin(n \frac{\pi x}{L}))
\]
where:
- \( a_0 = \frac{1}{2L} \int_{-L}^{L} f(x) \, dx \)
- \( a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos(n \frac{\pi x}{L}) \, dx \)
- \( b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin(n \frac{\pi x}{L}) \, dx \)

3. Conditions for Fourier Series
- **Even or Odd Functions**: If \( f(x) \) is even, only cosine terms will appear; if odd, only sine terms.
- **Differentiability**: For better convergence properties, the function should be differentiable at most points.

Conclusion
In summary, if the function \( f(x) = (t^2 - 1)^{\frac{1}{4}} \) is periodic, continuous, and satisfies the conditions mentioned above, it can indeed be represented as a Fourier series in the interval \([-π, π]\). Analyzing the specific form and behavior of the function will be necessary for practical computation of the coefficients \( a_n \) and \( b_n \).