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We establish formulas for the $b$-adic Walsh coefficients of functions in$C^\alpha[0,1]$ for an integer $\alpha \geq 1$ and give upper bounds on theWalsh coefficients of these functions. We also study the Walsh coefficients ofperiodic and non-periodic functions in reproducing kernel Hilbert spaces.

Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients