Bernstein–Riemann Interpolation Formula for Arbitrary Continuous Functions on an Interval

For arbitrary continuous functions on the interval [0, 1], we obtain an interpolation formula based on known values of these functions on some uniform grid. No additional assumptions about the functions are required. The construction of such a formula is connected with the properties of local Bernstein polynomials and the Riemann zeta function. Numerical results for the interpolation of functions of the Riemann, Weierstrass, Besicovitch, and Takagi types are presented.