## Abstract

The Fourier spectrum and its norms are given as explicit arithmetic expressions and evaluated, for Boolean functions computed by classes of constant depth, read-once circuits consisting of an arbitrary set of symmetric gates. Previous results of this nature estimate the spectral*L*
_{1} norm of functions computed by certain types of decision trees [20], [7], and in some cases, give randomized*procedures* that evaluate the spectrum by clever rounding [20]. One corollary of our results provides a large class of*AC*
^{0} functions whose spectral*L*
_{1} norm is exponential, thus generalizing the single example of such a function given in [9]. This shows that almost every read-once*AC*
^{0} function does not belong in the class*PL*
_{1} of functions with polynomially bounded spectral norms.

Implications of our results and technique are discussed, for estimating the spectral norms of*any* function in a constant depth circuit class, using the coding theoretic concept of weight distributions. Evaluating the spectral norms for any such function reduces to estimating certain non-trivial weight distributions of simple, linear codes.

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Sitharam, M. Evaluating spectral norms for constant depth circuits with symmetric gates.
*Comput Complexity* **5, **167–189 (1995). https://doi.org/10.1007/BF01268144

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### Key words

- Circuit complexity
- lower bounds
- Fourier transforms

### Subject classifications

- 68Q15
- 68Q99