# (0, 2, 3)–interpolation of continuous in generalized variation functions

Let $f$ be a real-valued $2\pi$-periodic function defined on $[-\pi,\pi]$, and for each open interval $I=(a,b)\subset [-\pi,\pi]$ set $f(I)=f(b)-f(a)$. We let $\Lambda=\{\lambda_k\}_{k=1}^\infty$ denote non-decreasing sequence of real numbers such that

\begin{equation*}

\sum_{k=1}^\infty \frac{1}{\lambda_k}=+\infty.

\end{equation*}

Then $f$ is said to be of $\Lambda$-bounded variation $(\Lambda BV)$ on $[-\pi,\pi]$ if

\begin{equation}

V(\Lambda,f):=\sum_{k=1}^\infty \frac{f(I_k)}{\lambda_k}\,{<}\,\infty,

\tag{*}\label{eq:*}

\end{equation}

where the supremum is taken over all sequences of non-overlapping open intervals $I_k \subset [-\pi,\pi]$, $k=1,2,\cdots$; if (\ref{eq:*}) holds when the supremum is taken over all sequences of open intervals $I_k \subset [-\pi,\pi]$, $k=1,2,\cdots$ for which either $I_k\,{<}\,I_{k+1}$, $k=1,2,\cdots$, or $I_k>I_{k+1}$, $k=1,2,\cdots$ (where $I\,{<}\,J$ means $I$ lies to the left of $J$), then $f$ is said to be of ordered $\Lambda$-bounded variation $(O\Lambda BV)$ on $[-\pi,\pi]$.

If the $\{\lambda_k\}_{k=1}^\infty$ are bounded, we have the classical Jordan bounded variation $(BV)$; if $\lambda_k=k$, we have harmonic (ordered harmonic) bounded variation, $HBV\;(OHBV)$. Let $\Lambda^m:=\{\lambda_k\}_{k=m+1}^\infty$, $m=0,1,\cdots$ A function $f$ of $\Lambda BV (O\Lambda BV)$ is said to be continuous in $\Lambda$-variation (ordered $\Lambda$-variation), if $V(\Lambda^m,f)\to 0$, as $m\to\infty$.

The above-mentioned classes were introduced in the seventies of the last century by Waterman. To date, there are many results on the properties of functions of generalized bounded variation. For instance in 1972 Waterman shown that we can replace $BV$ by $HBV$ in the classic Dirichlet–Jordan theorem: the Fourier series of function $f\in BV$ converge at every point of continuity of $f$ and the convergence is uniform on every closed interval of points of continuity of $f$. If one were to use $\Lambda BV$ instead $HBV$, where $\Lambda BV-HBV\neq\varnothing$,then the theorem would fail.

It is well known that there is a close analogy between Fourier series and Lagrange interpolation. As a result many facts of Fourier series were proved for interpolation processes. In particular interpolatory analog of Dirichlet–Jordan type theorem for $HBV$ was proved in 1986 by Kelzon.

Denote by $L_n(f,x), n=1,2,\cdots$, the Lagrange interpolation polynomial based on equidistant nodes

\begin{equation*}

x_{k,n}=\frac{2\pi k}{2n+1}, k=-n,\cdots,n,

\end{equation*}

and let $Q_n(f,x), n=1,2,\cdots$, be the $(0,2,3)$ Birkhoff (lacunary) interpolation polynomial such that

\begin{equation*}

Q_n(f,x_{k,n})=f(x_{k,n}),\; Q''_n(f,x_{k,n})=Q'''_n(f,x_{k,n})=0,\;k=-n,\cdots,n.

\end{equation*}

In present paper we prove a result: if $f$ is continuous in ordered harmonic variation on $[-\pi,\pi]$, then both $\{L_n(f,x)\}$ and $\{Q_n(f,x)\}$ converges to $f$ uniformly on $[-\pi,\pi]$.

Key words: Lagrange Interpolation, Birkhoff Interpolation, Lacunary Interpolation, Generalized Variation, Harmonic Variation.