Adjustment of functions and Lagrange interpolation based on the nodes close to the Legendre nodes

It is well known that the Lagrange interpolation of a continuous function based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere) like the Fourier series of a summable function. On the other hand any measurable almost everywhere finite function can be “adjusted” in a set of arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises: does the class of continuous functions have a similar property with respect to any interpolation process? In the present paper we prove that there exists a matrix of nodes $\mathfrak{M}_\gamma$ arbitrarily close to the Legendre matrix with the following property: any function $f\in{C[-1,1]}$ can be adjusted in a set of arbitrarily small measure such that the interpolation process of adjusted continuous function $g$ based on the nodes $\mathfrak{M}_\gamma$ will be uniformly convergent to $g$ on $[a,b]\subset(-1,1)$.