MATHEMATICS

V. F. NIKOLAEV

POLYNOMIAL OPERATIONS IN CERTAIN SPACES

(Presented by Academician A. N. Kolmogorov, 19 VIII 1957)

I. Consider a linear normed space \(E\) for which the following axioms are satisfied:

1. The elements of \(E\) are \(2\pi\)-periodic real functions of a real argument \(x\), summable on \([0, 2\pi]\).

2. Addition of elements of \(E\) and multiplication of an element by a real number are defined in the usual way.

3. The space \(E\) contains all trigonometric polynomials (the set of these polynomials will be denoted by \(T\), and the set of all trigonometric polynomials of order \(\leq n\) by \(T_n\)).

4. Membership of an element in \(E\) and the norm of an element are invariant with respect to an arbitrary shift of the argument, i.e., from \(f(x) \in E\) it follows that \(f(x+t) \in E\) and \(\|f(x+t)\|=\|f(x)\|\) for any \(t\).

5. The set \(T\) is everywhere dense in \(E\) in the sense of convergence in the norm (strong convergence).

Any space possessing these properties will be called, for brevity, a space of type \(E\). The simplest examples are the spaces \(\tilde C\) and \(\tilde L^p\) \((p \geq 1)\) of \(2\pi\)-periodic functions (with the usual definition of the norm).

II. Let \(E_1\) and \(E_2\) be spaces of type \(E\). A linear operation \(A_n(f,x)\) from \(E_1\) to \(E_2\) is called a trigonometric polynomial operation of order \(n\) (briefly, a polynomial operation) if it maps the space \(E_1\) into the subset \(T_n\) of the space \(E_2\).

For the norm \(\|A_n\|\) of the linear operation \(A_n(f,x)\) from \(E_1\) to \(E_2\) the notation \(\|A_n\|_{E_1}^{E_2}\) will also be used.

III. Let \(M\) be some subset of \(E_1\). An operation \(U(f,x)\) from \(E_1\) to \(E_2\) is called sliding on the set \(M\) if, for each element \(f(x)\) of the set \(M\), for any \(t\) the relation
\(U(f(x+t),x)=U(f(x),x+t)\) or \(U(f(x+t),x-t)=U(f(x),x)\) holds.

For example, the operation
\[ B_n(f,x)=\int_0^{2\pi} K_n(t)f(x+t)\,dt, \]
where \(K_n(x)\) is a fixed element of \(T_n\), is sliding on the whole set \(E_1\). An operation identical on \(M\) (i.e., leaving the elements of \(M\) unchanged) is sliding on \(M\).

IV. The fundamental formula for a trigonometric polynomial operation. For every linear operation \(A_n(f,x)\) from \(E_1\) to \(E_2\), mapping \(E_1\) onto \(T_n\), the formula
\[ \int_0^{2\pi} A_n(f(x+t),x-t)\,dt = \int_0^{2\pi} A_n(S_n(x+t),x-t)\,dt, \tag{1} \]
is valid for any value of \(x\),

where \(f(x)\) is any element of \(E_1\); \(S_n(x)=S_n(f,x)\) is the \(n\)-th partial sum of the Fourier series of the function \(f(x)\).

Particular cases. 1) If the operation \(A_n(f,x)\) is a sliding one on the whole set \(E_1\) (we denote this operation by \(B_n(f,x)\)), then

\[ B_n(f,x)=B_n(S_n,x). \tag{2} \]

It follows from this that every sliding polynomial operation from \(E_1\) to \(E_2\) is representable in the form

\[ B_n(f,x)=\int_0^{2\pi} K_n(t) f(x+t)\,dt, \tag{3} \]

where \(K_n(x)\) is a fixed element of \(T_n\) (namely, \(K_n(x)=B_n\left(\frac1\pi D_n(x),-x\right)\)).

2) If the operation \(A_n(f,x)\) coincides, on each element of the set \(T_n\), with the sliding operation (from \(E_1\) to \(E_2\)) \(B_n(f,x)\), then

\[ B_n(f,x)=\frac1{2\pi}\int_0^{2\pi} A_n(f(x+t),x-t)\,dt. \tag{4} \]

In particular, if \(A_n(f,x)\) coincides on \(T_n\) with the operation \(S_n(f,x)\), i.e. if \(A_n(f,x)\) leaves the elements of \(T_n\) unchanged, then

\[ S_n(f,x)=\frac1{2\pi}\int_0^{2\pi} A_n(f(x+t),x-t)\,dt. \tag{5} \]

If the operation \(A_n(f,x)\) takes each element \(\tau_n(x)\) of the set \(T\) into the conjugate polynomial \(\widetilde{\tau}_n(x)\), then

\[ \widetilde{S}_n(f,x)=\frac1{2\pi}\int_0^{2\pi} A_n(f(x+t),x-t)\,dt. \tag{6} \]

V. Extremal properties of the sliding polynomial operation. 1) Among all polynomial operations \(A_n(f,x)\) from \(E_1\) to \(E_2\) that coincide on the set \(T_n\) with a given polynomial operation \(B_n(f,x)\) sliding on all of \(E_1\), the operation \(B_n(f,x)\) has the least norm, i.e.

\[ \|B_n\|\leqslant \|A_n\|. \tag{7} \]

In particular, among all polynomial operations \(A_n(f,x)\) that preserve the set \(T_n\), the operation \(S_n(f,x)\) has the least norm, i.e.

\[ \|S_n\|\leqslant \|A_n\|. \tag{8} \]

Analogously for the operation occurring in (6).

2) Let \(A_n(f,x)\) be a polynomial operation from \(E\) to \(E\), coinciding on \(T_n\) with the sliding polynomial operation \(B_n(f,x)\) (from \(E\) to \(E\)). Let \(N(f)\) be a positive functional on \(E\) (not necessarily linear) having the property that \(N(f(x+t))=N(f(x))\) for any \(t\). If \(M\) is an arbitrary bounded subset of \(E\), moreover such that \(f(x)\in M\) implies \(f(x+t)\in M\), then the inequality holds

\[ \sup_{f\in M}\frac{\|f(x)-B_n(f,x)\|}{N(f)} \leqslant \sup_{f\in M}\frac{\|f(x)-A_n(f,x)\|}{N(f)}. \tag{9} \]

3) We shall consider the norms of the sliding operation \(B_n(f,x)\) from \(E\) to \(E\) for all possible spaces of type \(E\); the inequality holds

\[ \|B_n\|_{E}^{E} \leq \|B_n\|_{\widetilde{C}}^{\widetilde{C}}=\int_{0}^{2\pi}|K_n(t)|\,dt . \tag{10} \]

In particular, \(\|S_n\|_{E}^{E} \leq \|S_n\|_{\widetilde{C}}^{\widetilde{C}}=L_n\), where \(L_n\) is the Lebesgue constant for the ordinary Fourier series.

Remark. The results listed above are in part adjacent to the results of the works of S. M. Lozinskii \((^1)\) and contain as a special case the results of the papers \((^2)\) of D. L. Berman, who assumed, with respect to the space \(E\), besides axioms 1–5 of the present article, also a certain additional assumption on the structure of the norm in \(\widetilde{E}\).

Received
16 IX 1956

CITED LITERATURE

\(^1\) S. M. Lozinskii, DAN, 61, No. 2 (1948); 64, No. 4 (1949); 89, No. 4 (1953); 89, No. 5 (1953). \(^2\) D. L. Berman, DAN, 88, No. 1 (1952); 91, No. 6 (1953); 92, No. 4 (1953); 95, No. 2 (1954).