What are the functions in $L^1(\mathbb{R^+})$ verifying:

$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$

With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions but can we find other solutions?)

What are the functions verifying:

$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$

With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions but can we find other solutions?)

What are the functions in $L^1(\mathbb{R^+})$ verifying:

$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$

With $\lambda$ a constant ? (Functions $x^{\alpha}$ with $0<\alpha<1$ are solutions but can we find other solutions?)

What are the functions in $L^1(\mathbb{R^+})$ verifying:

$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$

With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions but can we find other solutions?)