A function with the following properties is called a Meyer scaling function: φ: ℝ → ℝ, its integral shifts φ(· + n), n ∈ ℤ, are orthonormal in L₂(ℝ), and its Fourier transform φ̂(y)=12π∫ℝφ(t)e−iytdt has the following properties: φ̂ is even, φ̂ = 0 outside [−π −ε, π +ε], φ̂=12π on [−π + ε, π − ε], where ε∈(0π3]. Here is the main result of the paper. Assume that ω:[0+∞)→[0+∞) and the function ω(x)x decreases. Then the following assertions are equivalent. 1. For every (or, equivalently, for some) ε∈(0π3], there exists x0 > 0 and a Meyer scaling function φ such that φ̂ = 0 outside [−π − ε, π + ε] and |φ(x)| ≤ e^−ω(|x|) for all |x| > x₀. 2. ∫1+∞ω(x)x2dx<+∞.