- Research Article
- Open Access
© Bogdan Szal. 2010
- Received: 26 January 2010
- Accepted: 13 April 2010
- Published: 23 May 2010
We introduce a new class of sequences called and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients to belong to the above class. This is a generalization of the result proved by M. Dyachenko and S. Tikhonov (2009). Then we discuss the relations among the weighted best approximation and the coefficients of trigonometric series. Moreover, we extend the results of B. Wei and D. Yu (2009) to the class .
- Positive Constant
- Weight Function
- Integrable Function
- Fourier Coefficient
- Elementary Calculation
In Subsection 2.1 we generalize the following result.
In Subsection 2.2 we generalize and extend the following results .
In order to formulate our new results we define the next class of sequences.
We formulate our results as follows.
If we take (and in Theorem 2.2), then the result of Dyachenko and Tikhonov [15, Theorems and ] follows from Theorems 2.2 and 2.3. By the embedding relations (1.8) and (1.15) we can also derive from Theorem 2.2 the result of You, Zhou, and Zhou .
2.2. Relations between The Best Approximation and Fourier Coefficients
If we restrict our attention to the class then by (1.8) and (1.15) Wei and Yu's result  follows from Theorems 2.5 and 2.6.
4.1. Proof of Theorem 2.1
4.2. Proof of Theorem 2.2
which completes the proof.
4.3. Proof of Theorem 2.3
This ends our proof.
4.4. Proof of Theorem 2.5
Combining (4.32) or (4.33), (4.35)–(4.43) and (4.45)–(4.50) we complete the proof of Theorem 2.5.
4.5. Proof of Theorem 2.6
This completes the proof.
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