Skip to main content
MathOverflow

Return to Question

deleted 1 characters in body
Source Link
Terry Tao
Terry Tao
  • 114.2k
  • 33
  • 462
  • 539

Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+2f(x)|\le |h|^{3/2}$$|f(x+2h)-2f(x+h)+f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies $|f'(x+h)-f'(x)|\le c |h|^{1/2}$ for all $x,h$?

Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+2f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies $|f'(x+h)-f'(x)|\le c |h|^{1/2}$ for all $x,h$?

Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies $|f'(x+h)-f'(x)|\le c |h|^{1/2}$ for all $x,h$?

Source Link
Loukas
Loukas
  • 83
  • 2

Second order difference implies differentiability

Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+2f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies $|f'(x+h)-f'(x)|\le c |h|^{1/2}$ for all $x,h$?