Hermitian-type splines (nonpolynomial in general) of second order (said to be quasi-Hermitian) are constructed on irregular grids. Extensions of the system of functionals which are biorthogonal to such splines are obtained, necessary and sufficient conditions for continuity of the splines are established, direct solutions of some generalized Hermite interpolating problems are given. The limit space of quasi-Hermitian splines is constructed under the condition of limiting transition from the spaces of B-splines when "pasting together" some knots of the grid

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