Thus [[tau].sub.n] consists of

trigonometric polynomials, dim [[tau].sub.n] = n.

where the supremum is taken with respect to all nonnegative

trigonometric polynomials h with [mathematical expression not reproducible].

In [26], some relations were established between the sequences of best approximations of continuous 2[pi]-periodic functions f (and also f [member of] [L.sub.p]) by

trigonometric polynomials of order [less than or equal to] n and the properties of their ([psi], [beta])-derivatives.

We will consider this problem in a concrete situation, like in approximation by [pi]-symmetric

trigonometric polynomials. In our previous papers (see, e.g.

The complete necessary mathematical background of this orthogonal

trigonometric polynomials can be found in [18].

For example, the positivity of

trigonometric polynomials are studied in geometric function theory by Gluchoff and Hartmann [1] and Ruscheweyh and Salinas [2].

Due to the well-known Weierstrass theorems on the uniform approximation of continuous functions on a finite interval by algebraic and

trigonometric polynomials, polynomial functions are very popular in "Approximation Theory" and one of the basic subjects of this theory is "interpolation of functions by polynomials".

We shall use the lemma to advance in getting other properties of the space A[P.sub.[phi]] (R, C), as mentioned above in this section: approximation by

trigonometric polynomials, the Bohr definition and existence of almost periodes, the last being Bochner property, stating the compactness of the translates of a given f [member of] A[P.sub.[phi]](R, C).

Among the topics are complex numbers, Fourier coefficients and first Fourier series,

trigonometric polynomials, the convergence of Fourier series, and the Fourier transform.

For example, a smooth periodic function can be approximated by

trigonometric polynomials; a square-integrable smooth function can be expanded into a wavelet series and be approximated by partial sum of the wavelet series; and a smooth function on a cube can be approximated well by polynomials.

In [3], a family of

trigonometric polynomials was introduced, which contains the trigonometric Lagrange and Bernstein polynomials.

Suppose that subdivision symbol P(z) has sum rules of order n (see [6, 7]), that is, if there exists a row vector y([xi]) of

trigonometric polynomials such that y(0) [not equal to] 0 and